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1.1 Machine Learning Overview | Welcome to machine learning!--[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=y8JgiWcUnU8
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Welcome to machine learning. What is machine learning? You probably use it many times a day without even knowing it. Anytime you want to find out something like, how do I make a sushi roll, you can do a web search on Google, Bing, or Baidu to find out. And that works so well because their machine learning software has figured out how to rank web pages. Or when you upload pictures to Instagram or Snapchat and think to yourself, I want to tag my friends so they can see their pictures. Well these apps can recognize your friends in your pictures and label them as well. That's also machine learning. Or if you've just finished watching a Star Wars movie on the video streaming service and you think, what other similar movies could I watch? Well the streaming service will likely use machine learning to recommend something that you might like. Each time you use voice-to-text on your phone to write a text message, hey Andrew, how's it going? Or tell your phone, hey Siri, play a song by Rihanna. Or ask your other phone, okay Google, show me Indian restaurants near me. That's also machine learning. Each time you receive an email titled, congratulations, you've won a million dollars. Well maybe you're rich, congratulations. Or more likely your email service will probably flag it as spam. Snapchat too is an application of machine learning. Beyond consumer applications that you might use, AI is also rapidly making its way into big companies and into industrial applications. For example, I'm deeply concerned about climate change and I'm glad to see that machine learning is already hoping to optimize wind turbine power generation. Or in healthcare, it's starting to make its way into hospitals to help doctors make accurate diagnoses. Or recently, at Landing AI, I've been doing a lot of work putting computer vision into factories to help inspect if something coming off the assembly line has any defects. That's machine learning. It's a science of getting computers to learn without being explicitly programmed. In this class, you learn about machine learning and get to implement machine learning in code yourself. Hundreds of others have taken the earlier version of this course, which is a course that led to the founding of Coursera. And many learners ended up building exciting machine learning systems or even pursuing very successful careers in AI. I'm excited that you're on this journey with me. Welcome and let's get started.
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1.1 Machine Learning Overview | Welcome to machine learning!--[Machine Learning | Andrew Ng]: Welcome to machine learning. What is machine learning? You probably use it many times a day without even knowing it. Anytime you want to find out something like, how do I make a sushi roll, you can do a web search on Google, Bing, or Baidu to find out. And that works so well because their machine learning software has figured out how to rank web pages. Or when you upload pictures to Instagram or Snapchat and think to yourself, I want to tag my friends so they can see their pictures. Well these apps can recognize your friends in your pictures and label them as well. That's also machine learning. Or if you've just finished watching a Star Wars movie on the video streaming service and you think, what other similar movies could I watch? Well the streaming service will likely use machine learning to recommend something that you might like. Each time you use voice-to-text on your phone to write a text message, hey Andrew, how's it going? Or tell your phone, hey Siri, play a song by Rihanna. Or ask your other phone, okay Google, show me Indian restaurants near me. That's also machine learning. Each time you receive an email titled, congratulations, you've won a million dollars. Well maybe you're rich, congratulations. Or more likely your email service will probably flag it as spam. Snapchat too is an application of machine learning. Beyond consumer applications that you might use, AI is also rapidly making its way into big companies and into industrial applications. For example, I'm deeply concerned about climate change and I'm glad to see that machine learning is already hoping to optimize wind turbine power generation. Or in healthcare, it's starting to make its way into hospitals to help doctors make accurate diagnoses. Or recently, at Landing AI, I've been doing a lot of work putting computer vision into factories to help inspect if something coming off the assembly line has any defects. That's machine learning. It's a science of getting computers to learn without being explicitly programmed. In this class, you learn about machine learning and get to implement machine learning in code yourself. Hundreds of others have taken the earlier version of this course, which is a course that led to the founding of Coursera. And many learners ended up building exciting machine learning systems or even pursuing very successful careers in AI. I'm excited that you're on this journey with me. Welcome and let's get started.
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1.2 Machine Learning Overview | What is machine learning? --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=AISftYVyS50
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So, what is machine learning? In this video you learn the definition of what it is, and also get a sense of when you might want to apply it. Let's take a look together. Here's the definition of what is machine learning that is attributed to Arthur Samuel. He defined machine learning as the feeble study that gives computers the ability to learn without being explicitly programmed. Arthur's claim to fame was that back in the 1950s he wrote a checkers playing program. And the amazing thing about this program was that Arthur Samuel himself wasn't a very good checkers player. What he did was he had programmed a computer to play maybe tens of thousands of games against itself. And by watching what sorts of board positions tend to lead to wins, and what positions tend to lead to losses, the checkers playing program learned over time what a good or bad board position is. By trying to get to good and avoid bad positions, his program learned to get better and better at playing checkers. Because the computer had the patience to play tens of thousands of games against itself, it was able to get so much checkers playing experience that eventually it became a better checkers player than Arthur Samuel himself. Now throughout these videos, besides me trying to talk about stuff, I'll occasionally ask you a question to help make sure you understand the content. Here's one about what happens if the computer had played far fewer games. Please take a look and pick whichever you think is a better answer. Thanks for looking at the quiz. And so if you had selected this answer, whether made it worse, then you got it right. In general, the more opportunities you give a learning algorithm to learn, the better it will perform. If you didn't select the correct answer the first time, that's totally okay too. The point of these quiz questions isn't to see if you can get them all correct on the first try. These questions are here just to help you practice the concepts you're learning. Arthur Samuel's definition was a rather informal one. But in the next two videos, we'll dive deeper together into what are the major types of machine learning algorithms. In this class, you learn about many different learning algorithms. The two main types of machine learning are supervised learning and unsupervised learning. We'll define what these terms mean more in the next couple videos. Of these two, supervised learning is the type of machine learning that is used most in many real world applications and that has seen the most rapid advancement and innovation. In this specialization, which has three causes in total, the first and second causes will focus on supervised learning and the third will focus on unsupervised learning, recommender systems and reinforcement learning. By far, the most used types of learning algorithms today are supervised learning, unsupervised learning and recommender systems. The other thing we're going
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1.2 Machine Learning Overview | What is machine learning? --[Machine Learning | Andrew Ng]: So, what is machine learning? In this video you learn the definition of what it is, and also get a sense of when you might want to apply it. Let's take a look together. Here's the definition of what is machine learning that is attributed to Arthur Samuel. He defined machine learning as the feeble study that gives computers the ability to learn without being explicitly programmed. Arthur's claim to fame was that back in the 1950s he wrote a checkers playing program. And the amazing thing about this program was that Arthur Samuel himself wasn't a very good checkers player. What he did was he had programmed a computer to play maybe tens of thousands of games against itself. And by watching what sorts of board positions tend to lead to wins, and what positions tend to lead to losses, the checkers playing program learned over time what a good or bad board position is. By trying to get to good and avoid bad positions, his program learned to get better and better at playing checkers. Because the computer had the patience to play tens of thousands of games against itself, it was able to get so much checkers playing experience that eventually it became a better checkers player than Arthur Samuel himself. Now throughout these videos, besides me trying to talk about stuff, I'll occasionally ask you a question to help make sure you understand the content. Here's one about what happens if the computer had played far fewer games. Please take a look and pick whichever you think is a better answer. Thanks for looking at the quiz. And so if you had selected this answer, whether made it worse, then you got it right. In general, the more opportunities you give a learning algorithm to learn, the better it will perform. If you didn't select the correct answer the first time, that's totally okay too. The point of these quiz questions isn't to see if you can get them all correct on the first try. These questions are here just to help you practice the concepts you're learning. Arthur Samuel's definition was a rather informal one. But in the next two videos, we'll dive deeper together into what are the major types of machine learning algorithms. In this class, you learn about many different learning algorithms. The two main types of machine learning are supervised learning and unsupervised learning. We'll define what these terms mean more in the next couple videos. Of these two, supervised learning is the type of machine learning that is used most in many real world applications and that has seen the most rapid advancement and innovation. In this specialization, which has three causes in total, the first and second causes will focus on supervised learning and the third will focus on unsupervised learning, recommender systems and reinforcement learning. By far, the most used types of learning algorithms today are supervised learning, unsupervised learning and recommender systems. The other thing we're going
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1.2 Machine Learning Overview | What is machine learning? --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=AISftYVyS50
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to spend a lot of time on in this specialization is practical advice for applying learning algorithms. This is something I feel pretty strongly about. Teaching about learning algorithms is like giving someone a set of tools and equally important or even more important than making sure you have great tools is making sure you know how to apply them. Because you know, what good is it if someone were to give you a state of the art hammer or a state of the art hand drill and say, good luck, now you have all the tools you need to build a three story house. It doesn't really work like that. And so too in machine learning, making sure you have the tools is really important. And so it's making sure that you know how to apply the tools of machine learning effectively. So that's what you get in this class, the tools as well as the skills of applying them effectively. I regularly visit with friends and teams in some of the top tech companies. And even today, I see experienced machine learning teams apply machine learning algorithms to some problems. And sometimes they've been going at it for six months without much success. And when I look at what they're doing, I sometimes feel like I could have told them six months ago that the current approach won't work. And there's a different way of using these tools that will give them a much better chance of success. So in this class, one of the relatively unique things you learn is you learn a lot about the best practices for how to actually develop a practical, valuable machine learning system. This way, you're less likely to end up in one of those teams that end up losing six months going in the wrong direction. In this class, you gain a sense of how the most skilled machine learning engineers build systems. I hope you finish this class as one of those very rare people in today's world that know how to design and build serious machine learning systems. So that's machine learning. In the next video, let's look more deeply at what is supervised learning and also what is unsupervised learning. In addition, you learn when you might want to use each of them, supervised and unsupervised learning. I'll see you in the next video.
| 397
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1.2 Machine Learning Overview | What is machine learning? --[Machine Learning | Andrew Ng]: to spend a lot of time on in this specialization is practical advice for applying learning algorithms. This is something I feel pretty strongly about. Teaching about learning algorithms is like giving someone a set of tools and equally important or even more important than making sure you have great tools is making sure you know how to apply them. Because you know, what good is it if someone were to give you a state of the art hammer or a state of the art hand drill and say, good luck, now you have all the tools you need to build a three story house. It doesn't really work like that. And so too in machine learning, making sure you have the tools is really important. And so it's making sure that you know how to apply the tools of machine learning effectively. So that's what you get in this class, the tools as well as the skills of applying them effectively. I regularly visit with friends and teams in some of the top tech companies. And even today, I see experienced machine learning teams apply machine learning algorithms to some problems. And sometimes they've been going at it for six months without much success. And when I look at what they're doing, I sometimes feel like I could have told them six months ago that the current approach won't work. And there's a different way of using these tools that will give them a much better chance of success. So in this class, one of the relatively unique things you learn is you learn a lot about the best practices for how to actually develop a practical, valuable machine learning system. This way, you're less likely to end up in one of those teams that end up losing six months going in the wrong direction. In this class, you gain a sense of how the most skilled machine learning engineers build systems. I hope you finish this class as one of those very rare people in today's world that know how to design and build serious machine learning systems. So that's machine learning. In the next video, let's look more deeply at what is supervised learning and also what is unsupervised learning. In addition, you learn when you might want to use each of them, supervised and unsupervised learning. I'll see you in the next video.
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1.3 Machine Learning Overview | Applications of machine learning -- [Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=hHYcNPfbBXQ
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In this class, you learn about the state of the art and also practice implementing machine learning algorithms yourself. You learn about the most important machine learning algorithms, some of which are exactly what's being used in large AI or large tech companies today, and you get a sense of what is the state of the art in AI. Beyond learning the algorithms though, in this class, you also learn all the important practical tips and tricks for making them perform well, and you get to implement them and see how they work for yourself. So why is machine learning so widely used today? Machine learning had grown up as a subfield of AI or artificial intelligence. We wanted to build intelligent machines, and it turns out that there are a few basic things that we could program a machine to do, such as how to find the shortest path from A to B like in your GPS. But for the most part, we just did not know how to write an explicit program to do many of the more interesting things, such as perform web search, recognize human speech, diagnose diseases from x-rays, or build a self-driving car. The only way we knew how to do these things was to have a machine learn to do it by itself. For me, when I founded and was leading the Google Brain team, I worked on problems like speech recognition, computer vision for Google Maps to view images, and advertising. Or leading AI by do, I worked on everything from AI for augmented reality to combating payment fraud to leading a self-driving car team. Most recently, at Landing AI, AI Fund, and Stanford University, I've been getting to work on AI applications in manufacturing, large-scale agriculture, healthcare, e-commerce, and other problems. Today, there are hundreds of thousands, perhaps millions of people, working on machine learning applications who could tell you similar stories about their work with machine learning. When you've learned these skills, I hope that you too will find it great fun to dabble in exciting different applications and maybe even different industries. In fact, I find it hard to think of any industry that machine learning is unlikely to touch in a significant way now or in the near future. Moving even further into the future, many people, including me, are excited about the AI dream of someday building machines as intelligent as you or me. This is sometimes called artificial general intelligence, or AGI. I think AGI has been overhyped and we're still a long way away from that goal. I don't know if it'll take 50 years or 500 years or longer to get there, but most AI researchers believe that the best way to get closer to what that goal is by using learning algorithms, maybe ones that take some inspiration from how the human brain works. You also hear a little more about this quest for AGI later in this course. According to a study
| 500
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1.3 Machine Learning Overview | Applications of machine learning -- [Machine Learning | Andrew Ng]: In this class, you learn about the state of the art and also practice implementing machine learning algorithms yourself. You learn about the most important machine learning algorithms, some of which are exactly what's being used in large AI or large tech companies today, and you get a sense of what is the state of the art in AI. Beyond learning the algorithms though, in this class, you also learn all the important practical tips and tricks for making them perform well, and you get to implement them and see how they work for yourself. So why is machine learning so widely used today? Machine learning had grown up as a subfield of AI or artificial intelligence. We wanted to build intelligent machines, and it turns out that there are a few basic things that we could program a machine to do, such as how to find the shortest path from A to B like in your GPS. But for the most part, we just did not know how to write an explicit program to do many of the more interesting things, such as perform web search, recognize human speech, diagnose diseases from x-rays, or build a self-driving car. The only way we knew how to do these things was to have a machine learn to do it by itself. For me, when I founded and was leading the Google Brain team, I worked on problems like speech recognition, computer vision for Google Maps to view images, and advertising. Or leading AI by do, I worked on everything from AI for augmented reality to combating payment fraud to leading a self-driving car team. Most recently, at Landing AI, AI Fund, and Stanford University, I've been getting to work on AI applications in manufacturing, large-scale agriculture, healthcare, e-commerce, and other problems. Today, there are hundreds of thousands, perhaps millions of people, working on machine learning applications who could tell you similar stories about their work with machine learning. When you've learned these skills, I hope that you too will find it great fun to dabble in exciting different applications and maybe even different industries. In fact, I find it hard to think of any industry that machine learning is unlikely to touch in a significant way now or in the near future. Moving even further into the future, many people, including me, are excited about the AI dream of someday building machines as intelligent as you or me. This is sometimes called artificial general intelligence, or AGI. I think AGI has been overhyped and we're still a long way away from that goal. I don't know if it'll take 50 years or 500 years or longer to get there, but most AI researchers believe that the best way to get closer to what that goal is by using learning algorithms, maybe ones that take some inspiration from how the human brain works. You also hear a little more about this quest for AGI later in this course. According to a study
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1.3 Machine Learning Overview | Applications of machine learning -- [Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=hHYcNPfbBXQ
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by McKinsey, AI and machine learning is estimated to create an additional 13 trillion US dollars of value annually by the year 2030. Even though machine learning is already creating tremendous amounts of value in the software industry, I think there could be even vastly greater value that has yet to be created outside the software industry in sectors such as retail, travel, transportation, automotive, materials, manufacturing, and so on. Because of the massive untapped opportunities across so many different sectors, today there is a vast unfulfilled demand for this skill set. That's why this is such a great time to be learning about machine learning. If you find machine learning applications exciting, I hope you stick with me through this course. I can almost guarantee that you'll find mastering these skills worthwhile. In the next video, we'll look at a more formal definition of what is machine learning and we'll begin to talk about the main types of machine learning problems and algorithms. You pick up some of the main machine learning terminology and start to get a sense of what are the different algorithms and when each one might be appropriate. So let's go on to the next video.
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1.3 Machine Learning Overview | Applications of machine learning -- [Machine Learning | Andrew Ng]: by McKinsey, AI and machine learning is estimated to create an additional 13 trillion US dollars of value annually by the year 2030. Even though machine learning is already creating tremendous amounts of value in the software industry, I think there could be even vastly greater value that has yet to be created outside the software industry in sectors such as retail, travel, transportation, automotive, materials, manufacturing, and so on. Because of the massive untapped opportunities across so many different sectors, today there is a vast unfulfilled demand for this skill set. That's why this is such a great time to be learning about machine learning. If you find machine learning applications exciting, I hope you stick with me through this course. I can almost guarantee that you'll find mastering these skills worthwhile. In the next video, we'll look at a more formal definition of what is machine learning and we'll begin to talk about the main types of machine learning problems and algorithms. You pick up some of the main machine learning terminology and start to get a sense of what are the different algorithms and when each one might be appropriate. So let's go on to the next video.
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1.4 Machine Learning Overview | Supervised learning part 1 --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=EZN_uM3J3kI
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Machine learning is creating tremendous economic value today. I think 99% of the economic value created by machine learning today is through one type of machine learning, which is called supervised learning. Let's take a look at what that means. Supervised machine learning, or more commonly supervised learning, refers to algorithms that learn x to y or input to output mappings. The key characteristic of supervised learning is that you give your learning algorithm examples to learn from that include the right answers, where by right answer I mean the correct label y for a given input x. And it's by seeing correct pairs of input x and desired output label y that the learning algorithm eventually learns to take just the input alone without the output label and gives a reasonably accurate prediction or guess of the output. Let's look at some examples. If the input x is an email and the output y is this email spam or not spam, this gives you your spam filter. Or if the input is an audio clip and the algorithm's job is to output the text transcript, then this is speech recognition. Or if you want to input English and have it output the corresponding Spanish, Arabic, Hindi, Chinese, Japanese or something else translation, then that's machine translation. Or the most lucrative form of supervised learning today is probably used in online advertising. Nearly all the large online ad platforms have a learning algorithm that inputs some information about an ad and some information about you and then tries to figure out if you will click on that ad or not. Because by showing you ads that you're slightly more likely to click on for these large online ad platforms, every click is revenue. This actually drives a lot of revenue for these companies. This is something that one's done a lot of work on, maybe not the most inspiring application, but it certainly has a significant economic impact in some companies today. Or if you want to build a self-driving car, the learning algorithm would take as input an image and some information from other sensors, such as a radar or other things, and then try to output the position of say other cars so that your self-driving car can safely drive around the other cars. Or take manufacturing. I've actually done a lot of work in this sector at Landing AI. You can have a learning algorithm take as input a picture of a manufactured product, say a cell phone that just rolled off the production line, and have the learning algorithm output whether or not there is a scratch, dent, or other defect in the product. This is called visual inspection and is helping manufacturers reduce or prevent defects in their products. In all of these applications, you would first train your model with examples of InputsX and the right answers, that is, the labels Y. After the model has learned from these Input Outputs or X and
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1.4 Machine Learning Overview | Supervised learning part 1 --[Machine Learning | Andrew Ng]: Machine learning is creating tremendous economic value today. I think 99% of the economic value created by machine learning today is through one type of machine learning, which is called supervised learning. Let's take a look at what that means. Supervised machine learning, or more commonly supervised learning, refers to algorithms that learn x to y or input to output mappings. The key characteristic of supervised learning is that you give your learning algorithm examples to learn from that include the right answers, where by right answer I mean the correct label y for a given input x. And it's by seeing correct pairs of input x and desired output label y that the learning algorithm eventually learns to take just the input alone without the output label and gives a reasonably accurate prediction or guess of the output. Let's look at some examples. If the input x is an email and the output y is this email spam or not spam, this gives you your spam filter. Or if the input is an audio clip and the algorithm's job is to output the text transcript, then this is speech recognition. Or if you want to input English and have it output the corresponding Spanish, Arabic, Hindi, Chinese, Japanese or something else translation, then that's machine translation. Or the most lucrative form of supervised learning today is probably used in online advertising. Nearly all the large online ad platforms have a learning algorithm that inputs some information about an ad and some information about you and then tries to figure out if you will click on that ad or not. Because by showing you ads that you're slightly more likely to click on for these large online ad platforms, every click is revenue. This actually drives a lot of revenue for these companies. This is something that one's done a lot of work on, maybe not the most inspiring application, but it certainly has a significant economic impact in some companies today. Or if you want to build a self-driving car, the learning algorithm would take as input an image and some information from other sensors, such as a radar or other things, and then try to output the position of say other cars so that your self-driving car can safely drive around the other cars. Or take manufacturing. I've actually done a lot of work in this sector at Landing AI. You can have a learning algorithm take as input a picture of a manufactured product, say a cell phone that just rolled off the production line, and have the learning algorithm output whether or not there is a scratch, dent, or other defect in the product. This is called visual inspection and is helping manufacturers reduce or prevent defects in their products. In all of these applications, you would first train your model with examples of InputsX and the right answers, that is, the labels Y. After the model has learned from these Input Outputs or X and
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1.4 Machine Learning Overview | Supervised learning part 1 --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=EZN_uM3J3kI
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Y pairs, it can then take a brand new Input X, something it's never seen before, and try to produce the appropriate corresponding output Y. Let's dive more deeply into one specific example. Say you want to predict housing prices based on the size of a house. You've collected some data and say you plot the data and it looks like this. Here on the horizontal axis is the size of the house in square feet. And yes, I live in the United States where we still use square feet. I know most of the world uses square meters. And here on the vertical axis is the price of the house in, say, thousands of dollars. So with this data, let's say a friend wants to know what's the price for their 750 square foot house. How can a learning algorithm help you? One thing a learning algorithm might be able to do is say, fit a straight line to the data and reading off the straight line, it looks like your friend's house could be sold for maybe about, I don't know, $150,000. But fitting a straight line isn't the only learning algorithm you can use. There are others that could work better for this application. For example, rather than fitting a straight line, you might decide that it's better to fit a curve, a function that's slightly more complicated or more complex than a straight line. If you do that and make a prediction here, then it looks like, well, your friend's house could be sold for closer to $200,000. One of the things you see later in this class is how you can decide whether to fit a straight line, a curve, or another function that is even more complex to the data. Now, it doesn't seem appropriate to pick the one that gives your friend the best price, but one thing you see is how to get an algorithm to systematically choose the most appropriate line or curve or other thing to fit to this data. What you're seeing in this slide is an example of supervised learning. This is the algorithm or data set in which the so-called right answer, that is, the label or the correct price y is given for every house on the plot. And the task of the learning algorithm is to produce more of these right answers, specifically predicting what is the likely price for other houses like your friend's house. That's why this is supervised learning. To define a little bit more terminology, this housing price prediction is a particular type of supervised learning called regression. And by regression, I mean we're trying to predict a number from infinitely many possible numbers such as the house prices in our example, which could be 150,000 or 70,000 or 183,000 or any other number in between. So that's supervised learning, learning input-output or x-to-y mappings. And you saw in this video an example of regression where the task is to predict
| 500
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1.4 Machine Learning Overview | Supervised learning part 1 --[Machine Learning | Andrew Ng]: Y pairs, it can then take a brand new Input X, something it's never seen before, and try to produce the appropriate corresponding output Y. Let's dive more deeply into one specific example. Say you want to predict housing prices based on the size of a house. You've collected some data and say you plot the data and it looks like this. Here on the horizontal axis is the size of the house in square feet. And yes, I live in the United States where we still use square feet. I know most of the world uses square meters. And here on the vertical axis is the price of the house in, say, thousands of dollars. So with this data, let's say a friend wants to know what's the price for their 750 square foot house. How can a learning algorithm help you? One thing a learning algorithm might be able to do is say, fit a straight line to the data and reading off the straight line, it looks like your friend's house could be sold for maybe about, I don't know, $150,000. But fitting a straight line isn't the only learning algorithm you can use. There are others that could work better for this application. For example, rather than fitting a straight line, you might decide that it's better to fit a curve, a function that's slightly more complicated or more complex than a straight line. If you do that and make a prediction here, then it looks like, well, your friend's house could be sold for closer to $200,000. One of the things you see later in this class is how you can decide whether to fit a straight line, a curve, or another function that is even more complex to the data. Now, it doesn't seem appropriate to pick the one that gives your friend the best price, but one thing you see is how to get an algorithm to systematically choose the most appropriate line or curve or other thing to fit to this data. What you're seeing in this slide is an example of supervised learning. This is the algorithm or data set in which the so-called right answer, that is, the label or the correct price y is given for every house on the plot. And the task of the learning algorithm is to produce more of these right answers, specifically predicting what is the likely price for other houses like your friend's house. That's why this is supervised learning. To define a little bit more terminology, this housing price prediction is a particular type of supervised learning called regression. And by regression, I mean we're trying to predict a number from infinitely many possible numbers such as the house prices in our example, which could be 150,000 or 70,000 or 183,000 or any other number in between. So that's supervised learning, learning input-output or x-to-y mappings. And you saw in this video an example of regression where the task is to predict
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1.5 Machine Learning Overview | Supervised learning part 2 --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=l16C3PKiHKg
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So, supervised learning algorithms learn to predict input-output or x-to-y mappings, and in the last video you saw that regression algorithms, which is a type of supervised learning algorithm, learn to predict numbers out of infinitely many possible numbers. There's a second major type of supervised learning algorithm called a classification algorithm. Let's take a look at what this means. Take breast cancer detection as an example of a classification problem. Say you're building a machine learning system so that doctors can have a diagnostic tool to detect breast cancer. This is important because early detection could potentially save a patient's life. Using a patient's medical records, your machine learning system tries to figure out if a tumor that is a lump is malignant, meaning cancerous or dangerous, or if that tumor, that lump, is benign, meaning that it's just a lump that isn't cancerous and isn't that dangerous. Some of my friends have actually been working on this specific problem. So maybe your dataset has tumors of various sizes, and these tumors are labeled as either benign, which I will designate in this example with a zero, or malignant, which I'll designate in this example with a one. You can then plot your data on a graph like this, where the horizontal axis represents the size of the tumor and the vertical axis takes on only two values, zero or one, depending on whether the tumor is benign, zero, or malignant, one. One reason that this is different from regression is that we're trying to predict only a small number of possible outputs or categories. In this case, two possible outputs, zero or one, benign or malignant. This is different from regression, which tries to predict any number out of an infinitely many number of possible numbers. And so the fact that there are only two possible outputs is what makes this classification. Because there are only two possible outputs or two possible categories in this example, you can also plot this dataset on a line like this, where now I'm going to use two different symbols to denote the category using a circle or an O to denote the benign examples and a cross to denote the malignant examples. And if a new patient walks in for a diagnosis and they have a lump that is this size, then the question is, will your system classify this tumor as benign or malignant? It turns out that in classification problems, you can also have more than two possible output categories. Maybe your learning algorithm can output multiple types of cancer diagnoses if it turns out to be malignant. So let's call two different types of cancer type one and type two. In this case, the algorithm would have three possible output categories it could predict. And by the way, in classification, the terms output classes and output categories are often used interchangeably, so when I say class or category, when referring to the output, it means the same thing.
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1.5 Machine Learning Overview | Supervised learning part 2 --[Machine Learning | Andrew Ng]: So, supervised learning algorithms learn to predict input-output or x-to-y mappings, and in the last video you saw that regression algorithms, which is a type of supervised learning algorithm, learn to predict numbers out of infinitely many possible numbers. There's a second major type of supervised learning algorithm called a classification algorithm. Let's take a look at what this means. Take breast cancer detection as an example of a classification problem. Say you're building a machine learning system so that doctors can have a diagnostic tool to detect breast cancer. This is important because early detection could potentially save a patient's life. Using a patient's medical records, your machine learning system tries to figure out if a tumor that is a lump is malignant, meaning cancerous or dangerous, or if that tumor, that lump, is benign, meaning that it's just a lump that isn't cancerous and isn't that dangerous. Some of my friends have actually been working on this specific problem. So maybe your dataset has tumors of various sizes, and these tumors are labeled as either benign, which I will designate in this example with a zero, or malignant, which I'll designate in this example with a one. You can then plot your data on a graph like this, where the horizontal axis represents the size of the tumor and the vertical axis takes on only two values, zero or one, depending on whether the tumor is benign, zero, or malignant, one. One reason that this is different from regression is that we're trying to predict only a small number of possible outputs or categories. In this case, two possible outputs, zero or one, benign or malignant. This is different from regression, which tries to predict any number out of an infinitely many number of possible numbers. And so the fact that there are only two possible outputs is what makes this classification. Because there are only two possible outputs or two possible categories in this example, you can also plot this dataset on a line like this, where now I'm going to use two different symbols to denote the category using a circle or an O to denote the benign examples and a cross to denote the malignant examples. And if a new patient walks in for a diagnosis and they have a lump that is this size, then the question is, will your system classify this tumor as benign or malignant? It turns out that in classification problems, you can also have more than two possible output categories. Maybe your learning algorithm can output multiple types of cancer diagnoses if it turns out to be malignant. So let's call two different types of cancer type one and type two. In this case, the algorithm would have three possible output categories it could predict. And by the way, in classification, the terms output classes and output categories are often used interchangeably, so when I say class or category, when referring to the output, it means the same thing.
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1.5 Machine Learning Overview | Supervised learning part 2 --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=l16C3PKiHKg
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So to summarize, classification algorithms predict categories. Categories don't have to be numbers. It could be non-numeric. For example, it can predict whether a picture is that of a cat or a dog. And it can predict if a tumor is benign or malignant. Categories can also be numbers like zero or one or zero or one or two. But what makes classification different from regression when you're interpreting the numbers is that classification predicts a small, finite, limited set of possible output categories such as zero, one and two, but not all possible numbers in between like 0.5 or 1.7. In the example of supervised learning that we've been looking at, we had only one input value, the size of the tumor. But you can also use more than one input value to predict an output. Here's an example, instead of just knowing the tumor size, say you also have each patient's age in years. Your new data set now has two inputs, age and tumor size. Plotting this new data set, we're going to use circles to show patients whose tumors are benign and crosses to show the patients with a tumor that was malignant. So when a new patient comes in, the doctor can measure the patient's tumor size and also record the patient's age. And so given this, how can we predict if this patient's tumor is benign or malignant? Well, given a data set like this, what the learning algorithm might do is find some boundary that separates out the malignant tumors from the benign ones. So the learning algorithm has to decide how to fit a boundary line to this data. The boundary line found by the learning algorithm will help the doctor with the diagnosis. In this case, the tumor is more likely to be benign. From this example, we've seen how two inputs, the patient's age and tumor size can be used. In other machine learning problems, often many more input values are required. My friends who worked on breast cancer detection use many additional inputs like the thickness of the tumor clump, uniformity of the cell size, uniformity of the cell shape, and so on. So to recap, supervised learning maps input X to output Y, where the learning algorithm learns from the quote, right answers. The two major types of supervised learning are regression and classification. In a regression application, like predicting prices of houses, the learning algorithm has to predict numbers from infinitely many possible output numbers. Whereas in classification, the learning algorithm has to make a prediction of a category, all of a small set of possible outputs. So you now know what is supervised learning, including both regression and classification. I hope you're having fun. Next, there's a second major type of machine learning called unsupervised learning. Let's go on to the next video to see what that is.
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1.5 Machine Learning Overview | Supervised learning part 2 --[Machine Learning | Andrew Ng]: So to summarize, classification algorithms predict categories. Categories don't have to be numbers. It could be non-numeric. For example, it can predict whether a picture is that of a cat or a dog. And it can predict if a tumor is benign or malignant. Categories can also be numbers like zero or one or zero or one or two. But what makes classification different from regression when you're interpreting the numbers is that classification predicts a small, finite, limited set of possible output categories such as zero, one and two, but not all possible numbers in between like 0.5 or 1.7. In the example of supervised learning that we've been looking at, we had only one input value, the size of the tumor. But you can also use more than one input value to predict an output. Here's an example, instead of just knowing the tumor size, say you also have each patient's age in years. Your new data set now has two inputs, age and tumor size. Plotting this new data set, we're going to use circles to show patients whose tumors are benign and crosses to show the patients with a tumor that was malignant. So when a new patient comes in, the doctor can measure the patient's tumor size and also record the patient's age. And so given this, how can we predict if this patient's tumor is benign or malignant? Well, given a data set like this, what the learning algorithm might do is find some boundary that separates out the malignant tumors from the benign ones. So the learning algorithm has to decide how to fit a boundary line to this data. The boundary line found by the learning algorithm will help the doctor with the diagnosis. In this case, the tumor is more likely to be benign. From this example, we've seen how two inputs, the patient's age and tumor size can be used. In other machine learning problems, often many more input values are required. My friends who worked on breast cancer detection use many additional inputs like the thickness of the tumor clump, uniformity of the cell size, uniformity of the cell shape, and so on. So to recap, supervised learning maps input X to output Y, where the learning algorithm learns from the quote, right answers. The two major types of supervised learning are regression and classification. In a regression application, like predicting prices of houses, the learning algorithm has to predict numbers from infinitely many possible output numbers. Whereas in classification, the learning algorithm has to make a prediction of a category, all of a small set of possible outputs. So you now know what is supervised learning, including both regression and classification. I hope you're having fun. Next, there's a second major type of machine learning called unsupervised learning. Let's go on to the next video to see what that is.
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1.6 Machine Learning Overview | Unsupervised learning part 1 --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=yzAFnfHYH9E
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After supervised learning, the most widely used form of machine learning is unsupervised learning. Let's take a look at what that means. We've talked about supervised learning, and this video is about unsupervised learning. But don't let the name unsupervised fool you. Unsupervised learning is, I think, just as super as supervised learning. When we're looking at supervised learning in the last video, recall that it looks something like this. In the case of a classification problem, each example was associated with an output label Y such as benign or malignant, designated by the O's and crosses. In unsupervised learning, we're given data that isn't associated with any output labels Y. Say you're given data on patients and their tumor size and the patient's age, but not whether the tumor was benign or malignant. So the data set looks like this on the right. We're not asked to diagnose whether the tumor is benign or malignant because we're not given any labels Y in the data set. Instead, our job is to find some structure or some pattern or just find something interesting in the data. This is unsupervised learning. We call it unsupervised because we're not trying to supervise the algorithm to give some quote right answer for every input. Instead, we ask the algorithm to figure out all by itself what's interesting or what patterns or structures there might be in this data. With this particular data set, an unsupervised learning algorithm might decide that the data can be assigned to two different groups or two different clusters. And so it might decide that there's one cluster or group over here and there's another cluster or group over here. This is a particular type of unsupervised learning called a clustering algorithm because it places the unlabeled data into different clusters. And this turns out to be used in many applications. For example, clustering is used in Google News. What Google News does is every day it goes and looks at hundreds of thousands of news articles on the internet and groups related stories together. For example, here's a sample from Google News where the headline of the top article is Giant Panda Gives Birth to Rare Twin Cubs at Japan's Older Zoo. This article actually caught my eye because my daughter loves pandas and so there are a lot of stuffed panda toys and watching of panda videos in my house. And looking at this, you might notice that below this are other related articles. Maybe from the headlines alone, you can start to guess what clustering might be doing. Notice that the word panda appears here, here, here, here, and here. And notice that the word twin also appears in all five articles. And the word zoo also appears in all of these articles. So the clustering algorithm is finding articles out of all the hundreds of thousands of news articles on the internet that day, finding the articles that mention similar words and grouping them into
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1.6 Machine Learning Overview | Unsupervised learning part 1 --[Machine Learning | Andrew Ng]: After supervised learning, the most widely used form of machine learning is unsupervised learning. Let's take a look at what that means. We've talked about supervised learning, and this video is about unsupervised learning. But don't let the name unsupervised fool you. Unsupervised learning is, I think, just as super as supervised learning. When we're looking at supervised learning in the last video, recall that it looks something like this. In the case of a classification problem, each example was associated with an output label Y such as benign or malignant, designated by the O's and crosses. In unsupervised learning, we're given data that isn't associated with any output labels Y. Say you're given data on patients and their tumor size and the patient's age, but not whether the tumor was benign or malignant. So the data set looks like this on the right. We're not asked to diagnose whether the tumor is benign or malignant because we're not given any labels Y in the data set. Instead, our job is to find some structure or some pattern or just find something interesting in the data. This is unsupervised learning. We call it unsupervised because we're not trying to supervise the algorithm to give some quote right answer for every input. Instead, we ask the algorithm to figure out all by itself what's interesting or what patterns or structures there might be in this data. With this particular data set, an unsupervised learning algorithm might decide that the data can be assigned to two different groups or two different clusters. And so it might decide that there's one cluster or group over here and there's another cluster or group over here. This is a particular type of unsupervised learning called a clustering algorithm because it places the unlabeled data into different clusters. And this turns out to be used in many applications. For example, clustering is used in Google News. What Google News does is every day it goes and looks at hundreds of thousands of news articles on the internet and groups related stories together. For example, here's a sample from Google News where the headline of the top article is Giant Panda Gives Birth to Rare Twin Cubs at Japan's Older Zoo. This article actually caught my eye because my daughter loves pandas and so there are a lot of stuffed panda toys and watching of panda videos in my house. And looking at this, you might notice that below this are other related articles. Maybe from the headlines alone, you can start to guess what clustering might be doing. Notice that the word panda appears here, here, here, here, and here. And notice that the word twin also appears in all five articles. And the word zoo also appears in all of these articles. So the clustering algorithm is finding articles out of all the hundreds of thousands of news articles on the internet that day, finding the articles that mention similar words and grouping them into
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1.6 Machine Learning Overview | Unsupervised learning part 1 --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=yzAFnfHYH9E
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clusters. Now what's cool is that this clustering algorithm figures out on its own which words suggest that certain articles are in the same group. What I mean is there isn't an employee at Google News who's telling the algorithm to find articles of the word panda and twins and zoo to put them into the same cluster. The news topics change every day and there are so many news stories it just isn't feasible to have people doing this every single day for all the topics the news covers. Instead, the algorithm has to figure out on its own without supervision what are the clusters of news articles today. So that's why this clustering algorithm is a type of unsupervised learning algorithm. Let's look at a second example of unsupervised learning applied to clustering genetic or DNA data. This image shows a picture of DNA microarray data. These look like tiny grids of a spreadsheet and each tiny column represents the genetic or DNA activity of one person. So for example, this entire column here is from one person's DNA and this other column is of another person. Each row represents a particular gene. So just as an example, perhaps this row here might represent a gene that affects eye color or this row here is a gene that affects how tall someone is. Researchers have even found a genetic link to whether someone dislikes certain vegetables such as broccoli or Brussels sprouts or asparagus. So next time someone asks you, why didn't you finish a salad? You can tell them, oh, maybe it's genetic. For DNA microarrays, the idea is to measure how much certain genes are expressed for each individual person. So these colors, red, green, gray, and so on, show the degree to which different individuals do or do not have a specific gene active. And what you can do is then run a clustering algorithm to group individuals into different categories or different types of people. Like maybe these individuals are grouped together and let's just call this type one. And these people are grouped into type two and these people are grouped as type three. This is unsupervised learning because we're not telling the algorithm in advance that there is a type one person with certain characteristics or a type two person with certain characteristics. Instead, what we're saying is, here's a bunch of data. I don't know what the different types of people are, but can you automatically find structure into data and automatically figure out what are the major types of individuals? Since we're not giving the algorithm the right answer for the examples in advance, this is unsupervised learning. Here's a third example. Many companies have huge databases of customer information. Given this data, can you automatically group your customers into different market segments so that you can more efficiently serve your customers? Concretely, the deeplearning.ai team did some research to better understand the deeplearning.ai community and why different individuals take
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1.6 Machine Learning Overview | Unsupervised learning part 1 --[Machine Learning | Andrew Ng]: clusters. Now what's cool is that this clustering algorithm figures out on its own which words suggest that certain articles are in the same group. What I mean is there isn't an employee at Google News who's telling the algorithm to find articles of the word panda and twins and zoo to put them into the same cluster. The news topics change every day and there are so many news stories it just isn't feasible to have people doing this every single day for all the topics the news covers. Instead, the algorithm has to figure out on its own without supervision what are the clusters of news articles today. So that's why this clustering algorithm is a type of unsupervised learning algorithm. Let's look at a second example of unsupervised learning applied to clustering genetic or DNA data. This image shows a picture of DNA microarray data. These look like tiny grids of a spreadsheet and each tiny column represents the genetic or DNA activity of one person. So for example, this entire column here is from one person's DNA and this other column is of another person. Each row represents a particular gene. So just as an example, perhaps this row here might represent a gene that affects eye color or this row here is a gene that affects how tall someone is. Researchers have even found a genetic link to whether someone dislikes certain vegetables such as broccoli or Brussels sprouts or asparagus. So next time someone asks you, why didn't you finish a salad? You can tell them, oh, maybe it's genetic. For DNA microarrays, the idea is to measure how much certain genes are expressed for each individual person. So these colors, red, green, gray, and so on, show the degree to which different individuals do or do not have a specific gene active. And what you can do is then run a clustering algorithm to group individuals into different categories or different types of people. Like maybe these individuals are grouped together and let's just call this type one. And these people are grouped into type two and these people are grouped as type three. This is unsupervised learning because we're not telling the algorithm in advance that there is a type one person with certain characteristics or a type two person with certain characteristics. Instead, what we're saying is, here's a bunch of data. I don't know what the different types of people are, but can you automatically find structure into data and automatically figure out what are the major types of individuals? Since we're not giving the algorithm the right answer for the examples in advance, this is unsupervised learning. Here's a third example. Many companies have huge databases of customer information. Given this data, can you automatically group your customers into different market segments so that you can more efficiently serve your customers? Concretely, the deeplearning.ai team did some research to better understand the deeplearning.ai community and why different individuals take
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1.6 Machine Learning Overview | Unsupervised learning part 1 --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=yzAFnfHYH9E
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these classes, subscribe to the batch weekly newsletter or attend our PyLAI events. Let's visualize the deeplearning.ai community as this collection of people. Running clustering, that is market segmentation, found a few distinct groups of individuals. One group's primary motivation is seeking knowledge to grow their skills. Perhaps this is you. And if so, that's great. A second group's primary motivation is looking for a way to develop their career. Maybe you want to get a promotion or a new job or make some career progression. If this strives you, that's great too. And yet another group wants to stay updated on how AI impacts their field of work. Perhaps this is you. That's great too. This is the clustering that our team use to try to better serve our community as we're trying to figure out what are the major categories of learners in the deeplearning.ai community. So if any of these is your top motivation for learning, that's great. And I hope I'll be able to help you on your journey. Or in case this is you and you want something totally different than the other three categories, that's fine too. And I want you to know I love you all the same. So to summarize, a clustering algorithm, which is a type of unsophisticated learning algorithm, takes data without labels and tries to automatically group them into clusters. And so maybe the next time you see or think of a panda, maybe you'll think of clustering as well. And besides clustering, there are other types of unsophisticated learning as well. Let's go on to the next video to take a look at some other types of unsophisticated learning algorithms.
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1.6 Machine Learning Overview | Unsupervised learning part 1 --[Machine Learning | Andrew Ng]: these classes, subscribe to the batch weekly newsletter or attend our PyLAI events. Let's visualize the deeplearning.ai community as this collection of people. Running clustering, that is market segmentation, found a few distinct groups of individuals. One group's primary motivation is seeking knowledge to grow their skills. Perhaps this is you. And if so, that's great. A second group's primary motivation is looking for a way to develop their career. Maybe you want to get a promotion or a new job or make some career progression. If this strives you, that's great too. And yet another group wants to stay updated on how AI impacts their field of work. Perhaps this is you. That's great too. This is the clustering that our team use to try to better serve our community as we're trying to figure out what are the major categories of learners in the deeplearning.ai community. So if any of these is your top motivation for learning, that's great. And I hope I'll be able to help you on your journey. Or in case this is you and you want something totally different than the other three categories, that's fine too. And I want you to know I love you all the same. So to summarize, a clustering algorithm, which is a type of unsophisticated learning algorithm, takes data without labels and tries to automatically group them into clusters. And so maybe the next time you see or think of a panda, maybe you'll think of clustering as well. And besides clustering, there are other types of unsophisticated learning as well. Let's go on to the next video to take a look at some other types of unsophisticated learning algorithms.
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1.7 Machine Learning Overview | Unsupervised learning part 2 --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=u7Y_b04upmQ
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In the last video, you saw what is unsupervised learning and one type of unsupervised learning called clustering. Let's give a slightly more formal definition of unsupervised learning and take a quick look at some other types of unsupervised learning other than clustering. Whereas in supervised learning, the data comes with both inputs x and output labels y, in unsupervised learning, the data comes only with inputs x but not output labels y, and the algorithm has to find some structure or some pattern or something interesting in the data. We've seen just one example of unsupervised learning called a clustering algorithm, which groups similar data points together. In this specialization, you learn about clustering as well as two other types of unsupervised learning. One is called anomaly detection, which is used to detect unusual events. This turns out to be really important for fraud detection in the financial system, where unusual events, unusual transactions could be a sign of fraud and for many other applications. And you also learn about dimensionality reduction. This lets you take a big data set and almost magically compress it to a much smaller data set while losing as little information as possible. In case anomaly detection and dimensionality reduction don't seem to make too much sense to you yet, don't worry about it. We'll get to this later in this specialization. Now, I'd like to ask you another question to help you check your understanding. And no pressure, if you don't get it right on the first try, it's totally fine. Please select any of the following that you think are examples of unsupervised learning. Two are unsupervised examples and two are supervised learning examples. So please take a look. Maybe you remember the spam filtering problem. If you have labeled data, you know, labeled as spam or non spam email, you can treat this as a supervised learning problem. The second example, the news story example. That's exactly the Google News and Tandem example that you saw in the last video. And so you can approach that using a clustering algorithm to group news articles together. So that would use unsupervised learning. The market segmentation example that I talked about a little bit earlier, you can do that as an unsupervised learning problem as well, because you can give your algorithm some data and ask it to discover market segments automatically. And the final example on diagnosing diabetes. Well, actually, that's a lot like our breast cancer example from the supervised learning videos. Only instead of benign or malignant tumors, we instead have diabetes or not diabetes. And so you can approach this as a supervised learning problem, just like we did for the breast tumor classification problem. Even though in this in the last video, we've talked mainly about clustering in later videos in this specialization, we'll dive much more deeply into anomaly detection and dimensionality reduction as well. So that's unsupervised learning. Before we wrap up this section, I want
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1.7 Machine Learning Overview | Unsupervised learning part 2 --[Machine Learning | Andrew Ng]: In the last video, you saw what is unsupervised learning and one type of unsupervised learning called clustering. Let's give a slightly more formal definition of unsupervised learning and take a quick look at some other types of unsupervised learning other than clustering. Whereas in supervised learning, the data comes with both inputs x and output labels y, in unsupervised learning, the data comes only with inputs x but not output labels y, and the algorithm has to find some structure or some pattern or something interesting in the data. We've seen just one example of unsupervised learning called a clustering algorithm, which groups similar data points together. In this specialization, you learn about clustering as well as two other types of unsupervised learning. One is called anomaly detection, which is used to detect unusual events. This turns out to be really important for fraud detection in the financial system, where unusual events, unusual transactions could be a sign of fraud and for many other applications. And you also learn about dimensionality reduction. This lets you take a big data set and almost magically compress it to a much smaller data set while losing as little information as possible. In case anomaly detection and dimensionality reduction don't seem to make too much sense to you yet, don't worry about it. We'll get to this later in this specialization. Now, I'd like to ask you another question to help you check your understanding. And no pressure, if you don't get it right on the first try, it's totally fine. Please select any of the following that you think are examples of unsupervised learning. Two are unsupervised examples and two are supervised learning examples. So please take a look. Maybe you remember the spam filtering problem. If you have labeled data, you know, labeled as spam or non spam email, you can treat this as a supervised learning problem. The second example, the news story example. That's exactly the Google News and Tandem example that you saw in the last video. And so you can approach that using a clustering algorithm to group news articles together. So that would use unsupervised learning. The market segmentation example that I talked about a little bit earlier, you can do that as an unsupervised learning problem as well, because you can give your algorithm some data and ask it to discover market segments automatically. And the final example on diagnosing diabetes. Well, actually, that's a lot like our breast cancer example from the supervised learning videos. Only instead of benign or malignant tumors, we instead have diabetes or not diabetes. And so you can approach this as a supervised learning problem, just like we did for the breast tumor classification problem. Even though in this in the last video, we've talked mainly about clustering in later videos in this specialization, we'll dive much more deeply into anomaly detection and dimensionality reduction as well. So that's unsupervised learning. Before we wrap up this section, I want
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1.8 Machine Learning Overview | Jupyter Notebooks --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=SRDUSIcnW8M
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So far in the videos, you've seen supervised learning and unsupervised learning, and also examples of both. For you to more deeply understand these concepts, I'd like to invite you in this class to see, run, and maybe later write code yourself to implement these concepts. The most widely used tool by machine learning and data science practitioners today is the Jupyter Notebook. This is the default environment that a lot of us use to code up and experiment and try things out. And so in this class, right here in your web browser, you'll be able to use a Jupyter Notebook environment to test out some of these ideas for yourself as well. This is not some made up, simplified environment. This is the exact same environment, the exact same tool, the Jupyter Notebook that developers are using in many large companies right now. One type of lab that you see throughout this class are optional labs, which are ones you can open and run one line at a time with usually not needing to write any code yourself. So optional labs are designed to be very easy, and I can guarantee you will get full marks for every single one of them, because there are no marks. All you need to do is open it up and just run the code we've provided. And by reading through and running the code in the optional labs, you see how machine learning code runs. And you should be able to complete them relatively quickly, just by running it one line at a time from top to bottom. Optional labs are completely optional, so you don't have to do them at all if you don't want to. But I hope you will take a look, because running through them will give you a deeper feel, give you a little bit more experience with what machine learning algorithms, what machine learning code actually looks like. Starting next week, there will also be some practice labs, which will give you an opportunity to write some of that code yourself. But we'll get to that next week, so don't worry about it for now. And I hope you just go through the next optional lab and get through the rest of the content for this week. So let's take a look at an example of a notebook. This is what you see when you go to the first optional lab. Feel free to scroll up and down and browse and mouse over the different menus and take a look at the different options here. You might notice that there are two types of these blocks, also called cells, in the notebook. And there are two types of cells. One is what's called a markdown cell, which means basically a bunch of text. So here you can actually edit the text if you don't like the text that we wrote. But this is text that describes the code. Then there's a second type of block
| 500
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1.8 Machine Learning Overview | Jupyter Notebooks --[Machine Learning | Andrew Ng]: So far in the videos, you've seen supervised learning and unsupervised learning, and also examples of both. For you to more deeply understand these concepts, I'd like to invite you in this class to see, run, and maybe later write code yourself to implement these concepts. The most widely used tool by machine learning and data science practitioners today is the Jupyter Notebook. This is the default environment that a lot of us use to code up and experiment and try things out. And so in this class, right here in your web browser, you'll be able to use a Jupyter Notebook environment to test out some of these ideas for yourself as well. This is not some made up, simplified environment. This is the exact same environment, the exact same tool, the Jupyter Notebook that developers are using in many large companies right now. One type of lab that you see throughout this class are optional labs, which are ones you can open and run one line at a time with usually not needing to write any code yourself. So optional labs are designed to be very easy, and I can guarantee you will get full marks for every single one of them, because there are no marks. All you need to do is open it up and just run the code we've provided. And by reading through and running the code in the optional labs, you see how machine learning code runs. And you should be able to complete them relatively quickly, just by running it one line at a time from top to bottom. Optional labs are completely optional, so you don't have to do them at all if you don't want to. But I hope you will take a look, because running through them will give you a deeper feel, give you a little bit more experience with what machine learning algorithms, what machine learning code actually looks like. Starting next week, there will also be some practice labs, which will give you an opportunity to write some of that code yourself. But we'll get to that next week, so don't worry about it for now. And I hope you just go through the next optional lab and get through the rest of the content for this week. So let's take a look at an example of a notebook. This is what you see when you go to the first optional lab. Feel free to scroll up and down and browse and mouse over the different menus and take a look at the different options here. You might notice that there are two types of these blocks, also called cells, in the notebook. And there are two types of cells. One is what's called a markdown cell, which means basically a bunch of text. So here you can actually edit the text if you don't like the text that we wrote. But this is text that describes the code. Then there's a second type of block
|
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1.8 Machine Learning Overview | Jupyter Notebooks --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=SRDUSIcnW8M
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or cell, which looks like this, which is a code cell. So here we've already provided the code. And if you want to run this code cell, hitting shift enter will run the code in this code cell. Oh, and by the way, if you click on a markdown cell, so it's showing all this formatting, go ahead and hit shift enter on your keyboard as well. And that will also convert it back to this nicely formatted text. This optional lab shows some common Python code. So you can go ahead and run this afterwards in your own Jupyter notebook. When you jump into this notebook yourself, what I'd like you to do is select the cells and hit shift enter. Well, read through the code, see if it makes sense, you know, try to make a prediction about what you think this code will do, and then hit shift enter and then see what the code actually does. And if you feel like it, feel free to go in and edit the code, you know, change the code and then run it and see what happens. And if you haven't played in a Jupyter notebook environment before, I hope you become more familiar with Python in a Jupyter notebook. I spend a lot of hours playing around in Jupyter notebooks, and so I hope you have fun with them too. And after that, I look forward to seeing you in the next video, where we'll take the supervised learning problem and start to flesh out our first supervised learning algorithm. I hope that'll be fun too, and look forward to seeing you there.
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1.8 Machine Learning Overview | Jupyter Notebooks --[Machine Learning | Andrew Ng]: or cell, which looks like this, which is a code cell. So here we've already provided the code. And if you want to run this code cell, hitting shift enter will run the code in this code cell. Oh, and by the way, if you click on a markdown cell, so it's showing all this formatting, go ahead and hit shift enter on your keyboard as well. And that will also convert it back to this nicely formatted text. This optional lab shows some common Python code. So you can go ahead and run this afterwards in your own Jupyter notebook. When you jump into this notebook yourself, what I'd like you to do is select the cells and hit shift enter. Well, read through the code, see if it makes sense, you know, try to make a prediction about what you think this code will do, and then hit shift enter and then see what the code actually does. And if you feel like it, feel free to go in and edit the code, you know, change the code and then run it and see what happens. And if you haven't played in a Jupyter notebook environment before, I hope you become more familiar with Python in a Jupyter notebook. I spend a lot of hours playing around in Jupyter notebooks, and so I hope you have fun with them too. And after that, I look forward to seeing you in the next video, where we'll take the supervised learning problem and start to flesh out our first supervised learning algorithm. I hope that'll be fun too, and look forward to seeing you there.
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1.9 Machine Learning Overview | Linear regression model part 1 --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=isx7QB_j4jY
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In this video, we'll look at what the overall process of supervised learning is like. Specifically, you see the first model of this course, a linear regression model. That just means filling a straight line to your data. It's probably the most widely used learning algorithm in the world today. And as you get familiar with linear regression, many of the concepts you see here will also apply to other machine learning models, models that you'll see later in this specialization. Let's start with a problem that you can address using linear regression. Say you want to predict the price of a house based on the size of a house. This is example we see in earlier this week. We're going to use a data set on house sizes and prices from Portland, a city in the United States. Here with a graph where the horizontal axis is the size of the house in square feet and the vertical axis is the price of the house in thousands of dollars. Let's go ahead and plot the data points for various houses in the data set. Here each data point, each of these little crosses is a house with a size and a price that it most recently was sold for. Now let's say you're a real estate agent in Portland and you're helping a client sell her house. And she's asking you, how much do you think you're going to get for this house? This data set might help you estimate the price she could get for it. You start by measuring the size of the house and it turns out that her house is 1250 square feet. How much do you think this house could sell for? One thing you could do is you can build a linear regression model from this data set. Your model will fit a straight line to the data, which might look like this. And based on this straight line fit to the data, you can kind of see that if a house is 1250 square feet, it will intersect the best fit line over here. And if you trace that to the vertical axis on the left, you can see the prices may be around here, say about $220,000. So this is an example of what's called a supervised learning model. We call this supervised learning because you are first training your model by giving it data that has the right answers. Because you give the model examples of houses with both the size of the house as well as the price that the model should predict for each house. Where here are the prices, that is the right answers are given for every house in the data set. This linear regression model is a particular type of supervised learning model. It's called a regression model because it predicts numbers as the output, like prices and dollars. Any supervised learning model that predicts a number, such as 220,000 or 1.5 or negative 33.2
| 500
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1.9 Machine Learning Overview | Linear regression model part 1 --[Machine Learning | Andrew Ng]: In this video, we'll look at what the overall process of supervised learning is like. Specifically, you see the first model of this course, a linear regression model. That just means filling a straight line to your data. It's probably the most widely used learning algorithm in the world today. And as you get familiar with linear regression, many of the concepts you see here will also apply to other machine learning models, models that you'll see later in this specialization. Let's start with a problem that you can address using linear regression. Say you want to predict the price of a house based on the size of a house. This is example we see in earlier this week. We're going to use a data set on house sizes and prices from Portland, a city in the United States. Here with a graph where the horizontal axis is the size of the house in square feet and the vertical axis is the price of the house in thousands of dollars. Let's go ahead and plot the data points for various houses in the data set. Here each data point, each of these little crosses is a house with a size and a price that it most recently was sold for. Now let's say you're a real estate agent in Portland and you're helping a client sell her house. And she's asking you, how much do you think you're going to get for this house? This data set might help you estimate the price she could get for it. You start by measuring the size of the house and it turns out that her house is 1250 square feet. How much do you think this house could sell for? One thing you could do is you can build a linear regression model from this data set. Your model will fit a straight line to the data, which might look like this. And based on this straight line fit to the data, you can kind of see that if a house is 1250 square feet, it will intersect the best fit line over here. And if you trace that to the vertical axis on the left, you can see the prices may be around here, say about $220,000. So this is an example of what's called a supervised learning model. We call this supervised learning because you are first training your model by giving it data that has the right answers. Because you give the model examples of houses with both the size of the house as well as the price that the model should predict for each house. Where here are the prices, that is the right answers are given for every house in the data set. This linear regression model is a particular type of supervised learning model. It's called a regression model because it predicts numbers as the output, like prices and dollars. Any supervised learning model that predicts a number, such as 220,000 or 1.5 or negative 33.2
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1.9 Machine Learning Overview | Linear regression model part 1 --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=isx7QB_j4jY
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is addressing what's called a regression problem. So linear regression is one example of a regression model, but there are other models for addressing regression problems too. And we'll see some of those later in course two of this specialization. And just to remind you, in contrast with the regression model, the other most common type of supervised learning model is called a classification model. A classification model predicts categories or discrete categories, such as predicting if a picture is of a cat meow or a dog woof. Or if given a medical record, it has to predict if a patient has a particular disease. You see more about classification models later in this course as well. So as a reminder about the difference between classification and regression, in classification, there are only a small number of possible outputs. If your model is recognizing cats versus dogs, that's two possible outputs. Or maybe you're trying to recognize any of 10 possible medical conditions in a patient. So there's a discrete, finite set of possible outputs. We call it a classification problem. Whereas in regression, there are infinitely many possible numbers that the model could output. In addition to visualizing this data as a plot here on the left, there's one other way of looking at the data that would be useful. And that's a data table here on the right. The data comprises a set of inputs. This would be the size of the house, which is this column here. It also has outputs, you're trying to predict the price, which is this column here. Notice that the horizontal and vertical axes correspond to these two columns, the size and the price. And so if you have, say, 47 rows in this data table, then there are 47 of these little crosses on the plot of the left, each cross corresponding to one row of the table. For example, the first row of the table is a house with size 2104 square feet. So that's around here. And this house, so for $400,000, which is around here. So this first row of the table is plotted as this data point over here. Now let's look at some notation for describing the data. This is notation that you find useful throughout your journey in machine learning. As you increasingly get familiar with machine learning terminology, this would be terminology that you can use to talk about machine learning concepts with others as well, since a lot of this is quite standard across AI. You'll be seeing this notation multiple times in this specialization. So it's okay if you don't remember everything the first time through, it will naturally become more familiar over time. The data set that you just saw and that is used to train the model is called a training set. Note that your client's house is not in this data set because it's not yet sold. So no one knows what his price is. So to predict the
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1.9 Machine Learning Overview | Linear regression model part 1 --[Machine Learning | Andrew Ng]: is addressing what's called a regression problem. So linear regression is one example of a regression model, but there are other models for addressing regression problems too. And we'll see some of those later in course two of this specialization. And just to remind you, in contrast with the regression model, the other most common type of supervised learning model is called a classification model. A classification model predicts categories or discrete categories, such as predicting if a picture is of a cat meow or a dog woof. Or if given a medical record, it has to predict if a patient has a particular disease. You see more about classification models later in this course as well. So as a reminder about the difference between classification and regression, in classification, there are only a small number of possible outputs. If your model is recognizing cats versus dogs, that's two possible outputs. Or maybe you're trying to recognize any of 10 possible medical conditions in a patient. So there's a discrete, finite set of possible outputs. We call it a classification problem. Whereas in regression, there are infinitely many possible numbers that the model could output. In addition to visualizing this data as a plot here on the left, there's one other way of looking at the data that would be useful. And that's a data table here on the right. The data comprises a set of inputs. This would be the size of the house, which is this column here. It also has outputs, you're trying to predict the price, which is this column here. Notice that the horizontal and vertical axes correspond to these two columns, the size and the price. And so if you have, say, 47 rows in this data table, then there are 47 of these little crosses on the plot of the left, each cross corresponding to one row of the table. For example, the first row of the table is a house with size 2104 square feet. So that's around here. And this house, so for $400,000, which is around here. So this first row of the table is plotted as this data point over here. Now let's look at some notation for describing the data. This is notation that you find useful throughout your journey in machine learning. As you increasingly get familiar with machine learning terminology, this would be terminology that you can use to talk about machine learning concepts with others as well, since a lot of this is quite standard across AI. You'll be seeing this notation multiple times in this specialization. So it's okay if you don't remember everything the first time through, it will naturally become more familiar over time. The data set that you just saw and that is used to train the model is called a training set. Note that your client's house is not in this data set because it's not yet sold. So no one knows what his price is. So to predict the
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1.9 Machine Learning Overview | Linear regression model part 1 --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=isx7QB_j4jY
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price of your client's house, you first train your model to learn from the training set and that model can then predict your client's house's price. In machine learning, the standard notation to denote the input here is lowercase x and we call this the input variable. It's also called a feature or an input feature. For example, for the first house in your training set, x is the size of the house, so x equals 2104. The standard notation to denote the output variable, which you're trying to predict, which is also sometimes called the target variable, is lowercase y. And so here, y is the price of the house and for the first training example, this is equal to 400. So y equals 400. So the data set has one row for each house and in this particular training set, there are 47 rows with each row representing a different training example. We're going to use lowercase m to refer to the total number of training examples and so here, m is equal to 47. To indicate a single training example, we're going to use the notation parentheses x comma y. So for the first training example, x comma y, this pair of numbers is 2104 comma 400. Now we have a lot of different training examples. We have 47 of them in fact. So to refer to a specific training example, this will correspond to a specific row in this table on the left. I'm going to use the notation x superscript in parentheses i comma y superscript in parentheses i. The superscript tells us that this is the i-th training example, such as the first, second or third up to the 47th training example. i here refers to a specific row in the table. So for instance, here is the first example when i equals 1 in the training set. And so x superscript 1 is equal to 2104 and y superscript 1 is equal to 400. And let's add the superscript 1 here as well. Just a note, this superscript i in parentheses is not exponentiation. So when I write this, this is not x squared. This is not x to the power of 2. It just refers to the second training example. So this i is just an index in the training set and refers to row i in the table. In this video, you saw what a training set is like as well as standard notation for describing this training set. In the next video, let's look at what it'll take to take this training set that you just saw and feed it to a learning algorithm so that the algorithm can learn from this data. We'll see that in the next video.
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1.9 Machine Learning Overview | Linear regression model part 1 --[Machine Learning | Andrew Ng]: price of your client's house, you first train your model to learn from the training set and that model can then predict your client's house's price. In machine learning, the standard notation to denote the input here is lowercase x and we call this the input variable. It's also called a feature or an input feature. For example, for the first house in your training set, x is the size of the house, so x equals 2104. The standard notation to denote the output variable, which you're trying to predict, which is also sometimes called the target variable, is lowercase y. And so here, y is the price of the house and for the first training example, this is equal to 400. So y equals 400. So the data set has one row for each house and in this particular training set, there are 47 rows with each row representing a different training example. We're going to use lowercase m to refer to the total number of training examples and so here, m is equal to 47. To indicate a single training example, we're going to use the notation parentheses x comma y. So for the first training example, x comma y, this pair of numbers is 2104 comma 400. Now we have a lot of different training examples. We have 47 of them in fact. So to refer to a specific training example, this will correspond to a specific row in this table on the left. I'm going to use the notation x superscript in parentheses i comma y superscript in parentheses i. The superscript tells us that this is the i-th training example, such as the first, second or third up to the 47th training example. i here refers to a specific row in the table. So for instance, here is the first example when i equals 1 in the training set. And so x superscript 1 is equal to 2104 and y superscript 1 is equal to 400. And let's add the superscript 1 here as well. Just a note, this superscript i in parentheses is not exponentiation. So when I write this, this is not x squared. This is not x to the power of 2. It just refers to the second training example. So this i is just an index in the training set and refers to row i in the table. In this video, you saw what a training set is like as well as standard notation for describing this training set. In the next video, let's look at what it'll take to take this training set that you just saw and feed it to a learning algorithm so that the algorithm can learn from this data. We'll see that in the next video.
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1.10 Machine Learning Overview | Linear regression model part 2 --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=vrTHO5zRq6s
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Let's look in this video at the process of how supervised learning works. Supervised learning algorithm will input a data set and then what exactly does it do and what does it output? Let's find out in this video. Recall that a training set in supervised learning includes both the input features, such as the size of the hulls, and also the output targets, such as the price of the hulls. The output targets are the right answers that the model will learn from. To train the model, you feed the training set, both the input features and the output targets, to your learning algorithm. Then your supervised learning algorithm will produce some function. We'll write this function as lowercase f, where f stands for function. Historically, this function used to be called a hypothesis, but I'm just going to call it a function f in this class. The job of f is to take a new input x and output an estimate or prediction, which I'm going to call y hat, and it's written like the variable y with this little hat symbol on top. In machine learning, the convention is that y hat is the estimate or the prediction for y. The function f is called the model. X is called the input or the input feature, and the output of the model is the prediction y hat. The model's prediction is the estimated value of y. When the symbol is just a letter y, then that refers to the target, which is the actual true value in the training set. In contrast, y hat is an estimate. It may or may not be the actual true value. Well, if you're helping your client to sell their hulls, well, the true price of the hulls is unknown until they sell it. So your model f, given the size, outputs a price which is the estimated, that is the prediction of what the true price will be. Now, when we design a learning algorithm, a key question is, how are we going to represent the function f? Or in other words, what is the math formula we're going to use to compute f? For now, let's stick with f being a straight line. So your function can be written as f subscript w comma b of x equals, I'm going to use w times x plus b. I'll define w and b soon, but for now, just know that w and b are numbers, and the values chosen for w and b will determine the prediction y hat based on the input feature x. So this f w b of x means f is a function that takes x's input, and depending on the values of w and b, f will output some value of a prediction y hat. As an alternative to writing this f w comma b of x, I'll sometimes just write f of x without explicitly including w and b in the subscript. It's just
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1.10 Machine Learning Overview | Linear regression model part 2 --[Machine Learning | Andrew Ng]: Let's look in this video at the process of how supervised learning works. Supervised learning algorithm will input a data set and then what exactly does it do and what does it output? Let's find out in this video. Recall that a training set in supervised learning includes both the input features, such as the size of the hulls, and also the output targets, such as the price of the hulls. The output targets are the right answers that the model will learn from. To train the model, you feed the training set, both the input features and the output targets, to your learning algorithm. Then your supervised learning algorithm will produce some function. We'll write this function as lowercase f, where f stands for function. Historically, this function used to be called a hypothesis, but I'm just going to call it a function f in this class. The job of f is to take a new input x and output an estimate or prediction, which I'm going to call y hat, and it's written like the variable y with this little hat symbol on top. In machine learning, the convention is that y hat is the estimate or the prediction for y. The function f is called the model. X is called the input or the input feature, and the output of the model is the prediction y hat. The model's prediction is the estimated value of y. When the symbol is just a letter y, then that refers to the target, which is the actual true value in the training set. In contrast, y hat is an estimate. It may or may not be the actual true value. Well, if you're helping your client to sell their hulls, well, the true price of the hulls is unknown until they sell it. So your model f, given the size, outputs a price which is the estimated, that is the prediction of what the true price will be. Now, when we design a learning algorithm, a key question is, how are we going to represent the function f? Or in other words, what is the math formula we're going to use to compute f? For now, let's stick with f being a straight line. So your function can be written as f subscript w comma b of x equals, I'm going to use w times x plus b. I'll define w and b soon, but for now, just know that w and b are numbers, and the values chosen for w and b will determine the prediction y hat based on the input feature x. So this f w b of x means f is a function that takes x's input, and depending on the values of w and b, f will output some value of a prediction y hat. As an alternative to writing this f w comma b of x, I'll sometimes just write f of x without explicitly including w and b in the subscript. It's just
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1.10 Machine Learning Overview | Linear regression model part 2 --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=vrTHO5zRq6s
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a simple notation, but means exactly the same thing as f w b of x. Let's plot the training set on the graph where the input feature x is on the horizontal axis, and the output target y is on the vertical axis. Remember, the algorithm learns from this data and generates a best fit line like maybe this one here. This straight line is the linear function f w b of x equals w times x plus b. Or more simply, we can drop w and b and just write f of x equals w x plus b. Here's what this function is doing, it's making predictions for the value of y using a straight line function of x. So you may ask, why are we choosing a linear function where linear function is just a fancy term for a straight line, instead of some nonlinear function like a curve or a parabola? Well, sometimes you want to fit more complex nonlinear functions as well, like a curve like this, but since this linear function is relatively simple and easy to work with, let's use a line as a foundation that will eventually help you to get to more complex models that are nonlinear. This particular model has a name, it's called linear regression. More specifically, this is linear regression with one variable, where the phrase one variable means that there's a single input variable or feature x, namely the size of the house. Another name for a linear model with one input variable is univariate linear regression, where uni means one in Latin and where variate means variable. So univariate is just a fancy way of saying one variable. In a later video, you'll also see a variation of regression where you want to make a prediction based not just on the size of a house, but on a bunch of other things that you may know about the house, such as number of bedrooms and other features. And by the way, when you're done with this video, there is another optional lab. You don't need to write any code, just review it, run the code and see what it does. That will show you how to define in Python a straight line function. And the lab will let you choose the values of W and B to try to fit the training data. You don't have to do the lab if you don't want to, but I hope you play of it when you're done watching this video. So that's linear regression. In order for you to make this work, one of the most important things you have to do is construct a cost function. The idea of a cost function is one of the most universal and important ideas in machine learning and is used in both linear regression and in training many of the most advanced AI models in the world. So let's go on to the next video and take a look at how
| 500
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1.10 Machine Learning Overview | Linear regression model part 2 --[Machine Learning | Andrew Ng]: a simple notation, but means exactly the same thing as f w b of x. Let's plot the training set on the graph where the input feature x is on the horizontal axis, and the output target y is on the vertical axis. Remember, the algorithm learns from this data and generates a best fit line like maybe this one here. This straight line is the linear function f w b of x equals w times x plus b. Or more simply, we can drop w and b and just write f of x equals w x plus b. Here's what this function is doing, it's making predictions for the value of y using a straight line function of x. So you may ask, why are we choosing a linear function where linear function is just a fancy term for a straight line, instead of some nonlinear function like a curve or a parabola? Well, sometimes you want to fit more complex nonlinear functions as well, like a curve like this, but since this linear function is relatively simple and easy to work with, let's use a line as a foundation that will eventually help you to get to more complex models that are nonlinear. This particular model has a name, it's called linear regression. More specifically, this is linear regression with one variable, where the phrase one variable means that there's a single input variable or feature x, namely the size of the house. Another name for a linear model with one input variable is univariate linear regression, where uni means one in Latin and where variate means variable. So univariate is just a fancy way of saying one variable. In a later video, you'll also see a variation of regression where you want to make a prediction based not just on the size of a house, but on a bunch of other things that you may know about the house, such as number of bedrooms and other features. And by the way, when you're done with this video, there is another optional lab. You don't need to write any code, just review it, run the code and see what it does. That will show you how to define in Python a straight line function. And the lab will let you choose the values of W and B to try to fit the training data. You don't have to do the lab if you don't want to, but I hope you play of it when you're done watching this video. So that's linear regression. In order for you to make this work, one of the most important things you have to do is construct a cost function. The idea of a cost function is one of the most universal and important ideas in machine learning and is used in both linear regression and in training many of the most advanced AI models in the world. So let's go on to the next video and take a look at how
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1.11 Machine Learning Overview | Cost function formula --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=ZzeDtSmrRoU
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In order to implement linear regression, the first key step is for us to define something called a cost function. This is something we'll build in this video. And the cost function will tell us how well the model is doing so that we can try to get it to do better. Let's look at what this means. Recall that you have a training set that contains input features x and output targets y. And the model you're going to use to fit this training set is this linear function fwb of x equals w times x plus b. To introduce a little bit more terminology, the w and b are called the parameters of the model. In machine learning, parameters of a model are the variables you can adjust during training in order to improve the model. Sometimes, you also hear the parameters w and b referred to as coefficients or as weights. Now, let's take a look at what these parameters w and b do. Depending on the values you've chosen for w and b, you get a different function f of x, which generates a different line on the graph. And remember that we can write f of x as a shorthand for fwb of x. We're going to take a look at some plots of f of x on a chart. Maybe you're already familiar with drawing lines on charts, but even if this is a review for you, I hope this will help you build intuition on how w and b, the parameters, determine f. When w is equal to 0 and b is equal to 1.5, then f looks like this horizontal line. In this case, the function f of x is 0 times x plus 1.5, so f is always a constant value. It always predicts 1.5 for the estimated value of y. So y hat is always equal to b. And here, b is also called the y-intercept, because that's where it crosses the vertical axis or the y-axis on this graph. As a second example, if w is 0.5 and b is equal to 0, then f of x is 0.5 times x. When x is 0, the prediction is also 0. And when x is 2, then the prediction is 0.5 times 2, which is 1. So you get a line that looks like this. And notice that the slope is 0.5 divided by 1, so the value of w gives you the slope of the line, which is 0.5. And finally, if w equals 0.5 and b equals 1, then f of x is 0.5 times x plus 1. And when x is 0, then f of x equals b, which is 1, so the line intersects the vertical axis at b, the y-intercept. Also when x is 2, then f of x is 2, so the line looks like this. Again, the slope is 0.5 divided by 1, so the value of w gives you the slope, which
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1.11 Machine Learning Overview | Cost function formula --[Machine Learning | Andrew Ng]: In order to implement linear regression, the first key step is for us to define something called a cost function. This is something we'll build in this video. And the cost function will tell us how well the model is doing so that we can try to get it to do better. Let's look at what this means. Recall that you have a training set that contains input features x and output targets y. And the model you're going to use to fit this training set is this linear function fwb of x equals w times x plus b. To introduce a little bit more terminology, the w and b are called the parameters of the model. In machine learning, parameters of a model are the variables you can adjust during training in order to improve the model. Sometimes, you also hear the parameters w and b referred to as coefficients or as weights. Now, let's take a look at what these parameters w and b do. Depending on the values you've chosen for w and b, you get a different function f of x, which generates a different line on the graph. And remember that we can write f of x as a shorthand for fwb of x. We're going to take a look at some plots of f of x on a chart. Maybe you're already familiar with drawing lines on charts, but even if this is a review for you, I hope this will help you build intuition on how w and b, the parameters, determine f. When w is equal to 0 and b is equal to 1.5, then f looks like this horizontal line. In this case, the function f of x is 0 times x plus 1.5, so f is always a constant value. It always predicts 1.5 for the estimated value of y. So y hat is always equal to b. And here, b is also called the y-intercept, because that's where it crosses the vertical axis or the y-axis on this graph. As a second example, if w is 0.5 and b is equal to 0, then f of x is 0.5 times x. When x is 0, the prediction is also 0. And when x is 2, then the prediction is 0.5 times 2, which is 1. So you get a line that looks like this. And notice that the slope is 0.5 divided by 1, so the value of w gives you the slope of the line, which is 0.5. And finally, if w equals 0.5 and b equals 1, then f of x is 0.5 times x plus 1. And when x is 0, then f of x equals b, which is 1, so the line intersects the vertical axis at b, the y-intercept. Also when x is 2, then f of x is 2, so the line looks like this. Again, the slope is 0.5 divided by 1, so the value of w gives you the slope, which
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1.11 Machine Learning Overview | Cost function formula --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=ZzeDtSmrRoU
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is 0.5. Recall that you have a training set, like the one shown here, with linear regression, what you want to do is to choose values for the parameters w and b so that the straight line you get from the function f somehow fits the data well, like maybe this line shown here. And when I say that the line fits the data, visually you can think of this to mean that the line defined by f is roughly passing through or somewhat close to the training examples as compared to other possible lines that are not as close to these points. And just to remind you of some notation, a training example like this point here is defined by x super strip i, y super strip i, where y is the target. For a given input x i, the function f also makes a predicted value for y, and the value that it predicts for y is y hat i, shown here. For our choice of a model, f of x i is w times x i plus b. Stated differently, the prediction y hat i is f of w b of x i, where for the model where using f of x i is equal to w x i plus b. So now the question is, how do you find values for w and b so that the prediction y hat i is close to the true target y i for many or maybe all training examples x i y i? To answer that question, let's first take a look at how to measure how well a line fits the training data. To do that, we're going to construct our cost function. The cost function takes the prediction y hat and compares it to the target y by taking y hat minus y. This difference is called the error. We're measuring how far off the prediction is from the target. Next let's compute the square of this error. Also we're going to want to compute this term for different training examples i in the training set. So when measuring the error for example i, we'll compute this squared error term. Finally, we want to measure the error across the entire training set. In particular, let's sum up the squared errors like this. We'll sum from i equals 1, 2, 3, all the way up to m. And remember that m is the number of training examples, which is 47 for this data set. Notice that if we have more training examples, m is larger and your cost function will calculate a bigger number since it's summing over more examples. So to build a cost function that doesn't automatically get bigger as the training set size gets larger, by convention, we will compute the average squared error instead of the total squared error. And we do that by dividing by m like this. Okay, we're nearly there. Just one last thing. By convention, the cost function that machine
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1.11 Machine Learning Overview | Cost function formula --[Machine Learning | Andrew Ng]: is 0.5. Recall that you have a training set, like the one shown here, with linear regression, what you want to do is to choose values for the parameters w and b so that the straight line you get from the function f somehow fits the data well, like maybe this line shown here. And when I say that the line fits the data, visually you can think of this to mean that the line defined by f is roughly passing through or somewhat close to the training examples as compared to other possible lines that are not as close to these points. And just to remind you of some notation, a training example like this point here is defined by x super strip i, y super strip i, where y is the target. For a given input x i, the function f also makes a predicted value for y, and the value that it predicts for y is y hat i, shown here. For our choice of a model, f of x i is w times x i plus b. Stated differently, the prediction y hat i is f of w b of x i, where for the model where using f of x i is equal to w x i plus b. So now the question is, how do you find values for w and b so that the prediction y hat i is close to the true target y i for many or maybe all training examples x i y i? To answer that question, let's first take a look at how to measure how well a line fits the training data. To do that, we're going to construct our cost function. The cost function takes the prediction y hat and compares it to the target y by taking y hat minus y. This difference is called the error. We're measuring how far off the prediction is from the target. Next let's compute the square of this error. Also we're going to want to compute this term for different training examples i in the training set. So when measuring the error for example i, we'll compute this squared error term. Finally, we want to measure the error across the entire training set. In particular, let's sum up the squared errors like this. We'll sum from i equals 1, 2, 3, all the way up to m. And remember that m is the number of training examples, which is 47 for this data set. Notice that if we have more training examples, m is larger and your cost function will calculate a bigger number since it's summing over more examples. So to build a cost function that doesn't automatically get bigger as the training set size gets larger, by convention, we will compute the average squared error instead of the total squared error. And we do that by dividing by m like this. Okay, we're nearly there. Just one last thing. By convention, the cost function that machine
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1.11 Machine Learning Overview | Cost function formula --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=ZzeDtSmrRoU
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learning people use actually divides by two times m. The extra division by two is just meant to make some of our later calculations a little bit neater. But the cost function still works whether you include this division by two or not. So this expression right here is the cost function. And we're going to write J of WB to refer to the cost function. This is also called the squared error cost function. And it's called this because you're taking the square of these error terms. In machine learning, different people will use different cost functions for different applications. But the squared error cost function is by far the most commonly used one for linear regression. And for that matter, for all regression problems, where it seems to give good results for many applications. So just as a reminder, the prediction y hat is equal to the output of the model f at x. So we can rewrite the cost function J of WB as one over two m times the sum from i equals one to m of f of x i minus y i, the quantity squared. Eventually, we're going to want to find values of W and B that make the cost function small. But before going there, let's first gain more intuition about what J of WB is really computing. At this point, you might be thinking we've done a whole lot of math to define the cost function. But what exactly is it doing? Let's go on to the next video, where we'll step through one example of what the cost function is really computing. That I hope will help you build intuition about what it means if J of WB is large versus if the cost J is small. Let's go on to the next video.
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1.11 Machine Learning Overview | Cost function formula --[Machine Learning | Andrew Ng]: learning people use actually divides by two times m. The extra division by two is just meant to make some of our later calculations a little bit neater. But the cost function still works whether you include this division by two or not. So this expression right here is the cost function. And we're going to write J of WB to refer to the cost function. This is also called the squared error cost function. And it's called this because you're taking the square of these error terms. In machine learning, different people will use different cost functions for different applications. But the squared error cost function is by far the most commonly used one for linear regression. And for that matter, for all regression problems, where it seems to give good results for many applications. So just as a reminder, the prediction y hat is equal to the output of the model f at x. So we can rewrite the cost function J of WB as one over two m times the sum from i equals one to m of f of x i minus y i, the quantity squared. Eventually, we're going to want to find values of W and B that make the cost function small. But before going there, let's first gain more intuition about what J of WB is really computing. At this point, you might be thinking we've done a whole lot of math to define the cost function. But what exactly is it doing? Let's go on to the next video, where we'll step through one example of what the cost function is really computing. That I hope will help you build intuition about what it means if J of WB is large versus if the cost J is small. Let's go on to the next video.
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1.12 Machine Learning Overview | Cost function intuition --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=33VvmIZof0E
|
We've seen the mathematical definition of the cost function. Now let's build some intuition about what the cost function is really doing. In this video, we'll walk through one example to see how the cost function can be used to find the best parameters for your model. I know this video is a little bit longer than the others, but bear with me, I think it'll be worth it. To recap, here's what we've seen about the cost function so far. You want to fit a straight line to the training data, so you have this model fwb of x is w times x plus b. And here the model's parameters are w and b. Now depending on the values chosen for these parameters, you get different straight lines like this. And you want to find values for w and b so that the straight line fits the training data well. To measure how well a choice of w and b fits the training data, you have a cost function j. And what the cost function j does is it measures the difference between the model's predictions and the actual true values for y. What you see later is that linear regression would try to find values for w and b that make a j of w be as small as possible. In math, we write it like this. We want to minimize j as a function of w and b. So now, in order for us to better visualize the cost function j, let's work with a simplified version of the linear regression model. We're going to use the model fw of x is w times x. You can think of this as taking the original model on the left and getting rid of the parameter b or setting the parameter b equal to zero. So it just goes away from the equation. So f is now just w times x. So you now have just one parameter w and your cost function j looks similar to what it was before, taking the difference and squaring it, except that now f is equal to w times xi and j is now a function of just w. And the goal becomes a little bit different as well because you have just one parameter w, not w and b. So with this simplified model, the goal is to find a value for w that minimizes j of w. To see this visually, what this means is that if b is set to zero, then f defines a line that looks like this. And you see that the line passes through the origin here because when x is zero, well, f of x is zero too. Now using this simplified model, let's see how the cost function changes as you choose different values for the parameter w. In particular, let's look at graphs of the model f of x and the cost function j. I'm going to plot these side by side
| 500
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1.12 Machine Learning Overview | Cost function intuition --[Machine Learning | Andrew Ng]: We've seen the mathematical definition of the cost function. Now let's build some intuition about what the cost function is really doing. In this video, we'll walk through one example to see how the cost function can be used to find the best parameters for your model. I know this video is a little bit longer than the others, but bear with me, I think it'll be worth it. To recap, here's what we've seen about the cost function so far. You want to fit a straight line to the training data, so you have this model fwb of x is w times x plus b. And here the model's parameters are w and b. Now depending on the values chosen for these parameters, you get different straight lines like this. And you want to find values for w and b so that the straight line fits the training data well. To measure how well a choice of w and b fits the training data, you have a cost function j. And what the cost function j does is it measures the difference between the model's predictions and the actual true values for y. What you see later is that linear regression would try to find values for w and b that make a j of w be as small as possible. In math, we write it like this. We want to minimize j as a function of w and b. So now, in order for us to better visualize the cost function j, let's work with a simplified version of the linear regression model. We're going to use the model fw of x is w times x. You can think of this as taking the original model on the left and getting rid of the parameter b or setting the parameter b equal to zero. So it just goes away from the equation. So f is now just w times x. So you now have just one parameter w and your cost function j looks similar to what it was before, taking the difference and squaring it, except that now f is equal to w times xi and j is now a function of just w. And the goal becomes a little bit different as well because you have just one parameter w, not w and b. So with this simplified model, the goal is to find a value for w that minimizes j of w. To see this visually, what this means is that if b is set to zero, then f defines a line that looks like this. And you see that the line passes through the origin here because when x is zero, well, f of x is zero too. Now using this simplified model, let's see how the cost function changes as you choose different values for the parameter w. In particular, let's look at graphs of the model f of x and the cost function j. I'm going to plot these side by side
|
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1.12 Machine Learning Overview | Cost function intuition --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=33VvmIZof0E
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and you'll be able to see how the two are related. First notice that for f subscript w, when the parameter w is fixed, that is, is always a constant value, then f w is only a function of x, which means that the estimated value of y depends on the value of the input x. In contrast, looking to the right, the cost function j is a function of w, where w controls the slope of the line defined by f w, so the cost defined by j depends on a parameter, in this case, the parameter w. So let's go ahead and plot these functions f w of x and j of w side by side so you can see how they are related. We'll start to the model that is the function f w of x on the left. Here the input feature x is on the horizontal axis and the output value y is on the vertical axis. Here's a plot of three points representing the trading set at positions 1 1, 2 2, and 3 3. Let's pick a value for w, say w is 1. So for this choice of w, the function f w looks like this straight line with a slope of 1. Now what you can do next is calculate the cost j when w equals 1. So you may recall that the cost function is defined as follows, it's the squared error cost function. So if you substitute f w of x i with w times x i, the cost function looks like this, where this expression is now w times x i minus y i. So for this value of w, it turns out that the error term inside the cost function, this w times x i minus y i is equal to zero for each of the three data points. Notice for this data set, when x is 1, then y is 1. When w is also 1, then f of x equals 1. So f of x equals y for this first training example, and the difference is zero. Plugging this into the cost function j, you get zero squared. Similarly, when x is 2, then y is 2, and f of x is also 2. So again, f of x equals y for the second training example. In the cost function, the squared error for the second example is also zero squared. Finally, when x is 3, then y is 3, and f of 3 is also 3. In the cost function, the third squared error term is also zero squared. So for all three examples in this training set, f of x i equals y i for each training example i. So f of x i minus y i is zero. So for this particular data set, when w is 1, then the cost j is equal to zero. Now what you can do on the right is plot the cost function j. And notice
| 500
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1.12 Machine Learning Overview | Cost function intuition --[Machine Learning | Andrew Ng]: and you'll be able to see how the two are related. First notice that for f subscript w, when the parameter w is fixed, that is, is always a constant value, then f w is only a function of x, which means that the estimated value of y depends on the value of the input x. In contrast, looking to the right, the cost function j is a function of w, where w controls the slope of the line defined by f w, so the cost defined by j depends on a parameter, in this case, the parameter w. So let's go ahead and plot these functions f w of x and j of w side by side so you can see how they are related. We'll start to the model that is the function f w of x on the left. Here the input feature x is on the horizontal axis and the output value y is on the vertical axis. Here's a plot of three points representing the trading set at positions 1 1, 2 2, and 3 3. Let's pick a value for w, say w is 1. So for this choice of w, the function f w looks like this straight line with a slope of 1. Now what you can do next is calculate the cost j when w equals 1. So you may recall that the cost function is defined as follows, it's the squared error cost function. So if you substitute f w of x i with w times x i, the cost function looks like this, where this expression is now w times x i minus y i. So for this value of w, it turns out that the error term inside the cost function, this w times x i minus y i is equal to zero for each of the three data points. Notice for this data set, when x is 1, then y is 1. When w is also 1, then f of x equals 1. So f of x equals y for this first training example, and the difference is zero. Plugging this into the cost function j, you get zero squared. Similarly, when x is 2, then y is 2, and f of x is also 2. So again, f of x equals y for the second training example. In the cost function, the squared error for the second example is also zero squared. Finally, when x is 3, then y is 3, and f of 3 is also 3. In the cost function, the third squared error term is also zero squared. So for all three examples in this training set, f of x i equals y i for each training example i. So f of x i minus y i is zero. So for this particular data set, when w is 1, then the cost j is equal to zero. Now what you can do on the right is plot the cost function j. And notice
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1.12 Machine Learning Overview | Cost function intuition --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=33VvmIZof0E
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that because the cost function is a function of the parameter w, the horizontal axis is now labeled w and not x. And the vertical axis is now j and not y. So you have j of 1 equals to zero. In other words, when w equals 1, j of w is zero. So let me go ahead and plot that. Now let's look at how f and j change for different values of w. w can take on a range of values, right? So w can take on negative values, w can be zero, and it can take on positive values too. So what if w is equal to 0.5 instead of 1? What would these graphs look like then? Let's go ahead and plot that. So let's set w to be equal to 0.5. And in this case, the function f of x now looks like this, is aligned with a slope equal to 0.5. And let's also compute the cost j when w is 0.5. Recall that the cost function is measuring the squared error or difference between the estimated value that is y hat i, which is f of x i, and the true value that is y i for each example i. So visually you can see that the error or difference is equal to the height of this vertical line here when x is equal to 1, because this little line is the gap between the actual value of y and the value that the function f predicted, which is a bit further down here. So for this first example, when x is 1, f of x is 0.5, so the squared error on the first example is 0.5 minus 1 squared. Remember the cost function will sum over all the training examples in the training set. So let's go on to the second training example. When x is 2, the model is predicting f of x is 1, and the actual value of y is 2. So the error for the second example is equal to the height of this little line segment here, and the squared error is the square of the length of this line segment, so you get 1 minus 2 squared. Let's do the third example. Repeating this process, the error here, also shown by this line segment, is 1.5 minus 3 squared. Next we sum up all of these terms, which turns out to be equal to 3.5. Then we multiply this term by 1 over 2m, where m is the number of training examples. Since there are three training examples, m equals 3, so this is equal to 1 over 2 times 3, where this m here is 3. If we work out the math, this turns out to be 3.5 divided by 6, so the cost j is about 0.58. Let's go ahead and plot that over there on the right. Now let's try one more value for w. How about if w equals 0? What
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1.12 Machine Learning Overview | Cost function intuition --[Machine Learning | Andrew Ng]: that because the cost function is a function of the parameter w, the horizontal axis is now labeled w and not x. And the vertical axis is now j and not y. So you have j of 1 equals to zero. In other words, when w equals 1, j of w is zero. So let me go ahead and plot that. Now let's look at how f and j change for different values of w. w can take on a range of values, right? So w can take on negative values, w can be zero, and it can take on positive values too. So what if w is equal to 0.5 instead of 1? What would these graphs look like then? Let's go ahead and plot that. So let's set w to be equal to 0.5. And in this case, the function f of x now looks like this, is aligned with a slope equal to 0.5. And let's also compute the cost j when w is 0.5. Recall that the cost function is measuring the squared error or difference between the estimated value that is y hat i, which is f of x i, and the true value that is y i for each example i. So visually you can see that the error or difference is equal to the height of this vertical line here when x is equal to 1, because this little line is the gap between the actual value of y and the value that the function f predicted, which is a bit further down here. So for this first example, when x is 1, f of x is 0.5, so the squared error on the first example is 0.5 minus 1 squared. Remember the cost function will sum over all the training examples in the training set. So let's go on to the second training example. When x is 2, the model is predicting f of x is 1, and the actual value of y is 2. So the error for the second example is equal to the height of this little line segment here, and the squared error is the square of the length of this line segment, so you get 1 minus 2 squared. Let's do the third example. Repeating this process, the error here, also shown by this line segment, is 1.5 minus 3 squared. Next we sum up all of these terms, which turns out to be equal to 3.5. Then we multiply this term by 1 over 2m, where m is the number of training examples. Since there are three training examples, m equals 3, so this is equal to 1 over 2 times 3, where this m here is 3. If we work out the math, this turns out to be 3.5 divided by 6, so the cost j is about 0.58. Let's go ahead and plot that over there on the right. Now let's try one more value for w. How about if w equals 0? What
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1.12 Machine Learning Overview | Cost function intuition --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=33VvmIZof0E
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do the graphs for f and j look like when w is equal to 0? It turns out that if w is equal to 0, then f of x is just this horizontal line that is exactly on the x-axis, and so the error for each example is a line that goes from each point down to the horizontal line that represents f of x equals 0. So the cost j when w equals 0 is 1 over 2m times the quantity 1 squared plus 2 squared plus 3 squared, and that's equal to 1 over 6 times 14, which is about 2.33. So let's plot this point where w is 0 and j of 0 is 2.33 over here. And you can keep doing this for other values of w. Since w can be any number, it can also be a negative value, so if w is negative 0.5, then the line f is a downward sloping line like this. It turns out that when w is negative 0.5, then you end up with an even higher cost around 5.25, which is this point up here. And you can continue computing the cost function for different values of w and so on and plot these, right? So it turns out that by computing a range of values, you can slowly trace out what the cost function j looks like. And that's what j is. To recap, each value of parameter w corresponds to a different straight line fit, f of x, on the graph to the left. And for the given training set, that choice for a value of w corresponds to a single point, a single point, on the graph on the right. Because for each value of w, you can calculate a cost j of w. For example, when w equals 1, this corresponds to this straight line fit through the data. And it also corresponds to this point on the graph of j, where w equals 1 and the cost j of 1 equals 0. Whereas when w equals 0.5, this gives you this line, which has a smaller slope. And this line, in combination with the training set, corresponds to this point on the cost function graph at w equals 0.5. So for each value of w, you wind up with a different line and its corresponding cost j of w. And you can use these points to trace out this plot on the right. Given this, how can you choose the value of w that results in the function f fitting the data well? Well, as you can imagine, choosing a value of w that causes j of w to be as small as possible seems like a good bet. J is the cost function that measures how big the squared errors are. So choosing w that minimizes these squared errors, makes them as small as possible, would give us a good model. In this example, if you were to choose the value of
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1.12 Machine Learning Overview | Cost function intuition --[Machine Learning | Andrew Ng]: do the graphs for f and j look like when w is equal to 0? It turns out that if w is equal to 0, then f of x is just this horizontal line that is exactly on the x-axis, and so the error for each example is a line that goes from each point down to the horizontal line that represents f of x equals 0. So the cost j when w equals 0 is 1 over 2m times the quantity 1 squared plus 2 squared plus 3 squared, and that's equal to 1 over 6 times 14, which is about 2.33. So let's plot this point where w is 0 and j of 0 is 2.33 over here. And you can keep doing this for other values of w. Since w can be any number, it can also be a negative value, so if w is negative 0.5, then the line f is a downward sloping line like this. It turns out that when w is negative 0.5, then you end up with an even higher cost around 5.25, which is this point up here. And you can continue computing the cost function for different values of w and so on and plot these, right? So it turns out that by computing a range of values, you can slowly trace out what the cost function j looks like. And that's what j is. To recap, each value of parameter w corresponds to a different straight line fit, f of x, on the graph to the left. And for the given training set, that choice for a value of w corresponds to a single point, a single point, on the graph on the right. Because for each value of w, you can calculate a cost j of w. For example, when w equals 1, this corresponds to this straight line fit through the data. And it also corresponds to this point on the graph of j, where w equals 1 and the cost j of 1 equals 0. Whereas when w equals 0.5, this gives you this line, which has a smaller slope. And this line, in combination with the training set, corresponds to this point on the cost function graph at w equals 0.5. So for each value of w, you wind up with a different line and its corresponding cost j of w. And you can use these points to trace out this plot on the right. Given this, how can you choose the value of w that results in the function f fitting the data well? Well, as you can imagine, choosing a value of w that causes j of w to be as small as possible seems like a good bet. J is the cost function that measures how big the squared errors are. So choosing w that minimizes these squared errors, makes them as small as possible, would give us a good model. In this example, if you were to choose the value of
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1.12 Machine Learning Overview | Cost function intuition --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=33VvmIZof0E
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w that results in the smallest possible value of j of w, you'd end up picking w equals 1. And as you can see, that's actually a pretty good choice. This results in a line that fits the training data very well. So that's how in linear regression, you use the cost function to find the value of w that minimizes j. Or in the more general case, when we had parameters w and b rather than just w, you find the values of w and b that minimize j. So to summarize, you saw plots of both f and j and work through how the two are related. As you vary w or vary w and b, you end up with different straight lines. And when that straight line passes close to data, the cost j is small. So the goal of linear regression is to find the parameters w or w and b that results in the smallest possible value for the cost function j. Now in this video, we worked through our example with a simplified problem using only w. In the next video, let's visualize what the cost function looks like for the full version of linear regression using both w and b. And you see some cool 3D plots. Let's go to the next video.
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1.12 Machine Learning Overview | Cost function intuition --[Machine Learning | Andrew Ng]: w that results in the smallest possible value of j of w, you'd end up picking w equals 1. And as you can see, that's actually a pretty good choice. This results in a line that fits the training data very well. So that's how in linear regression, you use the cost function to find the value of w that minimizes j. Or in the more general case, when we had parameters w and b rather than just w, you find the values of w and b that minimize j. So to summarize, you saw plots of both f and j and work through how the two are related. As you vary w or vary w and b, you end up with different straight lines. And when that straight line passes close to data, the cost j is small. So the goal of linear regression is to find the parameters w or w and b that results in the smallest possible value for the cost function j. Now in this video, we worked through our example with a simplified problem using only w. In the next video, let's visualize what the cost function looks like for the full version of linear regression using both w and b. And you see some cool 3D plots. Let's go to the next video.
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1.13 Machine Learning Overview | Visualizing the cost function --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=75eDNh4A07Q
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In the last video, you saw one visualization of the cost function J of w or J of wb. Let's look at some further, richer visualizations so you can get an even better intuition about what the cost function is doing. Here is what we've seen so far. There's the model, the model's parameters w and b, the cost function J of w and b, as well as the goal of linear regression, which is to minimize the cost function J of w and b over parameters w and b. In the last video, we had temporarily set b to zero in order to simplify the visualizations. But now, let's go back to the original model with both parameters w and b without setting b to be equal to zero. Same as last time, we want to get a visual understanding of the model function f of x, shown here on the left, and how it relates to the cost function J of wb, shown here on the right. Here's a training set of house sizes and prices. Let's see pick one possible function of x, like this one. Here I've set w to 0.06 and b to 50. So f of x is 0.06 times x plus 50. Note that this is not a particularly good model for this training set. It's actually a pretty bad model. It seems to consistently underestimate housing prices. Given these values for w and b, let's look at what the cost function J of w and b may look like. Recall what we saw last time was when you had only w because we temporarily set b to zero to simplify things. Back then, we had come up with a plot of the cost function that looked like this as a function of w only. So when we had only one parameter w, the cost function had this u-shaped curve, shaped a bit like a soup bowl. That sounds delicious. Now, in this housing price example that we have on this slide, we have two parameters, w and b, and so the plot becomes a little more complex. It turns out that the cost function also has a similar shape, like a soup bowl, except in three dimensions instead of two. In fact, depending on your training set, the cost function will look something like this. To me, this looks like a soup bowl, maybe because I'm a little bit hungry. Or maybe to you, it looks like a curved dinner plate or a hammock. Actually, that sounds relaxing too. And there's your coconut drink. Maybe when you're done with this course, you should treat yourself to vacation and relax in a hammock like this. What you see here is a 3D surface plot where the axes are labeled w and b. So as you vary w and b, which are the two parameters of the model, you get different values for the cost function j of w and b. This is a
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1.13 Machine Learning Overview | Visualizing the cost function --[Machine Learning | Andrew Ng]: In the last video, you saw one visualization of the cost function J of w or J of wb. Let's look at some further, richer visualizations so you can get an even better intuition about what the cost function is doing. Here is what we've seen so far. There's the model, the model's parameters w and b, the cost function J of w and b, as well as the goal of linear regression, which is to minimize the cost function J of w and b over parameters w and b. In the last video, we had temporarily set b to zero in order to simplify the visualizations. But now, let's go back to the original model with both parameters w and b without setting b to be equal to zero. Same as last time, we want to get a visual understanding of the model function f of x, shown here on the left, and how it relates to the cost function J of wb, shown here on the right. Here's a training set of house sizes and prices. Let's see pick one possible function of x, like this one. Here I've set w to 0.06 and b to 50. So f of x is 0.06 times x plus 50. Note that this is not a particularly good model for this training set. It's actually a pretty bad model. It seems to consistently underestimate housing prices. Given these values for w and b, let's look at what the cost function J of w and b may look like. Recall what we saw last time was when you had only w because we temporarily set b to zero to simplify things. Back then, we had come up with a plot of the cost function that looked like this as a function of w only. So when we had only one parameter w, the cost function had this u-shaped curve, shaped a bit like a soup bowl. That sounds delicious. Now, in this housing price example that we have on this slide, we have two parameters, w and b, and so the plot becomes a little more complex. It turns out that the cost function also has a similar shape, like a soup bowl, except in three dimensions instead of two. In fact, depending on your training set, the cost function will look something like this. To me, this looks like a soup bowl, maybe because I'm a little bit hungry. Or maybe to you, it looks like a curved dinner plate or a hammock. Actually, that sounds relaxing too. And there's your coconut drink. Maybe when you're done with this course, you should treat yourself to vacation and relax in a hammock like this. What you see here is a 3D surface plot where the axes are labeled w and b. So as you vary w and b, which are the two parameters of the model, you get different values for the cost function j of w and b. This is a
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1.13 Machine Learning Overview | Visualizing the cost function --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=75eDNh4A07Q
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lot like the u-shaped curve you saw in the last video, except instead of having one parameter, w, as input in the j, you now have two parameters, w and b, as inputs into this soup bowl or this hammock-shaped function j. And I just want to point out that any single point on the surface represents a particular choice of w and b. For example, if w was minus 10 and b was minus 15, then the height of the surface above this point is the value of j when w is minus 10 and b is minus 15. Now in order to look even more closely at specific points, there's another way of plotting the cost function j that would be useful for visualization, which is rather than using these 3D surface plots. I'd like to take this exact same function j, so I'm not changing the function j at all, and plot it using something called a contour plot. And if you've ever seen a topographical map showing how high different mountains are, the contours in a topographical map are basically horizontal slices of the landscape of, say, a mountain. This image is of Mount Fuji in Japan. I still remember my family visiting Mount Fuji when I was a teenager. It was a beautiful sight. And if you fly directly above the mountain, that's what this contour map looks like. It shows all the points that are at the same height for different heights. At the bottom of this slide is a 3D surface plot of the cost function j. I know it doesn't look very bow-shaped, but it is actually a bow just very stretched out, which is why it looks like that. In an optional lab that is shortly to follow, you'll be able to see this in 3D and spin around the surface yourself, and it'll look more obviously bow-shaped there. Next, here on the upper right is a contour plot of this exact same cost function as that shown at the bottom. The two axes on this contour plot are B on the vertical axis and W on the horizontal axis. What each of these ovals, also called ellipses, shows is the set of points on the 3D surface which are at the exact same height. In other words, the set of points which have the same value for the cost function j. So, to get a contour plot, you take the 3D surface at the bottom and you use a knife to slice it horizontally. You take horizontal slices of that 3D surface and get all the points that are at the same height. Therefore, each horizontal slice ends up being shown as one of these ellipses, or one of these ovals. So concretely, if you take that point, and that point, and that point, all of these three points have the same value for the cost function j, even though they have different values for W and B. And in
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1.13 Machine Learning Overview | Visualizing the cost function --[Machine Learning | Andrew Ng]: lot like the u-shaped curve you saw in the last video, except instead of having one parameter, w, as input in the j, you now have two parameters, w and b, as inputs into this soup bowl or this hammock-shaped function j. And I just want to point out that any single point on the surface represents a particular choice of w and b. For example, if w was minus 10 and b was minus 15, then the height of the surface above this point is the value of j when w is minus 10 and b is minus 15. Now in order to look even more closely at specific points, there's another way of plotting the cost function j that would be useful for visualization, which is rather than using these 3D surface plots. I'd like to take this exact same function j, so I'm not changing the function j at all, and plot it using something called a contour plot. And if you've ever seen a topographical map showing how high different mountains are, the contours in a topographical map are basically horizontal slices of the landscape of, say, a mountain. This image is of Mount Fuji in Japan. I still remember my family visiting Mount Fuji when I was a teenager. It was a beautiful sight. And if you fly directly above the mountain, that's what this contour map looks like. It shows all the points that are at the same height for different heights. At the bottom of this slide is a 3D surface plot of the cost function j. I know it doesn't look very bow-shaped, but it is actually a bow just very stretched out, which is why it looks like that. In an optional lab that is shortly to follow, you'll be able to see this in 3D and spin around the surface yourself, and it'll look more obviously bow-shaped there. Next, here on the upper right is a contour plot of this exact same cost function as that shown at the bottom. The two axes on this contour plot are B on the vertical axis and W on the horizontal axis. What each of these ovals, also called ellipses, shows is the set of points on the 3D surface which are at the exact same height. In other words, the set of points which have the same value for the cost function j. So, to get a contour plot, you take the 3D surface at the bottom and you use a knife to slice it horizontally. You take horizontal slices of that 3D surface and get all the points that are at the same height. Therefore, each horizontal slice ends up being shown as one of these ellipses, or one of these ovals. So concretely, if you take that point, and that point, and that point, all of these three points have the same value for the cost function j, even though they have different values for W and B. And in
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1.13 Machine Learning Overview | Visualizing the cost function --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=75eDNh4A07Q
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the figure on the upper left, you see also that these three points correspond to different functions f, all three of which are actually pretty bad for predicting housing prices in this case. Now, the bottom of the bow, where the cost function j is at a minimum, is this point right here, at the center of this concentric ovals. If you haven't seen contour plots much before, I'd like you to imagine, if you will, that you are flying high up above the bow in an airplane or in a rocket ship, and you're looking straight down at it. So that is as if you set your computer monitor flat on your desk facing up, and the bow shape is coming directly out of your screen rising above your desk. Imagine that the bow shape grows out of your computer screen lying flat, like that, so that each of these ovals have the same height above your screen, and the minimum of the bow is right down there in the center of the smallest oval. So it turns out that the contour plots are a convenient way to visualize the 3D cost function j, but in a way that's plotted in just 2D. In this video, you saw how the 3D bow-shaped surface plot can also be visualized as a contour plot. Using this visualization tool, in the next video, let's visualize some specific choices of WMB in the linear regression model so that we can see how these different choices affect the straight line you're fitting to the data. Let's go on to the next video.
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1.13 Machine Learning Overview | Visualizing the cost function --[Machine Learning | Andrew Ng]: the figure on the upper left, you see also that these three points correspond to different functions f, all three of which are actually pretty bad for predicting housing prices in this case. Now, the bottom of the bow, where the cost function j is at a minimum, is this point right here, at the center of this concentric ovals. If you haven't seen contour plots much before, I'd like you to imagine, if you will, that you are flying high up above the bow in an airplane or in a rocket ship, and you're looking straight down at it. So that is as if you set your computer monitor flat on your desk facing up, and the bow shape is coming directly out of your screen rising above your desk. Imagine that the bow shape grows out of your computer screen lying flat, like that, so that each of these ovals have the same height above your screen, and the minimum of the bow is right down there in the center of the smallest oval. So it turns out that the contour plots are a convenient way to visualize the 3D cost function j, but in a way that's plotted in just 2D. In this video, you saw how the 3D bow-shaped surface plot can also be visualized as a contour plot. Using this visualization tool, in the next video, let's visualize some specific choices of WMB in the linear regression model so that we can see how these different choices affect the straight line you're fitting to the data. Let's go on to the next video.
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1.14 Machine Learning Overview | Visualizing examples --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=GY0KyF3h8hA
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Let's look at some more visualizations of w and b. Here's one example. Over here, you have a particular point on the graph J. For this point, w equals about negative 0.15 and b equals about 800. So this point corresponds to one pair of values for w and b that use a particular cost J. And in fact, this particular pair of values for w and b corresponds to this function f of x, which is this line that you can see on the left. This line intersects the vertical axis at 800 because b equals 800 and the slope of the line is negative 0.15 because w equals negative 0.15. Now if you look at the data points in the training set, you may notice that this line is not a good fit to the data. For this function f of x, with these values of w and b, many of the predictions for the value of y are quite far from the actual target value of y that is in the training data. Since this line is not a good fit, if you look at the graph of J, the cost of this line is out here, which is pretty far from the minimum. This is a pretty high cost because this choice of w and b is just not that good a fit to the training set. Now let's look at another example with a different choice of w and b. Now here is another function that is still not a great fit for the data, but maybe slightly less bad. So this point here represents the cost for this particular pair of w and b that creates that line. The value of w is equal to 0 and the value of b is about 360. This pair of parameters corresponds to this function, which is a flat line because f of x equals 0 times x plus 360. I hope that makes sense. Let's look at yet another example. Here's one more choice for w and b, and with these values, you end up with this line f of x. Again, not a great fit to the data. It is actually further away from the minimum compared to the previous example. And remember that the minimum is at the center of that smallest ellipse. This example, if you look at f of x on the left, this looks like a pretty good fit to the training set. You can see on the right, this point representing the cost is very close to the center of the small ellipse. It's not quite exactly the minimum, but it's pretty close. For this value of w and b, you get this line f of x. You can see that if you measure the vertical distances between the data points and the predicted values on the straight line, you get the error for each data point. The sum of squared errors for all of these data points is pretty
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1.14 Machine Learning Overview | Visualizing examples --[Machine Learning | Andrew Ng]: Let's look at some more visualizations of w and b. Here's one example. Over here, you have a particular point on the graph J. For this point, w equals about negative 0.15 and b equals about 800. So this point corresponds to one pair of values for w and b that use a particular cost J. And in fact, this particular pair of values for w and b corresponds to this function f of x, which is this line that you can see on the left. This line intersects the vertical axis at 800 because b equals 800 and the slope of the line is negative 0.15 because w equals negative 0.15. Now if you look at the data points in the training set, you may notice that this line is not a good fit to the data. For this function f of x, with these values of w and b, many of the predictions for the value of y are quite far from the actual target value of y that is in the training data. Since this line is not a good fit, if you look at the graph of J, the cost of this line is out here, which is pretty far from the minimum. This is a pretty high cost because this choice of w and b is just not that good a fit to the training set. Now let's look at another example with a different choice of w and b. Now here is another function that is still not a great fit for the data, but maybe slightly less bad. So this point here represents the cost for this particular pair of w and b that creates that line. The value of w is equal to 0 and the value of b is about 360. This pair of parameters corresponds to this function, which is a flat line because f of x equals 0 times x plus 360. I hope that makes sense. Let's look at yet another example. Here's one more choice for w and b, and with these values, you end up with this line f of x. Again, not a great fit to the data. It is actually further away from the minimum compared to the previous example. And remember that the minimum is at the center of that smallest ellipse. This example, if you look at f of x on the left, this looks like a pretty good fit to the training set. You can see on the right, this point representing the cost is very close to the center of the small ellipse. It's not quite exactly the minimum, but it's pretty close. For this value of w and b, you get this line f of x. You can see that if you measure the vertical distances between the data points and the predicted values on the straight line, you get the error for each data point. The sum of squared errors for all of these data points is pretty
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1.14 Machine Learning Overview | Visualizing examples --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=GY0KyF3h8hA
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close to the minimum possible sum of squared errors among all possible straight line fits. I hope that by looking at these figures, you can get a better sense of how different choices of the parameters affect the line f of x and how this corresponds to different values for the cost j. And hopefully, you can see how the better fit lines correspond to points on the graph of j that are closer to the minimum possible cost for this cost function j of w and b. In the optional lab that follows this video, you get to run some code. And remember, all of the code is given, so you just need to hit shift enter to run it and take a look at it. And the lab will show you how the cost function is implemented in code. And given a small training set and different choices for the parameters, you'll be able to see how the cost varies depending on how well the model fits the data. In the optional lab, you also can play with an interactive contour plot. Check this out. You can use your mouse cursor to click anywhere on the contour plot, and you're going to see the straight line defined by the values you chose for the parameters w and b. You see a dot appear also on the 3D surface plot showing the cost. Finally, the optional lab also has a 3D surface plot that you can manually rotate and spin around using your mouse cursor to take a better look at what the cost function looks like. I hope you enjoyed playing with the optional lab. Now, in linear regression, rather than having to manually try to read the contour plot for the best value for w and b, which isn't really a good procedure and also won't work once we get to more complex machine learning models, what you really want is an efficient algorithm that you can write in code for automatically finding the values of parameters w and b that gives you the best fit line that minimizes the cost function j. There is an algorithm for doing this called gradient descent. This algorithm is one of the most important algorithms in machine learning. Gradient descent and variations on gradient descent are used to train not just linear regression, but some of the biggest and most complex models in all of AI. So, let's go to the next video to dive into this really important algorithm called gradient descent.
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1.14 Machine Learning Overview | Visualizing examples --[Machine Learning | Andrew Ng]: close to the minimum possible sum of squared errors among all possible straight line fits. I hope that by looking at these figures, you can get a better sense of how different choices of the parameters affect the line f of x and how this corresponds to different values for the cost j. And hopefully, you can see how the better fit lines correspond to points on the graph of j that are closer to the minimum possible cost for this cost function j of w and b. In the optional lab that follows this video, you get to run some code. And remember, all of the code is given, so you just need to hit shift enter to run it and take a look at it. And the lab will show you how the cost function is implemented in code. And given a small training set and different choices for the parameters, you'll be able to see how the cost varies depending on how well the model fits the data. In the optional lab, you also can play with an interactive contour plot. Check this out. You can use your mouse cursor to click anywhere on the contour plot, and you're going to see the straight line defined by the values you chose for the parameters w and b. You see a dot appear also on the 3D surface plot showing the cost. Finally, the optional lab also has a 3D surface plot that you can manually rotate and spin around using your mouse cursor to take a better look at what the cost function looks like. I hope you enjoyed playing with the optional lab. Now, in linear regression, rather than having to manually try to read the contour plot for the best value for w and b, which isn't really a good procedure and also won't work once we get to more complex machine learning models, what you really want is an efficient algorithm that you can write in code for automatically finding the values of parameters w and b that gives you the best fit line that minimizes the cost function j. There is an algorithm for doing this called gradient descent. This algorithm is one of the most important algorithms in machine learning. Gradient descent and variations on gradient descent are used to train not just linear regression, but some of the biggest and most complex models in all of AI. So, let's go to the next video to dive into this really important algorithm called gradient descent.
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1.15 Machine Learning Overview | Gradient descent --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=6W3tzcOnWfQ
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Welcome back. In the last video we saw visualizations of the cost function j and how you can try different choices of the parameters w and b and see what cost value that gets you. It would be nice if we had a more systematic way to find the values of w and b that result in the smallest possible cost j of w, b. It turns out there's an algorithm called gradient descent that you can use to do that. Gradient descent is used all over the place in machine learning, not just for linear regression, but for training, for example, some of the most advanced neural network models, also called deep learning models. Deep learning models are something you learn about in the second course. So learning this tool of gradient descent will set you up with one of the most important building blocks in machine learning. Here's an overview of what we'll do with gradient descent. You have the cost function j of w, b right here that you want to minimize. In the example we've seen so far, this is a cost function for linear regression, but it turns out that gradient descent is an algorithm that you can use to try to minimize any function, not just a cost function for linear regression. Just to make this discussion on gradient descent more general, it turns out that gradient descent applies to more general functions, including other cost functions that work with models that have more than two parameters. So for instance, if you have a cost function j as a function of w1, w2, up to wn and b, your objective is to minimize j over the parameters w1 to wn and b. In other words, you want to pick values for w1 through wn and b that gives you the smallest possible value of j. It turns out that gradient descent is an algorithm that you can apply to try to minimize this cost function j as well. What you're going to do is just start off with some initial guesses for w and b. In linear regression, it won't matter too much what the initial values are. So a common choice is to set them both to zero. So for example, you can set w to zero and b to zero as the initial guess. With the gradient descent algorithm, what you're going to do is you'll keep on changing the parameters w and b a bit every time to try to reduce the cost j of wb until hopefully j settles at or near a minimum. One thing I should note is that for some functions j that may not be a bow shape or a hammer shape, it is possible for there to be more than one possible minimum. Let's take a look at an example of a more complex surface plot j to see what gradient descent is doing. This function is not a squared error cost function. For linear
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1.15 Machine Learning Overview | Gradient descent --[Machine Learning | Andrew Ng]: Welcome back. In the last video we saw visualizations of the cost function j and how you can try different choices of the parameters w and b and see what cost value that gets you. It would be nice if we had a more systematic way to find the values of w and b that result in the smallest possible cost j of w, b. It turns out there's an algorithm called gradient descent that you can use to do that. Gradient descent is used all over the place in machine learning, not just for linear regression, but for training, for example, some of the most advanced neural network models, also called deep learning models. Deep learning models are something you learn about in the second course. So learning this tool of gradient descent will set you up with one of the most important building blocks in machine learning. Here's an overview of what we'll do with gradient descent. You have the cost function j of w, b right here that you want to minimize. In the example we've seen so far, this is a cost function for linear regression, but it turns out that gradient descent is an algorithm that you can use to try to minimize any function, not just a cost function for linear regression. Just to make this discussion on gradient descent more general, it turns out that gradient descent applies to more general functions, including other cost functions that work with models that have more than two parameters. So for instance, if you have a cost function j as a function of w1, w2, up to wn and b, your objective is to minimize j over the parameters w1 to wn and b. In other words, you want to pick values for w1 through wn and b that gives you the smallest possible value of j. It turns out that gradient descent is an algorithm that you can apply to try to minimize this cost function j as well. What you're going to do is just start off with some initial guesses for w and b. In linear regression, it won't matter too much what the initial values are. So a common choice is to set them both to zero. So for example, you can set w to zero and b to zero as the initial guess. With the gradient descent algorithm, what you're going to do is you'll keep on changing the parameters w and b a bit every time to try to reduce the cost j of wb until hopefully j settles at or near a minimum. One thing I should note is that for some functions j that may not be a bow shape or a hammer shape, it is possible for there to be more than one possible minimum. Let's take a look at an example of a more complex surface plot j to see what gradient descent is doing. This function is not a squared error cost function. For linear
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1.15 Machine Learning Overview | Gradient descent --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=6W3tzcOnWfQ
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regression with the squared error cost function, you always end up with a bow shape or a hammer shape. But this is a type of cost function you might get if you're training a neural network model. Notice the axes. That is w and b on the bottom axis. For different values of w and b, you get different points on the surface j of wb, where the height of the surface at some point is the value of the cost function. Now, let's imagine that this surface plot is actually a view of a slightly hilly outdoor park or a golf course, where the high points are hills and the low points are valleys like so. And I'd like you to imagine, if you will, that you're physically standing at this point on the hill. And if it helps you to relax, imagine that this lots of really nice green grass and butterflies and flowers is a really nice hill. And your goal is to start up here and get to the bottom of one of these valleys as efficiently as possible. So what the gradient descent algorithm does is you're going to spin around 360 degrees and look around and ask yourself, if I were to take a tiny little baby step in one direction, and I want to go downhill as quickly as possible toward one of these valleys, what direction do I choose to take that baby step? Well, if you want to walk down this hill as efficiently as possible, it turns out that if you're standing at this point in the hill and you look around, you may notice that the best direction to take your next step downhill is roughly that direction. Mathematically, this is the direction of steepest descent. And it means that when you take a tiny baby little step, this takes you downhill faster than a tiny little baby step you could have taken in any other direction. So after taking this first step, you're now at this point on the hill over here. Now, let's repeat the process. Standing at this new point, you're going to again spin around 360 degrees and ask yourself, in what direction would I take the next little baby step in order to move downhill? And if you do that and take another step, you end up moving a bit in that direction and you can keep going. From this new point, you can again look around and decide what direction would take you downhill most quickly. Take another step, another step, and so on until you find yourself at the bottom of this valley at this local minimum right here. What you just did was go through multiple steps of gradient descent. It turns out gradient descent has an interesting property. Remember that you can choose a starting point at the surface by choosing starting values for the parameters w and b. When you perform gradient descent a moment ago, you had started
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1.15 Machine Learning Overview | Gradient descent --[Machine Learning | Andrew Ng]: regression with the squared error cost function, you always end up with a bow shape or a hammer shape. But this is a type of cost function you might get if you're training a neural network model. Notice the axes. That is w and b on the bottom axis. For different values of w and b, you get different points on the surface j of wb, where the height of the surface at some point is the value of the cost function. Now, let's imagine that this surface plot is actually a view of a slightly hilly outdoor park or a golf course, where the high points are hills and the low points are valleys like so. And I'd like you to imagine, if you will, that you're physically standing at this point on the hill. And if it helps you to relax, imagine that this lots of really nice green grass and butterflies and flowers is a really nice hill. And your goal is to start up here and get to the bottom of one of these valleys as efficiently as possible. So what the gradient descent algorithm does is you're going to spin around 360 degrees and look around and ask yourself, if I were to take a tiny little baby step in one direction, and I want to go downhill as quickly as possible toward one of these valleys, what direction do I choose to take that baby step? Well, if you want to walk down this hill as efficiently as possible, it turns out that if you're standing at this point in the hill and you look around, you may notice that the best direction to take your next step downhill is roughly that direction. Mathematically, this is the direction of steepest descent. And it means that when you take a tiny baby little step, this takes you downhill faster than a tiny little baby step you could have taken in any other direction. So after taking this first step, you're now at this point on the hill over here. Now, let's repeat the process. Standing at this new point, you're going to again spin around 360 degrees and ask yourself, in what direction would I take the next little baby step in order to move downhill? And if you do that and take another step, you end up moving a bit in that direction and you can keep going. From this new point, you can again look around and decide what direction would take you downhill most quickly. Take another step, another step, and so on until you find yourself at the bottom of this valley at this local minimum right here. What you just did was go through multiple steps of gradient descent. It turns out gradient descent has an interesting property. Remember that you can choose a starting point at the surface by choosing starting values for the parameters w and b. When you perform gradient descent a moment ago, you had started
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1.15 Machine Learning Overview | Gradient descent --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=6W3tzcOnWfQ
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at this point over here, right? Now, imagine if you try gradient descent again, but this time you choose a different starting point by choosing parameters that place your starting point just a couple steps to the right over here. If you then repeat the gradient descent process, which means you look around, take a little step in the direction of the steepest descent, so you end up here. Then you again look around, take another step and so on. And if you were to run gradient descent the second time, starting just a couple steps to the right of where we did it the first time, then you will end up in a totally different valley, this different minimum over here on the right. The bottoms of both the first and the second valleys are called local minima, because if you start going down the first valley, gradient descent won't lead you to the second valley. And the same is true if you started going down the second valley, you will stay in that second minimum and not find your way into the first local minimum. So this is an interesting property of the gradient descent algorithm, and you'll see more about this later. So in this video, you saw how gradient descent helps you go downhill. In the next video, let's look at the mathematical expressions that you can implement to make gradient descent work. Let's go on to the next video.
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1.15 Machine Learning Overview | Gradient descent --[Machine Learning | Andrew Ng]: at this point over here, right? Now, imagine if you try gradient descent again, but this time you choose a different starting point by choosing parameters that place your starting point just a couple steps to the right over here. If you then repeat the gradient descent process, which means you look around, take a little step in the direction of the steepest descent, so you end up here. Then you again look around, take another step and so on. And if you were to run gradient descent the second time, starting just a couple steps to the right of where we did it the first time, then you will end up in a totally different valley, this different minimum over here on the right. The bottoms of both the first and the second valleys are called local minima, because if you start going down the first valley, gradient descent won't lead you to the second valley. And the same is true if you started going down the second valley, you will stay in that second minimum and not find your way into the first local minimum. So this is an interesting property of the gradient descent algorithm, and you'll see more about this later. So in this video, you saw how gradient descent helps you go downhill. In the next video, let's look at the mathematical expressions that you can implement to make gradient descent work. Let's go on to the next video.
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1.16 Machine Learning Overview | Implementing gradient descent --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=Dz4JZzgn-hg
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Let's take a look at how you can actually implement the gradient descent algorithm. Let me write down the gradient descent algorithm. Here it is. On each step, w, the parameter, is updated to the old value of w minus alpha times this term d over dw of the cost function j of wb. So what this expression is saying is update your parameter w by taking the current value of w and adjusting it a small amount, which is this expression on the right, minus alpha times this term over here. So if you feel like there's a lot going on in this equation, it's okay. Don't worry about it. We'll unpack it together. First, this equal notation here. Notice I said we're assigning w a value using this equal sign. So in this context, this equal sign is the assignment operator. Specifically, in this context, if you write code that says a equals c, it means take the value of c and store it in your computer in the variable a. If you write a equals a plus one, it means set the value of a to be equal to a plus one or increment the value of a by one. So the assignment operator in coding is different than truth assertions in mathematics, where if I write a equals c, I'm asserting, that is, I am claiming that the values of a and c are equal to each other. And hopefully I will never write a truth assertion a equals a plus one because, you know, that just can't possibly be true. So in Python and in other programming languages, truth assertions are sometimes written as equals equals. So you may see code that says a equals equals c if you're testing whether a is equal to c. But in math notation, as we conventionally use it, like in these videos, the equal sign can be used for either assignments or for truth assertion. And so I try to make sure it's clear when I write an equal sign, whether we're assigning a value to a variable or whether we're asserting the truth of the equality of two values. Now let's dive more deeply into what the symbols in this equation means. The symbol here is the Greek alphabet alpha, and in this equation, alpha is also called the learning rate. The learning rate is usually a small positive number between zero and one, and it might be say 0.01. What alpha does is it basically controls how big of a step you take downhill. So if alpha is very large, then that corresponds to a very aggressive gradient descent procedure where you're trying to take huge steps downhill. And if alpha is very small, then you'd be taking small baby steps downhill. We'll come back later to delve more deeply into how to choose a good learning rate alpha. And finally, this term here, that's the derivative term of the cost function J. Let's not worry
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1.16 Machine Learning Overview | Implementing gradient descent --[Machine Learning | Andrew Ng]: Let's take a look at how you can actually implement the gradient descent algorithm. Let me write down the gradient descent algorithm. Here it is. On each step, w, the parameter, is updated to the old value of w minus alpha times this term d over dw of the cost function j of wb. So what this expression is saying is update your parameter w by taking the current value of w and adjusting it a small amount, which is this expression on the right, minus alpha times this term over here. So if you feel like there's a lot going on in this equation, it's okay. Don't worry about it. We'll unpack it together. First, this equal notation here. Notice I said we're assigning w a value using this equal sign. So in this context, this equal sign is the assignment operator. Specifically, in this context, if you write code that says a equals c, it means take the value of c and store it in your computer in the variable a. If you write a equals a plus one, it means set the value of a to be equal to a plus one or increment the value of a by one. So the assignment operator in coding is different than truth assertions in mathematics, where if I write a equals c, I'm asserting, that is, I am claiming that the values of a and c are equal to each other. And hopefully I will never write a truth assertion a equals a plus one because, you know, that just can't possibly be true. So in Python and in other programming languages, truth assertions are sometimes written as equals equals. So you may see code that says a equals equals c if you're testing whether a is equal to c. But in math notation, as we conventionally use it, like in these videos, the equal sign can be used for either assignments or for truth assertion. And so I try to make sure it's clear when I write an equal sign, whether we're assigning a value to a variable or whether we're asserting the truth of the equality of two values. Now let's dive more deeply into what the symbols in this equation means. The symbol here is the Greek alphabet alpha, and in this equation, alpha is also called the learning rate. The learning rate is usually a small positive number between zero and one, and it might be say 0.01. What alpha does is it basically controls how big of a step you take downhill. So if alpha is very large, then that corresponds to a very aggressive gradient descent procedure where you're trying to take huge steps downhill. And if alpha is very small, then you'd be taking small baby steps downhill. We'll come back later to delve more deeply into how to choose a good learning rate alpha. And finally, this term here, that's the derivative term of the cost function J. Let's not worry
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1.16 Machine Learning Overview | Implementing gradient descent --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=Dz4JZzgn-hg
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about the details of this derivative right now, but later on you get to see more about the derivative term. But for now, you can think of this derivative term that I drew a magenta box around as telling you in which direction you want to take your baby step. And in combination with the learning rate alpha, it also determines the size of the steps you want to take downhill. Now I do want to mention that derivatives come from calculus, and even if you aren't familiar with calculus, don't worry about it. Even without knowing any calculus, you'll be able to figure out all you need to know about this derivative term in this video and the next. One more thing. Remember your model has two parameters, not just W, but also B. So you also have an assignment operation to update the parameter B that looks very similar. B is assigned the O value of B minus the learning rate alpha times this slightly different derivative term D over dB of J of WB. So remember in the graph of the surface plot where you're taking baby steps until you get to the bottom of the valley? Well, for the gradient descent algorithm, you're going to repeat these two update steps until the algorithm converges. And by converges, I mean that you reach the point at a local minimum where the parameters W and B no longer change much with each additional step that you take. Now there's one more subtle detail about how to correctly implement gradient descent. You're going to update two parameters W and B, right? So this update takes place for both parameters W and B. One important detail is that for gradient descent, you want to simultaneously update W and B, meaning you want to update both parameters at the same time. What I mean by that is that in this expression, you're going to update W from the old W to a new W, and you're also updating B from its old value to a new value B. And the way to implement this is to compute the right side, computing this thing for W and B and simultaneously at the same time, update W and B to the new values. So let's take a look at what this means. Here's the correct way to implement gradient descent, which does a simultaneous update. This sets a variable tempW equal to that expression, which is W minus that term here. Let's also set another variable tempB to that, which is B minus that term. So you compute both right-hand sides, both updates, and store them into variables tempW and tempB. Then you copy the value of tempW into W, and you also copy the value of tempB into B. Now one thing you may notice is that this value of W is from before W gets updated. Here notice that the pre-update W is what goes into the derivative term over here,
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1.16 Machine Learning Overview | Implementing gradient descent --[Machine Learning | Andrew Ng]: about the details of this derivative right now, but later on you get to see more about the derivative term. But for now, you can think of this derivative term that I drew a magenta box around as telling you in which direction you want to take your baby step. And in combination with the learning rate alpha, it also determines the size of the steps you want to take downhill. Now I do want to mention that derivatives come from calculus, and even if you aren't familiar with calculus, don't worry about it. Even without knowing any calculus, you'll be able to figure out all you need to know about this derivative term in this video and the next. One more thing. Remember your model has two parameters, not just W, but also B. So you also have an assignment operation to update the parameter B that looks very similar. B is assigned the O value of B minus the learning rate alpha times this slightly different derivative term D over dB of J of WB. So remember in the graph of the surface plot where you're taking baby steps until you get to the bottom of the valley? Well, for the gradient descent algorithm, you're going to repeat these two update steps until the algorithm converges. And by converges, I mean that you reach the point at a local minimum where the parameters W and B no longer change much with each additional step that you take. Now there's one more subtle detail about how to correctly implement gradient descent. You're going to update two parameters W and B, right? So this update takes place for both parameters W and B. One important detail is that for gradient descent, you want to simultaneously update W and B, meaning you want to update both parameters at the same time. What I mean by that is that in this expression, you're going to update W from the old W to a new W, and you're also updating B from its old value to a new value B. And the way to implement this is to compute the right side, computing this thing for W and B and simultaneously at the same time, update W and B to the new values. So let's take a look at what this means. Here's the correct way to implement gradient descent, which does a simultaneous update. This sets a variable tempW equal to that expression, which is W minus that term here. Let's also set another variable tempB to that, which is B minus that term. So you compute both right-hand sides, both updates, and store them into variables tempW and tempB. Then you copy the value of tempW into W, and you also copy the value of tempB into B. Now one thing you may notice is that this value of W is from before W gets updated. Here notice that the pre-update W is what goes into the derivative term over here,
|
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1.16 Machine Learning Overview | Implementing gradient descent --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=Dz4JZzgn-hg
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okay? In contrast, here's an incorrect implementation of gradient descent that does not do a simultaneous update. In this incorrect implementation, we compute tempW same as before, so far that's okay. And now here's where things start to differ. We then update W with the value in tempW before calculating the new value for the other parameter B. Next, we calculate tempB as B minus that term here. And finally, we update B with the value in tempB. The difference between the right-hand side and the left-hand side implementations is that if you look over here, this W has already been updated to this new value, and it's this updated W that actually goes into the cost function J of WB. It means that this term here on the right is not the same as this term over here that you see on the left. And that also means this tempB term on the right is not quite the same as the tempB term on the left, and thus this updated value for B on the right is not the same as this updated value for variable B on the left. The way that gradient descent is implemented in code, it actually turns out to be more natural to implement it the correct way with simultaneous updates. When you hear someone talk about gradient descent, they always mean the gradient descent where you perform a simultaneous update of the parameters. If however, you were to implement non-simultaneous updates, it turns out it will probably work more or less anyway, but doing it this way isn't really the correct way to implement it, it's actually some other algorithm with different properties. So I would advise you to just stick to the correct simultaneous update and not use this incorrect version on the right. So that's gradient descent. In the next video, we'll go into details of the derivative term, which you saw in this video, but that we didn't really talk about in detail. Calculus are part of calculus, and again, if you're not familiar with calculus, don't worry about it. You won't need to know calculus at all in order to complete this course or this specialization, and you'll have all the information you need in order to implement gradient descent. Coming up in the next video, we'll go over derivatives together, and you come away with the intuition and knowledge you need to be able to implement and apply gradient descent yourself. I think that'll be an exciting thing for you to know how to implement. So let's go on to the next video to see how to do that.
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1.16 Machine Learning Overview | Implementing gradient descent --[Machine Learning | Andrew Ng]: okay? In contrast, here's an incorrect implementation of gradient descent that does not do a simultaneous update. In this incorrect implementation, we compute tempW same as before, so far that's okay. And now here's where things start to differ. We then update W with the value in tempW before calculating the new value for the other parameter B. Next, we calculate tempB as B minus that term here. And finally, we update B with the value in tempB. The difference between the right-hand side and the left-hand side implementations is that if you look over here, this W has already been updated to this new value, and it's this updated W that actually goes into the cost function J of WB. It means that this term here on the right is not the same as this term over here that you see on the left. And that also means this tempB term on the right is not quite the same as the tempB term on the left, and thus this updated value for B on the right is not the same as this updated value for variable B on the left. The way that gradient descent is implemented in code, it actually turns out to be more natural to implement it the correct way with simultaneous updates. When you hear someone talk about gradient descent, they always mean the gradient descent where you perform a simultaneous update of the parameters. If however, you were to implement non-simultaneous updates, it turns out it will probably work more or less anyway, but doing it this way isn't really the correct way to implement it, it's actually some other algorithm with different properties. So I would advise you to just stick to the correct simultaneous update and not use this incorrect version on the right. So that's gradient descent. In the next video, we'll go into details of the derivative term, which you saw in this video, but that we didn't really talk about in detail. Calculus are part of calculus, and again, if you're not familiar with calculus, don't worry about it. You won't need to know calculus at all in order to complete this course or this specialization, and you'll have all the information you need in order to implement gradient descent. Coming up in the next video, we'll go over derivatives together, and you come away with the intuition and knowledge you need to be able to implement and apply gradient descent yourself. I think that'll be an exciting thing for you to know how to implement. So let's go on to the next video to see how to do that.
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1.17 Machine Learning Overview | Gradient descent intuition --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=rGvoO8U2Ozc
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Now, let's dive more deeply into gradient descent to gain better intuition about what it's doing and why it might make sense. Here's the gradient descent algorithm that you saw in the previous video. And as a reminder, this variable, this Greek symbol alpha is the learning rate. And the learning rate controls how big of a step you take when updating the model's parameters w and b. So this term here, this d over dw, this is a derivative term. And by convention in math, this d is written with this funny font here. And in case anyone watching this has a PhD in math or is an expert in multivariate calculus, they may be wondering, that's not the derivative, that's the partial derivative. And yes, they'd be right. But for the purposes of implementing a machine learning algorithm, I'm just going to call it derivative, and don't worry about these little distinctions. So what we're going to focus on now is get more intuition about what this learning rate and what this derivative are doing, and why when multiplied together like this, it results in updates to parameters w and b that make sense. In order to do this, let's use a slightly simpler example, where we work on minimizing just one parameter. So let's say that you have a cost function j of just one parameter w, where w is a number. This means that gradient descent now looks like this. w is updated to w minus the learning rate alpha times d over dw of j of w. And you're trying to minimize the cost by adjusting the parameter w. So this is like our previous example, where we had temporarily set b equal to zero. With one parameter w instead of two, you can look at two dimensional graphs of the cost function j instead of three dimensional graphs. Let's look at what gradient descent does on this function j of w. Here on the horizontal axis is parameter w, and on the vertical axis is the cost j of w. Now, let's initialize gradient descent with some starting value for w. Let's initialize it at this location. So imagine that you start off at this point right here on the function j. What gradient descent will do is it will update w to be w minus learning rate alpha times d over dw of j of w. Let's look at what this derivative term here means. A way to think about the derivative at this point on the line is to draw a tangent line, which is a straight line that touches this curve at that point. In math, the slope of this line is the derivative of the function j at this point. And to get the slope, you can draw a little triangle like this. And if you compute the height divided by the width of this triangle, that is the slope. So for example, the slope might be, you know, 2
| 500
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1.17 Machine Learning Overview | Gradient descent intuition --[Machine Learning | Andrew Ng]: Now, let's dive more deeply into gradient descent to gain better intuition about what it's doing and why it might make sense. Here's the gradient descent algorithm that you saw in the previous video. And as a reminder, this variable, this Greek symbol alpha is the learning rate. And the learning rate controls how big of a step you take when updating the model's parameters w and b. So this term here, this d over dw, this is a derivative term. And by convention in math, this d is written with this funny font here. And in case anyone watching this has a PhD in math or is an expert in multivariate calculus, they may be wondering, that's not the derivative, that's the partial derivative. And yes, they'd be right. But for the purposes of implementing a machine learning algorithm, I'm just going to call it derivative, and don't worry about these little distinctions. So what we're going to focus on now is get more intuition about what this learning rate and what this derivative are doing, and why when multiplied together like this, it results in updates to parameters w and b that make sense. In order to do this, let's use a slightly simpler example, where we work on minimizing just one parameter. So let's say that you have a cost function j of just one parameter w, where w is a number. This means that gradient descent now looks like this. w is updated to w minus the learning rate alpha times d over dw of j of w. And you're trying to minimize the cost by adjusting the parameter w. So this is like our previous example, where we had temporarily set b equal to zero. With one parameter w instead of two, you can look at two dimensional graphs of the cost function j instead of three dimensional graphs. Let's look at what gradient descent does on this function j of w. Here on the horizontal axis is parameter w, and on the vertical axis is the cost j of w. Now, let's initialize gradient descent with some starting value for w. Let's initialize it at this location. So imagine that you start off at this point right here on the function j. What gradient descent will do is it will update w to be w minus learning rate alpha times d over dw of j of w. Let's look at what this derivative term here means. A way to think about the derivative at this point on the line is to draw a tangent line, which is a straight line that touches this curve at that point. In math, the slope of this line is the derivative of the function j at this point. And to get the slope, you can draw a little triangle like this. And if you compute the height divided by the width of this triangle, that is the slope. So for example, the slope might be, you know, 2
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1.17 Machine Learning Overview | Gradient descent intuition --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=rGvoO8U2Ozc
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over 1, for instance. And when the tangent line is pointing up into the right, the slope is positive, which means that this derivative is a positive number, so it's greater than zero. And so the updated w is going to be w minus the learning rate times some positive number. The learning rate is always a positive number. So if you take w minus a positive number, you end up with a new value for w that is smaller. So on the graph, we are moving to the left, we are decreasing the value of w. And you may notice that this is the right thing to do if your goal is to decrease the cost j, because when we move toward the left on this curve, the cost j decreases and you're getting closer to the minimum for j, which is over here. So so far, gradient descent seems to be doing the right thing. Now, let's look at another example. Let's take the same function j of w as above. And now let's say that you initialize gradient descent at a different location, say by choosing a starting value for w that's over here on the left. So that's this point of the function j. Now the derivative term, remember is d over dw of j of w. And when we look at the tangent line at this point over here, the slope of this line is the derivative of j at this point. But this tangent line is sloping down into the right. And so this line sloping down into the right has a negative slope. In other words, the derivative j at this point is a negative number. For instance, if you draw a triangle, then the height like this is negative two and the width is one. So the slope is negative two divided by one, which is negative two, which is a negative number. So when you update w, you get w minus the learning rate times a negative number. And so this means you subtract from w a negative number. But subtracting a negative number means adding a positive number. And so you end up increasing w, because subtracting a negative number is the same as adding a positive number to w. So this step of gradient descent causes w to increase, which means you're moving to the right of the graph, and your cos j has decreased down to here. And again, it looks like gradient descent is doing something reasonable, it's getting you closer to the minimum. So hopefully, these last two examples show some of the intuition behind what the derivative term is doing, and why this helps gradient descent change w to get you closer to the minimum. I hope this video gave you some sense for why the derivative term in gradient descent makes sense. One other key quantity in the gradient descent algorithm is the learning rate alpha. How do you choose alpha? What happens if
| 500
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1.17 Machine Learning Overview | Gradient descent intuition --[Machine Learning | Andrew Ng]: over 1, for instance. And when the tangent line is pointing up into the right, the slope is positive, which means that this derivative is a positive number, so it's greater than zero. And so the updated w is going to be w minus the learning rate times some positive number. The learning rate is always a positive number. So if you take w minus a positive number, you end up with a new value for w that is smaller. So on the graph, we are moving to the left, we are decreasing the value of w. And you may notice that this is the right thing to do if your goal is to decrease the cost j, because when we move toward the left on this curve, the cost j decreases and you're getting closer to the minimum for j, which is over here. So so far, gradient descent seems to be doing the right thing. Now, let's look at another example. Let's take the same function j of w as above. And now let's say that you initialize gradient descent at a different location, say by choosing a starting value for w that's over here on the left. So that's this point of the function j. Now the derivative term, remember is d over dw of j of w. And when we look at the tangent line at this point over here, the slope of this line is the derivative of j at this point. But this tangent line is sloping down into the right. And so this line sloping down into the right has a negative slope. In other words, the derivative j at this point is a negative number. For instance, if you draw a triangle, then the height like this is negative two and the width is one. So the slope is negative two divided by one, which is negative two, which is a negative number. So when you update w, you get w minus the learning rate times a negative number. And so this means you subtract from w a negative number. But subtracting a negative number means adding a positive number. And so you end up increasing w, because subtracting a negative number is the same as adding a positive number to w. So this step of gradient descent causes w to increase, which means you're moving to the right of the graph, and your cos j has decreased down to here. And again, it looks like gradient descent is doing something reasonable, it's getting you closer to the minimum. So hopefully, these last two examples show some of the intuition behind what the derivative term is doing, and why this helps gradient descent change w to get you closer to the minimum. I hope this video gave you some sense for why the derivative term in gradient descent makes sense. One other key quantity in the gradient descent algorithm is the learning rate alpha. How do you choose alpha? What happens if
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1.17 Machine Learning Overview | Gradient descent intuition --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=rGvoO8U2Ozc
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it's too small? What happens if it's too big? In the next video, let's take a deeper look at the parameter alpha to help build intuitions about what it does, as well as how to make a good choice for a good value of alpha for your implementation of gradient descent.
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1.17 Machine Learning Overview | Gradient descent intuition --[Machine Learning | Andrew Ng]: it's too small? What happens if it's too big? In the next video, let's take a deeper look at the parameter alpha to help build intuitions about what it does, as well as how to make a good choice for a good value of alpha for your implementation of gradient descent.
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1.18 Machine Learning Overview | Learning rate --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=Eu8lt4j9xiU
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The choice of the learning rate alpha will have a huge impact on the efficiency of your implementation of gradient descent, and if alpha, the learning rate, is chosen poorly, gradient descent may not even work at all. In this video, let's take a deeper look at the learning rate. This will also help you choose better learning rates for your implementations of gradient descent. So, here again is the gradient descent rule. W is updated to be W minus the learning rate alpha times the derivative term. To learn more about what the learning rate alpha is doing, let's see what could happen if the learning rate alpha is either too small, or if it is too large. For the case where the learning rate is too small, here's a graph where the horizontal axis is W and the vertical axis is the cos j, and here's a graph of the function j of W. Let's start gradient descent at this point here. If the learning rate is too small, then what happens is that you multiply your derivative term by some really, really, really small number. So, you're going to be multiplying by a number alpha that's really small, like 0.0000001, and so you end up taking a very small baby step like that. Then from this point, you're going to take another, you know, tiny, tiny little baby step, but because the learning rate is so small, the second step is also just minus Q. The outcome of this process is that you do end up decreasing the cos j, but incredibly slowly. So, here's another step, and another step, another tiny step, until you finally approach the minimum. But, as you may notice, you're going to need a lot of steps to get to the minimum. So, to summarize, if the learning rate is too small, then gradient descent will work, but it will be slow. It will take a very long time, because it's going to take these tiny, tiny baby steps, and it's going to need a lot of steps before it gets anywhere close to the minimum. Now, let's look at a different case. What happens if the learning rate is too large? Here's another graph of the cos function, and let's say we start gradient descent with W at this value here. So, it's actually already pretty close to the minimum. So, the derivative points to the right, but if the learning rate is too large, then you update W via a giant step to be all the way over here, and that's this point here on the function j. So, you move from this point on the left all the way to this point on the right, and now the cost has actually gotten worse. It has increased because it has started out at this value here, and after one step, it actually increased to this value here. Now, the derivative at this new point says to decrease W, but when
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1.18 Machine Learning Overview | Learning rate --[Machine Learning | Andrew Ng]: The choice of the learning rate alpha will have a huge impact on the efficiency of your implementation of gradient descent, and if alpha, the learning rate, is chosen poorly, gradient descent may not even work at all. In this video, let's take a deeper look at the learning rate. This will also help you choose better learning rates for your implementations of gradient descent. So, here again is the gradient descent rule. W is updated to be W minus the learning rate alpha times the derivative term. To learn more about what the learning rate alpha is doing, let's see what could happen if the learning rate alpha is either too small, or if it is too large. For the case where the learning rate is too small, here's a graph where the horizontal axis is W and the vertical axis is the cos j, and here's a graph of the function j of W. Let's start gradient descent at this point here. If the learning rate is too small, then what happens is that you multiply your derivative term by some really, really, really small number. So, you're going to be multiplying by a number alpha that's really small, like 0.0000001, and so you end up taking a very small baby step like that. Then from this point, you're going to take another, you know, tiny, tiny little baby step, but because the learning rate is so small, the second step is also just minus Q. The outcome of this process is that you do end up decreasing the cos j, but incredibly slowly. So, here's another step, and another step, another tiny step, until you finally approach the minimum. But, as you may notice, you're going to need a lot of steps to get to the minimum. So, to summarize, if the learning rate is too small, then gradient descent will work, but it will be slow. It will take a very long time, because it's going to take these tiny, tiny baby steps, and it's going to need a lot of steps before it gets anywhere close to the minimum. Now, let's look at a different case. What happens if the learning rate is too large? Here's another graph of the cos function, and let's say we start gradient descent with W at this value here. So, it's actually already pretty close to the minimum. So, the derivative points to the right, but if the learning rate is too large, then you update W via a giant step to be all the way over here, and that's this point here on the function j. So, you move from this point on the left all the way to this point on the right, and now the cost has actually gotten worse. It has increased because it has started out at this value here, and after one step, it actually increased to this value here. Now, the derivative at this new point says to decrease W, but when
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1.18 Machine Learning Overview | Learning rate --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=Eu8lt4j9xiU
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the learning rate is too big, then you may take a huge step going from here all the way out here. So, now you've gotten to this point here, and again, if the learning rate is too big, then you take another huge step at the next iteration and way overshoot the minimum again. So, now you're at this point on the right, and one more time, you do another update and end up all the way here, and so you're now at this point here. So, as you may notice, you're actually getting further and further away from the minimum. So, if the learning rate is too large, then gradient descent may overshoot and may never reach the minimum. And another way to say that is that gradient descent may fail to converge and may even diverge. So, here's another question you may be wondering. What if your parameter W is already at this point here, so that your cost J is already at a local minimum? What do you think one step of gradient descent will do if you've already reached a minimum? So, this is a tricky one. When I was first learning this stuff, it actually took me a long time to figure it out, but let's work through this together. Let's suppose you have some cost function J, and the one you see here isn't a square error cost function, and this cost function has two local minima corresponding to the two values that you see here. Now, let's suppose that after some number of steps of gradient descent, your parameter W is over here, say equal to 5, and so this is the current value of W. This means that you're at this point on the cost function J, and that happens to be a local minimum. Turns out, if you draw a tangent to the function at this point, the slope of this line is zero, and thus, the derivative term here is equal to zero for the current value of W, and so your gradient descent update becomes W is updated to W minus the learning rate times zero, where here, that's because the derivative term is equal to zero, and this is the same as saying let's set W to be equal to W. So, this means that if you're already at a local minimum, gradient descent leaves W unchanged because it just updates the new value of W to be the exact same old value of W. So, concretely, let's say if the current value of W is 5, and alpha is 0.1, after one iteration, you update W as W minus alpha times zero, and it is still equal to 5. So, if your parameters have already brought you to a local minimum, then further gradient descent steps to absolutely nothing. It doesn't change the parameters, which is what you want, because it keeps the solution at that local minimum. This also explains why gradient descent can reach a
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1.18 Machine Learning Overview | Learning rate --[Machine Learning | Andrew Ng]: the learning rate is too big, then you may take a huge step going from here all the way out here. So, now you've gotten to this point here, and again, if the learning rate is too big, then you take another huge step at the next iteration and way overshoot the minimum again. So, now you're at this point on the right, and one more time, you do another update and end up all the way here, and so you're now at this point here. So, as you may notice, you're actually getting further and further away from the minimum. So, if the learning rate is too large, then gradient descent may overshoot and may never reach the minimum. And another way to say that is that gradient descent may fail to converge and may even diverge. So, here's another question you may be wondering. What if your parameter W is already at this point here, so that your cost J is already at a local minimum? What do you think one step of gradient descent will do if you've already reached a minimum? So, this is a tricky one. When I was first learning this stuff, it actually took me a long time to figure it out, but let's work through this together. Let's suppose you have some cost function J, and the one you see here isn't a square error cost function, and this cost function has two local minima corresponding to the two values that you see here. Now, let's suppose that after some number of steps of gradient descent, your parameter W is over here, say equal to 5, and so this is the current value of W. This means that you're at this point on the cost function J, and that happens to be a local minimum. Turns out, if you draw a tangent to the function at this point, the slope of this line is zero, and thus, the derivative term here is equal to zero for the current value of W, and so your gradient descent update becomes W is updated to W minus the learning rate times zero, where here, that's because the derivative term is equal to zero, and this is the same as saying let's set W to be equal to W. So, this means that if you're already at a local minimum, gradient descent leaves W unchanged because it just updates the new value of W to be the exact same old value of W. So, concretely, let's say if the current value of W is 5, and alpha is 0.1, after one iteration, you update W as W minus alpha times zero, and it is still equal to 5. So, if your parameters have already brought you to a local minimum, then further gradient descent steps to absolutely nothing. It doesn't change the parameters, which is what you want, because it keeps the solution at that local minimum. This also explains why gradient descent can reach a
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1.18 Machine Learning Overview | Learning rate --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=Eu8lt4j9xiU
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local minimum even with a fixed learning rate alpha. Here's what I mean. To illustrate this, let's look at another example. Here's the cost function J of W that we want to minimize. Let's initialize gradient descent up here at this point. If we take one update step, maybe it'll take us to that point. And because this derivative is pretty large, gradient descent takes a relatively big step, right? Now, we're at this second point, where we take another step. And you may notice that the slope is not as steep as it was at the first point, so the derivative isn't as large. And so, the next update step will not be as large as that first step. Now, we're at this third point here, and the derivative is smaller than it was at the previous step and we'll take an even smaller step. As we approach the minimum, the derivative gets closer and closer to zero. So as we run gradient descent, eventually we're taking very small steps until you finally reach a local minimum. So just to recap, as we get nearer a local minimum, gradient descent will automatically take smaller steps. And that's because as we approach the local minimum, the derivative automatically gets smaller and that means the update steps also automatically get smaller, even if the learning rate alpha is kept at some fixed value. So that's the gradient descent algorithm. You can use it to try to minimize any cost function j, not just the mean squared error cost function that we're using for linear regression. In the next video, we're going to take the function j and set that back to be exactly the linear regression model's cost function, the mean squared error cost function that we had come up with earlier. And putting together gradient descent with this cost function that will give you your first learning algorithm, the linear regression algorithm.
| 322
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1.18 Machine Learning Overview | Learning rate --[Machine Learning | Andrew Ng]: local minimum even with a fixed learning rate alpha. Here's what I mean. To illustrate this, let's look at another example. Here's the cost function J of W that we want to minimize. Let's initialize gradient descent up here at this point. If we take one update step, maybe it'll take us to that point. And because this derivative is pretty large, gradient descent takes a relatively big step, right? Now, we're at this second point, where we take another step. And you may notice that the slope is not as steep as it was at the first point, so the derivative isn't as large. And so, the next update step will not be as large as that first step. Now, we're at this third point here, and the derivative is smaller than it was at the previous step and we'll take an even smaller step. As we approach the minimum, the derivative gets closer and closer to zero. So as we run gradient descent, eventually we're taking very small steps until you finally reach a local minimum. So just to recap, as we get nearer a local minimum, gradient descent will automatically take smaller steps. And that's because as we approach the local minimum, the derivative automatically gets smaller and that means the update steps also automatically get smaller, even if the learning rate alpha is kept at some fixed value. So that's the gradient descent algorithm. You can use it to try to minimize any cost function j, not just the mean squared error cost function that we're using for linear regression. In the next video, we're going to take the function j and set that back to be exactly the linear regression model's cost function, the mean squared error cost function that we had come up with earlier. And putting together gradient descent with this cost function that will give you your first learning algorithm, the linear regression algorithm.
|
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1.19 Machine Learning Overview| Gradient descent for Linear Regression--[Machine Learning|Andrew Ng]
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https://www.youtube.com/watch?v=ZSg_NglG3aA
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So, previously you took a look at the linear regression model and then the cost function and then the gradient descent algorithm. In this video, we're going to put it all together and use the square error cost function for the linear regression model with gradient descent. This will allow us to train the linear regression model to fit a straight line to our training data. Let's get to it. This is the linear regression model and to the right is the squared error cost function and below is the gradient descent algorithm. It turns out if you calculate these derivatives, these are the terms you would get. The derivative with respect to w is this. One over m summed from i equals one through m. Then the error term, that is the difference between the predicted and the actual values, times the input feature x i. And the derivative with respect to b is this formula over here, which looks the same as the equation above except that it doesn't have that x i term at the end. And if you use these formulas to compute these two derivatives and implement gradient descent this way, it will work. Now you may be wondering, where did I get these formulas from? They're derived using calculus. If you want to see the full derivation, I'll quickly run through the derivation on the next slide. But if you don't remember or aren't interested in the calculus, don't worry about it. You can skip the materials on the next slide entirely and still be able to implement gradient descent and finish this class and everything will work just fine. So in this slide, which is one of the most mathematical slides of the entire specialization and again is completely optional, we'll show you how to calculate the derivative terms. Let's start with the first term. The derivative of the cost function j with respect to w. We'll start by plugging in the definition of the cost function j. So j of w b is this. So one over two m times this sum of the squared error terms. And now remember also that f of w b of x i is equal to this term over here, which is w x i plus b. And so what we would like to do is compute the derivative also called the partial derivative with respect to w of this equation right here on the right. If you've taken a calculus class before, and again is totally fine if you haven't, you may know that by the rules of calculus, the derivative is equal to this term over here, which is why the two here and the two here cancel out, leaving us with this equation that you saw on the previous slide. So this, by the way, is why we had to find the cost function with the one half earlier this week is because it makes the partial derivative meter. It cancels out the
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1.19 Machine Learning Overview| Gradient descent for Linear Regression--[Machine Learning|Andrew Ng]: So, previously you took a look at the linear regression model and then the cost function and then the gradient descent algorithm. In this video, we're going to put it all together and use the square error cost function for the linear regression model with gradient descent. This will allow us to train the linear regression model to fit a straight line to our training data. Let's get to it. This is the linear regression model and to the right is the squared error cost function and below is the gradient descent algorithm. It turns out if you calculate these derivatives, these are the terms you would get. The derivative with respect to w is this. One over m summed from i equals one through m. Then the error term, that is the difference between the predicted and the actual values, times the input feature x i. And the derivative with respect to b is this formula over here, which looks the same as the equation above except that it doesn't have that x i term at the end. And if you use these formulas to compute these two derivatives and implement gradient descent this way, it will work. Now you may be wondering, where did I get these formulas from? They're derived using calculus. If you want to see the full derivation, I'll quickly run through the derivation on the next slide. But if you don't remember or aren't interested in the calculus, don't worry about it. You can skip the materials on the next slide entirely and still be able to implement gradient descent and finish this class and everything will work just fine. So in this slide, which is one of the most mathematical slides of the entire specialization and again is completely optional, we'll show you how to calculate the derivative terms. Let's start with the first term. The derivative of the cost function j with respect to w. We'll start by plugging in the definition of the cost function j. So j of w b is this. So one over two m times this sum of the squared error terms. And now remember also that f of w b of x i is equal to this term over here, which is w x i plus b. And so what we would like to do is compute the derivative also called the partial derivative with respect to w of this equation right here on the right. If you've taken a calculus class before, and again is totally fine if you haven't, you may know that by the rules of calculus, the derivative is equal to this term over here, which is why the two here and the two here cancel out, leaving us with this equation that you saw on the previous slide. So this, by the way, is why we had to find the cost function with the one half earlier this week is because it makes the partial derivative meter. It cancels out the
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1.19 Machine Learning Overview| Gradient descent for Linear Regression--[Machine Learning|Andrew Ng]
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https://www.youtube.com/watch?v=ZSg_NglG3aA
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two that appears from computing the derivative. For the other derivative with respect to b, this is quite similar. I can write it out like this and once again, plug in the definition of f of x i giving this equation. And by the rules of calculus, this is equal to this, where there's no x i anymore at the end. And so the twos cancel once more and you end up with this expression for the derivative with respect to b. And so now you have these two expressions for the derivatives and you can plug them into the gradient descent algorithm. So here's the gradient descent algorithm for linear regression. You repeatedly carry out these updates to w and b until convergence. Remember that this f of x is a linear regression model, so it's equal to w times x plus b. This expression here is the derivative of the cost function with respect to w and this expression is the derivative of the cost function with respect to b. And just as a reminder, you want to update w and b simultaneously on each step. Now let's get familiar with how gradient descent works. One issue we saw with gradient descent is that it can lead to a local minimum instead of a global minimum, where the global minimum means the point that has the lowest possible value for the cost function j out of all possible points. You may recall this surface plot that looks like an outdoor park with a few hills with the grass and the birds is a relaxing outdoor hill. This function has more than one local minimum. Remember depending on where you initialize the parameters w and b, you can end up at different local minima. You can end up here or you can end up here. But it turns out when you're using a squared error cost function with linear regression, the cost function does not and will never have multiple local minima. It has a single global minimum because of this bow shape. The technical term for this is that this cost function is a convex function. Informally, a convex function is a bow shaped function and it cannot have any local minima other than the single global minimum. So when you implement gradient descent on a convex function, one nice property is that so long as your learning rate is chosen appropriately, it will always converge to the global minimum. Congratulations, you now know how to implement gradient descent for linear regression. We have just one last video for this week. In that video, we'll see this algorithm in action. Let's go to that last video.
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1.19 Machine Learning Overview| Gradient descent for Linear Regression--[Machine Learning|Andrew Ng]: two that appears from computing the derivative. For the other derivative with respect to b, this is quite similar. I can write it out like this and once again, plug in the definition of f of x i giving this equation. And by the rules of calculus, this is equal to this, where there's no x i anymore at the end. And so the twos cancel once more and you end up with this expression for the derivative with respect to b. And so now you have these two expressions for the derivatives and you can plug them into the gradient descent algorithm. So here's the gradient descent algorithm for linear regression. You repeatedly carry out these updates to w and b until convergence. Remember that this f of x is a linear regression model, so it's equal to w times x plus b. This expression here is the derivative of the cost function with respect to w and this expression is the derivative of the cost function with respect to b. And just as a reminder, you want to update w and b simultaneously on each step. Now let's get familiar with how gradient descent works. One issue we saw with gradient descent is that it can lead to a local minimum instead of a global minimum, where the global minimum means the point that has the lowest possible value for the cost function j out of all possible points. You may recall this surface plot that looks like an outdoor park with a few hills with the grass and the birds is a relaxing outdoor hill. This function has more than one local minimum. Remember depending on where you initialize the parameters w and b, you can end up at different local minima. You can end up here or you can end up here. But it turns out when you're using a squared error cost function with linear regression, the cost function does not and will never have multiple local minima. It has a single global minimum because of this bow shape. The technical term for this is that this cost function is a convex function. Informally, a convex function is a bow shaped function and it cannot have any local minima other than the single global minimum. So when you implement gradient descent on a convex function, one nice property is that so long as your learning rate is chosen appropriately, it will always converge to the global minimum. Congratulations, you now know how to implement gradient descent for linear regression. We have just one last video for this week. In that video, we'll see this algorithm in action. Let's go to that last video.
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1.20 Machine Learning Overview | Running gradient descent --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=3xyYI4wPuTs
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Let's see what happens when you run gradient descent for linear regression. Let's go see the algorithm in action. Here's a plot of the model and data on the upper left, and a contour plot of the cost function on the upper right. And at the bottom is the surface plot of the same cost function. Often W and B will both be initialized to zero, but for this demonstration, let's initialize W to be equal to negative 0.1 and B to be 900. So this corresponds to f of x equals negative 0.1 x plus 900. Now if we take one step using gradient descent, we end up going from this point of the cost function out here to this point just down and to the right. And notice that the straight line fit has also changed a bit. Let's take another step. The cost function has now moved to this third point, and again the function f of x has also changed a bit. As you take more of these steps, the cost is decreasing at each update, so the parameters W and B are following this trajectory. And if you look on the left, you get this corresponding straight line fit that, you know, fits the data better and better until we've reached the global minimum. The global minimum corresponds to this straight line fit, which is a relatively good fit to the data. I mean, isn't that cool? And so that's gradient descent, and we're going to use this to fit a model to the housing data. And you can now use this f of x model to predict the price of your client's house or anyone else's house. For instance, if your friend's house size is 1250 square feet, you can now read off the value and predict that maybe they could get, I don't know, $250,000 for the house. To be more precise, this gradient descent process is called batch gradient descent. The term batch gradient descent refers to the fact that on every step of gradient descent, we're looking at all of the training examples instead of just a subset of the training data. So in computing gradient descent, when computing derivatives, we're computing the sum from i equals one to m, and batch gradient descent is looking at the entire batch of training examples at each update. I know that batch gradient descent may not be the most intuitive name, but this is what people in the machine learning community call it. If you've heard of the newsletter, The Batch, that's published by deeplearning.ai, the newsletter, The Batch, was also named for this concept in machine learning. And then it turns out that there are other versions of gradient descent that do not look at the entire training set, but instead looks at smaller subsets of the training data at each update step. But we'll use batch gradient descent for linear regression. So that's it for linear regression. Congratulations on getting through
| 500
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1.20 Machine Learning Overview | Running gradient descent --[Machine Learning | Andrew Ng]: Let's see what happens when you run gradient descent for linear regression. Let's go see the algorithm in action. Here's a plot of the model and data on the upper left, and a contour plot of the cost function on the upper right. And at the bottom is the surface plot of the same cost function. Often W and B will both be initialized to zero, but for this demonstration, let's initialize W to be equal to negative 0.1 and B to be 900. So this corresponds to f of x equals negative 0.1 x plus 900. Now if we take one step using gradient descent, we end up going from this point of the cost function out here to this point just down and to the right. And notice that the straight line fit has also changed a bit. Let's take another step. The cost function has now moved to this third point, and again the function f of x has also changed a bit. As you take more of these steps, the cost is decreasing at each update, so the parameters W and B are following this trajectory. And if you look on the left, you get this corresponding straight line fit that, you know, fits the data better and better until we've reached the global minimum. The global minimum corresponds to this straight line fit, which is a relatively good fit to the data. I mean, isn't that cool? And so that's gradient descent, and we're going to use this to fit a model to the housing data. And you can now use this f of x model to predict the price of your client's house or anyone else's house. For instance, if your friend's house size is 1250 square feet, you can now read off the value and predict that maybe they could get, I don't know, $250,000 for the house. To be more precise, this gradient descent process is called batch gradient descent. The term batch gradient descent refers to the fact that on every step of gradient descent, we're looking at all of the training examples instead of just a subset of the training data. So in computing gradient descent, when computing derivatives, we're computing the sum from i equals one to m, and batch gradient descent is looking at the entire batch of training examples at each update. I know that batch gradient descent may not be the most intuitive name, but this is what people in the machine learning community call it. If you've heard of the newsletter, The Batch, that's published by deeplearning.ai, the newsletter, The Batch, was also named for this concept in machine learning. And then it turns out that there are other versions of gradient descent that do not look at the entire training set, but instead looks at smaller subsets of the training data at each update step. But we'll use batch gradient descent for linear regression. So that's it for linear regression. Congratulations on getting through
|
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1.20 Machine Learning Overview | Running gradient descent --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=3xyYI4wPuTs
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your first machine learning model. I hope you go and celebrate or, I don't know, maybe take a nap in your hammock. In the optional lab that follows this video, you see a review of the gradient descent algorithm as well as how to implement it in code. You also see a plot that shows how the cost decreases as you continue training more iterations. And you also see a contour plot, seeing how the cost gets closer to the global minimum as gradient descent finds better and better values for the parameters W and B. So remember that to do the optional lab, you just need to read and run this code. You won't need to write any code yourself. And I hope you take a few moments to do that and also become familiar with the gradient descent code because this will help you to implement this and similar algorithms in the future yourself. Thanks for sticking with me through the end of this last video for the first week and congratulations for making it all the way here. You're on your way to becoming a machine learning person. In addition to the optional labs, if you haven't done so yet, I hope you also check out the practice quizzes, which are a nice way that you can double check your own understanding of the concepts. It's also totally fine if you don't get them all right the first time. And you can also take the quizzes multiple times until you get the score that you want. You now know how to implement linear regression with one variable. And that brings us to the close of this week. Next week, we'll learn to make linear regression much more powerful. Instead of one feature like size of a house, you learn how to get it to work with lots of features. You also learn how to get it to fit nonlinear curves. These improvements will make the algorithm much more useful and valuable. Lastly, we'll also go over some practical tips that will really help for getting linear regression to work on practical applications. I'm really happy to have you here with me in this class and I look forward to seeing you next week.
| 376
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1.20 Machine Learning Overview | Running gradient descent --[Machine Learning | Andrew Ng]: your first machine learning model. I hope you go and celebrate or, I don't know, maybe take a nap in your hammock. In the optional lab that follows this video, you see a review of the gradient descent algorithm as well as how to implement it in code. You also see a plot that shows how the cost decreases as you continue training more iterations. And you also see a contour plot, seeing how the cost gets closer to the global minimum as gradient descent finds better and better values for the parameters W and B. So remember that to do the optional lab, you just need to read and run this code. You won't need to write any code yourself. And I hope you take a few moments to do that and also become familiar with the gradient descent code because this will help you to implement this and similar algorithms in the future yourself. Thanks for sticking with me through the end of this last video for the first week and congratulations for making it all the way here. You're on your way to becoming a machine learning person. In addition to the optional labs, if you haven't done so yet, I hope you also check out the practice quizzes, which are a nice way that you can double check your own understanding of the concepts. It's also totally fine if you don't get them all right the first time. And you can also take the quizzes multiple times until you get the score that you want. You now know how to implement linear regression with one variable. And that brings us to the close of this week. Next week, we'll learn to make linear regression much more powerful. Instead of one feature like size of a house, you learn how to get it to work with lots of features. You also learn how to get it to fit nonlinear curves. These improvements will make the algorithm much more useful and valuable. Lastly, we'll also go over some practical tips that will really help for getting linear regression to work on practical applications. I'm really happy to have you here with me in this class and I look forward to seeing you next week.
|
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2.1 Linear Regression with Multiple Variables | Multiple features --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=fX86EFWljY0
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Welcome back. In this week, we'll learn to make linear regression much faster and much more powerful, and by the end of this week, you'll be two-thirds of the way to finishing this first course. Let's start by looking at a version of linear regression that can look at not just one feature, but a lot of different features. Let's take a look. In the original version of linear regression, you had a single feature x, the size of the house, and you're able to predict y, the price of the house. So the model was FWB of x equals Wx plus b. But now, what if you did not only have the size of the house as a feature with which to try to predict the price, but if you also knew the number of bedrooms, the number of floors, and the age of the home in years, it seems like this would give you a lot more information with which to predict the price. To introduce a little bit of new notation, we're going to use the variables x subscript 1, x subscript 2, x subscript 3, and x subscript 4 to denote the four features. And for simplicity, let's introduce a little bit more notation. We'll write x subscript j, or sometimes I'll just say for short x sub j to represent the list of features. So here, j will go from 1 through 4 because we have four features. I'm going to use lowercase n to denote the total number of features. So in this example, n is equal to 4. As before, we'll use x super stripped i to denote the I've training example. So here, x super strip i is actually going to be a list of four numbers, or sometimes we'll call this a vector that includes all the features of the I've training example. So as a concrete example, x super strip in parentheses 2 will be a vector of the features for the second training example. So it will equal to this 1416, 32, and 40. And technically, I'm writing these numbers in a row, so sometimes this is called a row vector rather than a column vector. But if you don't know what the difference is, don't worry about it. It's not that important for this purpose. And to refer to a specific feature in the I've training example, I will write x super strip i subscript j. So for example, x super strip 2 subscript 3 will be the value of the third feature, that is the number of flaws in the second training example. And so that's going to be equal to two. Sometimes in order to emphasize that this x two is not a number, but it's actually a list of numbers that is a vector, we'll draw an arrow on top of that, just to visually show that is a vector. And over here as well. But you don't have to draw this arrow in your
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2.1 Linear Regression with Multiple Variables | Multiple features --[Machine Learning | Andrew Ng]: Welcome back. In this week, we'll learn to make linear regression much faster and much more powerful, and by the end of this week, you'll be two-thirds of the way to finishing this first course. Let's start by looking at a version of linear regression that can look at not just one feature, but a lot of different features. Let's take a look. In the original version of linear regression, you had a single feature x, the size of the house, and you're able to predict y, the price of the house. So the model was FWB of x equals Wx plus b. But now, what if you did not only have the size of the house as a feature with which to try to predict the price, but if you also knew the number of bedrooms, the number of floors, and the age of the home in years, it seems like this would give you a lot more information with which to predict the price. To introduce a little bit of new notation, we're going to use the variables x subscript 1, x subscript 2, x subscript 3, and x subscript 4 to denote the four features. And for simplicity, let's introduce a little bit more notation. We'll write x subscript j, or sometimes I'll just say for short x sub j to represent the list of features. So here, j will go from 1 through 4 because we have four features. I'm going to use lowercase n to denote the total number of features. So in this example, n is equal to 4. As before, we'll use x super stripped i to denote the I've training example. So here, x super strip i is actually going to be a list of four numbers, or sometimes we'll call this a vector that includes all the features of the I've training example. So as a concrete example, x super strip in parentheses 2 will be a vector of the features for the second training example. So it will equal to this 1416, 32, and 40. And technically, I'm writing these numbers in a row, so sometimes this is called a row vector rather than a column vector. But if you don't know what the difference is, don't worry about it. It's not that important for this purpose. And to refer to a specific feature in the I've training example, I will write x super strip i subscript j. So for example, x super strip 2 subscript 3 will be the value of the third feature, that is the number of flaws in the second training example. And so that's going to be equal to two. Sometimes in order to emphasize that this x two is not a number, but it's actually a list of numbers that is a vector, we'll draw an arrow on top of that, just to visually show that is a vector. And over here as well. But you don't have to draw this arrow in your
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2.1 Linear Regression with Multiple Variables | Multiple features --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=fX86EFWljY0
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notation. You can think of the arrow as an optional signifier that sometimes use just to emphasize that this is a vector and not a number. Now that we have multiple features, let's take a look at what our model would look like. Previously, this is how we defined the model where x was a single feature. So a single number. But now with multiple features, we're going to define it differently. Instead, the model will be f WB of x equals W1 x1 plus W2 x2 plus W3 x3 plus W4 x4 plus B. Concretely, for housing price prediction, one possible model may be that we estimate the price of the house as 0.1 times x1, the size of the house, plus four times x2, the number of bedrooms, plus 10 times x3, the number of floors, minus two times x4, the age of the house in years plus 80. Let's think a bit about how you might interpret these parameters. If the model is trying to predict the price of the house in thousands of dollars, you can think of this B equals 80 as saying that the base price of a house starts off at maybe $80,000, assuming it has no size, no bedrooms, no floor and no age. And you can think of this 0.1 as saying that maybe for every additional square foot, the price will increase by $0.1,000 or by $100 because we're saying that for each square foot, the price increases by 0.1, you know, times $1,000, which is $100. And maybe for each additional bathroom, the price increases by $4,000. And for each additional floor, the price may increase by $10,000. And for each additional year of the house's age, the price may decrease by $2,000 because the parameter is negative two. And in general, if you have n features, then the model will look like this. Here again is the definition of the model with n features. What we're going to do next is introduce a little bit of notation to rewrite this expression in a simpler but equivalent way. Let's define w as a list of numbers that lists the parameters w1, w2, w3, all the way through wn. In mathematics, this is called a vector. And sometimes to designate that this is a vector, which just means a list of numbers, I'm going to draw a little arrow on top. You don't always have to draw this arrow. And you can do so or not in your own notation. So you can think of this little arrow as just an optional signifier to remind us that this is a vector. If you've taken a linear algebra class before, you might recognize that this is a row vector as opposed to a column vector. But if you don't know what those terms means, you don't need to worry about it. Next, same as before, b is a single number and not a vector. And so this vector w together with
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2.1 Linear Regression with Multiple Variables | Multiple features --[Machine Learning | Andrew Ng]: notation. You can think of the arrow as an optional signifier that sometimes use just to emphasize that this is a vector and not a number. Now that we have multiple features, let's take a look at what our model would look like. Previously, this is how we defined the model where x was a single feature. So a single number. But now with multiple features, we're going to define it differently. Instead, the model will be f WB of x equals W1 x1 plus W2 x2 plus W3 x3 plus W4 x4 plus B. Concretely, for housing price prediction, one possible model may be that we estimate the price of the house as 0.1 times x1, the size of the house, plus four times x2, the number of bedrooms, plus 10 times x3, the number of floors, minus two times x4, the age of the house in years plus 80. Let's think a bit about how you might interpret these parameters. If the model is trying to predict the price of the house in thousands of dollars, you can think of this B equals 80 as saying that the base price of a house starts off at maybe $80,000, assuming it has no size, no bedrooms, no floor and no age. And you can think of this 0.1 as saying that maybe for every additional square foot, the price will increase by $0.1,000 or by $100 because we're saying that for each square foot, the price increases by 0.1, you know, times $1,000, which is $100. And maybe for each additional bathroom, the price increases by $4,000. And for each additional floor, the price may increase by $10,000. And for each additional year of the house's age, the price may decrease by $2,000 because the parameter is negative two. And in general, if you have n features, then the model will look like this. Here again is the definition of the model with n features. What we're going to do next is introduce a little bit of notation to rewrite this expression in a simpler but equivalent way. Let's define w as a list of numbers that lists the parameters w1, w2, w3, all the way through wn. In mathematics, this is called a vector. And sometimes to designate that this is a vector, which just means a list of numbers, I'm going to draw a little arrow on top. You don't always have to draw this arrow. And you can do so or not in your own notation. So you can think of this little arrow as just an optional signifier to remind us that this is a vector. If you've taken a linear algebra class before, you might recognize that this is a row vector as opposed to a column vector. But if you don't know what those terms means, you don't need to worry about it. Next, same as before, b is a single number and not a vector. And so this vector w together with
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2.1 Linear Regression with Multiple Variables | Multiple features --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=fX86EFWljY0
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this number b are the parameters of the model. Let me also write x as a list or a vector, again, a row vector that lists all of the features x1, x2, x3 up through xn. This is again a vector. So I'm going to add a little arrow up on top to signify. So in the notation up on top, we can also add little arrows here and here to signify that that w and that x are actually these lists of numbers that they're actually these vectors. So with this notation, the model can now be rewritten more succinctly as f of x equals the vector w dot and this dot refers to a dot product from linear algebra of x, the vector plus the number b. So what is this dot product thing? Well, the dot products of two vectors of two lists of numbers w and x is computed by taking the corresponding pairs of numbers w1 and x1, multiplying that w2 x2, multiplying that w3 x3, multiplying that all the way up to wn xn, multiplying that and then summing up all of these products. Writing that out, this means that the dot product is equal to w1 x1 plus w2 x2 plus w3 x3 plus all the way up to wn xn. And then finally we add back in the b on top. And you notice that this gives us exactly the same expression as we had on top. So the dot product notation lets you write the model in a more compact form with fewer characters. The name for this type of linear regression model with multiple input features is multiple linear regression. This is in contrast to univariate regression, which had just one feature. And by the way, you might think this algorithm is called multivariate regression, but that term actually refers to something else that we won't be using here. So I'm going to refer to this model as multiple linear regression. And so that's it for linear regression with multiple features, which is also called multiple linear regression. In order to implement this, there's a really neat trick called vectorization, which will make it much simpler to implement this and many other learning algorithms. Let's go on to the next video to take a look at what is vectorization.
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2.1 Linear Regression with Multiple Variables | Multiple features --[Machine Learning | Andrew Ng]: this number b are the parameters of the model. Let me also write x as a list or a vector, again, a row vector that lists all of the features x1, x2, x3 up through xn. This is again a vector. So I'm going to add a little arrow up on top to signify. So in the notation up on top, we can also add little arrows here and here to signify that that w and that x are actually these lists of numbers that they're actually these vectors. So with this notation, the model can now be rewritten more succinctly as f of x equals the vector w dot and this dot refers to a dot product from linear algebra of x, the vector plus the number b. So what is this dot product thing? Well, the dot products of two vectors of two lists of numbers w and x is computed by taking the corresponding pairs of numbers w1 and x1, multiplying that w2 x2, multiplying that w3 x3, multiplying that all the way up to wn xn, multiplying that and then summing up all of these products. Writing that out, this means that the dot product is equal to w1 x1 plus w2 x2 plus w3 x3 plus all the way up to wn xn. And then finally we add back in the b on top. And you notice that this gives us exactly the same expression as we had on top. So the dot product notation lets you write the model in a more compact form with fewer characters. The name for this type of linear regression model with multiple input features is multiple linear regression. This is in contrast to univariate regression, which had just one feature. And by the way, you might think this algorithm is called multivariate regression, but that term actually refers to something else that we won't be using here. So I'm going to refer to this model as multiple linear regression. And so that's it for linear regression with multiple features, which is also called multiple linear regression. In order to implement this, there's a really neat trick called vectorization, which will make it much simpler to implement this and many other learning algorithms. Let's go on to the next video to take a look at what is vectorization.
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2.2 Linear Regression with Multiple Variables | Vectorization part 1 --[Machine Learning |Andrew Ng]
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https://www.youtube.com/watch?v=G8yfD_Xu7Ko
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In this video, you see a very useful idea called vectorization. When you're implementing a learning algorithm, using vectorization will both make your code shorter and also make it run much more efficiently. Learning how to write vectorized code will allow you to also take advantage of modern numerical linear algebra libraries, as well as maybe even GPU hardware that stands for graphics processor unit. This is hardware originally designed to speed up computer graphics on your computer, but turns out can be used when you write vectorized code to also help you execute your code much more quickly. Let's look at a concrete example of what vectorization means. Here's an example with parameters w and b, where w is a vector with three numbers, and you also have a vector of features x with also three numbers. So here, n is equal to three. So notice that in linear algebra, the index or the counting starts from one, and so the first value is subscripted w1 and x1. In Python code, you can define these variables w, b, and x using arrays like this. Here I'm actually using a numerical linear algebra library in Python called NumPy, which is by far the most widely used numerical linear algebra library in Python and in machine learning. Because in Python, the indexing of arrays or counting in arrays starts from zero, you would access the first value of w using w square bracket zero, the second value using w square bracket one, and the third using w square bracket two. So the indexing here goes from zero, one, to two, rather than one, two, to three. Similarly, to access individual features of x, you would use x0, x1, and x2. Many programming languages, including Python, start counting from zero rather than one. Now, let's look at an implementation without vectorization for computing the model's prediction. In code, it would look like this. You take each parameter w and multiply it by its associated feature. Now, you could write your code like this, but what if n is in three, but instead n is 100 or 100,000? It's both inefficient for you to code and inefficient for your computer to compute. So here's another way, still without using vectorization, but using a for loop. In math, you can use a summation operator to add all the products of wj and xj for j equals one through n. Then outside the summation, you add b at the end. So the summation goes from j equals one up to and including n for n equals three, j therefore goes from one, two, to three. In code, you can initialize f to zero, then for j in range from zero to n, this actually makes j go from zero to n minus one. So from zero, one to two, you can then add to f the product of wj times xj. Finally, outside the for loop, you add b. Notice that in Python, the range
| 500
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2.2 Linear Regression with Multiple Variables | Vectorization part 1 --[Machine Learning |Andrew Ng]: In this video, you see a very useful idea called vectorization. When you're implementing a learning algorithm, using vectorization will both make your code shorter and also make it run much more efficiently. Learning how to write vectorized code will allow you to also take advantage of modern numerical linear algebra libraries, as well as maybe even GPU hardware that stands for graphics processor unit. This is hardware originally designed to speed up computer graphics on your computer, but turns out can be used when you write vectorized code to also help you execute your code much more quickly. Let's look at a concrete example of what vectorization means. Here's an example with parameters w and b, where w is a vector with three numbers, and you also have a vector of features x with also three numbers. So here, n is equal to three. So notice that in linear algebra, the index or the counting starts from one, and so the first value is subscripted w1 and x1. In Python code, you can define these variables w, b, and x using arrays like this. Here I'm actually using a numerical linear algebra library in Python called NumPy, which is by far the most widely used numerical linear algebra library in Python and in machine learning. Because in Python, the indexing of arrays or counting in arrays starts from zero, you would access the first value of w using w square bracket zero, the second value using w square bracket one, and the third using w square bracket two. So the indexing here goes from zero, one, to two, rather than one, two, to three. Similarly, to access individual features of x, you would use x0, x1, and x2. Many programming languages, including Python, start counting from zero rather than one. Now, let's look at an implementation without vectorization for computing the model's prediction. In code, it would look like this. You take each parameter w and multiply it by its associated feature. Now, you could write your code like this, but what if n is in three, but instead n is 100 or 100,000? It's both inefficient for you to code and inefficient for your computer to compute. So here's another way, still without using vectorization, but using a for loop. In math, you can use a summation operator to add all the products of wj and xj for j equals one through n. Then outside the summation, you add b at the end. So the summation goes from j equals one up to and including n for n equals three, j therefore goes from one, two, to three. In code, you can initialize f to zero, then for j in range from zero to n, this actually makes j go from zero to n minus one. So from zero, one to two, you can then add to f the product of wj times xj. Finally, outside the for loop, you add b. Notice that in Python, the range
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2.2 Linear Regression with Multiple Variables | Vectorization part 1 --[Machine Learning |Andrew Ng]
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https://www.youtube.com/watch?v=G8yfD_Xu7Ko
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zero to n means that j goes from zero all the way to n minus one and does not include n itself. And more commonly, this is written range n in Python. But in this video, I added a zero here just to emphasize that it starts from zero. While this implementation is a bit better than the first one, it still doesn't use vectorization and isn't that efficient. Now let's look at how you can do this using vectorization. This is the math expression that the function f, which is the dot product of w and x plus b. And now you can implement this with a single line of code. By computing fp equals np dot dot, I said dot dot because the first dot is the period and the second dot is the function or the method called dot. But it's fp equals np dot dot w comma x and this implements the mathematical dot products between the vectors w and x. And then finally, you can add b to it at the end. This NumPy dot function is a vectorized implementation of the dot product operation between two vectors. And especially when n is large, this will run much faster than the two previous code examples. I want to emphasize that vectorization actually has two distinct benefits. First, it makes the code shorter. It's now just one line of code. Isn't that cool? And second, it also results in your code running much faster than either of the two previous implementations that did not use vectorization. And the reason that the vectorized implementation is much faster is behind the scenes, the NumPy dot function is able to use parallel hardware in your computer. And this is true whether you're running this on a normal computer that is on a normal computer CPU, or if you are using a GPU, a graphics processor unit that's often used to accelerate machine learning jobs. And the ability of the NumPy dot function to use parallel hardware makes it much more efficient than the for loop or the sequential calculation that we saw previously. Now this version is much more practical when n is large because you are not typing w0 times x0 plus w1 times x1 plus lots of additional terms like you would have had for the previous version. But while this saves a lot on the typing, it's still not that computationally efficient because it still doesn't use vectorization. So to recap, vectorization makes your code shorter, so hopefully easier to write and easier for you or others to read. And it also makes it run much faster. But what is this magic behind vectorization that makes this run so much faster? Let's take a look at what your computer is actually doing behind the scenes to make vectorized code run so much faster.
| 477
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2.2 Linear Regression with Multiple Variables | Vectorization part 1 --[Machine Learning |Andrew Ng]: zero to n means that j goes from zero all the way to n minus one and does not include n itself. And more commonly, this is written range n in Python. But in this video, I added a zero here just to emphasize that it starts from zero. While this implementation is a bit better than the first one, it still doesn't use vectorization and isn't that efficient. Now let's look at how you can do this using vectorization. This is the math expression that the function f, which is the dot product of w and x plus b. And now you can implement this with a single line of code. By computing fp equals np dot dot, I said dot dot because the first dot is the period and the second dot is the function or the method called dot. But it's fp equals np dot dot w comma x and this implements the mathematical dot products between the vectors w and x. And then finally, you can add b to it at the end. This NumPy dot function is a vectorized implementation of the dot product operation between two vectors. And especially when n is large, this will run much faster than the two previous code examples. I want to emphasize that vectorization actually has two distinct benefits. First, it makes the code shorter. It's now just one line of code. Isn't that cool? And second, it also results in your code running much faster than either of the two previous implementations that did not use vectorization. And the reason that the vectorized implementation is much faster is behind the scenes, the NumPy dot function is able to use parallel hardware in your computer. And this is true whether you're running this on a normal computer that is on a normal computer CPU, or if you are using a GPU, a graphics processor unit that's often used to accelerate machine learning jobs. And the ability of the NumPy dot function to use parallel hardware makes it much more efficient than the for loop or the sequential calculation that we saw previously. Now this version is much more practical when n is large because you are not typing w0 times x0 plus w1 times x1 plus lots of additional terms like you would have had for the previous version. But while this saves a lot on the typing, it's still not that computationally efficient because it still doesn't use vectorization. So to recap, vectorization makes your code shorter, so hopefully easier to write and easier for you or others to read. And it also makes it run much faster. But what is this magic behind vectorization that makes this run so much faster? Let's take a look at what your computer is actually doing behind the scenes to make vectorized code run so much faster.
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2.3 Linear Regression with Multiple Variables | Vectorization part 2 --[Machine Learning |Andrew Ng]
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https://www.youtube.com/watch?v=nRGG50GDNAA
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I remember when I first learned about vectorization, I spent many hours on my computer taking an un-vectorized version of an algorithm, running it, see how long it ran, and then running a vectorized version of the code and seeing how much faster that ran. And I just spent hours playing with that. And it frankly blew my mind that the same algorithm, vectorized, would run so much faster. It felt almost like a magic trick to me. In this video, let's figure out how this magic trick really works. Let's take a deeper look at how a vectorized implementation may work on your computer behind the scenes. Let's look at this for loop. A for loop like this runs without vectorization. So if j ranges from 0 to, say, 15, this piece of code performs operations one after another. On the first time step, which I'm going to write as time 0 or t0, it first operates on the values at index 0. At the next time step, it calculates values corresponding to index 1 and so on until the fifth theme step, where it computes that. In other words, it calculates these computations one step at a time, one step after another. In contrast, this function in NumPy is implemented in the computer hardware with vectorization. So the computer can get all values of the vectors w and x, and in a single step, it multiplies each pair of w and x with each other all at the same time in parallel. Then after that, the computer takes these 16 numbers and uses specialized hardware to add them all together very efficiently, rather than needing to carry out distinct additions one after another to add up these 16 numbers. This means that code with vectorization can perform calculations in much less time than code without vectorization. And this matters more when you're running learning algorithms on large data sets or trying to train large models, which is often the case with machine learning. So that's why being able to write vectorized implementations of learning algorithms has been a key step to getting learning algorithms to run efficiently and therefore scale well to the large data sets that many modern machine learning algorithms now have to operate on. Now let's take a look at a concrete example of how this helps with implementing multiple linear regression, that is linear regression with multiple input features. Say you have a problem with 16 features and 16 parameters w1 through w16 in addition to the parameter b. You calculated 16 derivative terms for these 16 weights and in code, maybe you stored the values of w and d in two non-pyrotries with d storing the values of the derivatives. For this example, I'm just going to ignore the parameter b. Now you want to compute and update for each of these 16 parameters. So wj is updated to wj minus the learning rate, say 0.1 times dj for j from 1 through
| 500
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2.3 Linear Regression with Multiple Variables | Vectorization part 2 --[Machine Learning |Andrew Ng]: I remember when I first learned about vectorization, I spent many hours on my computer taking an un-vectorized version of an algorithm, running it, see how long it ran, and then running a vectorized version of the code and seeing how much faster that ran. And I just spent hours playing with that. And it frankly blew my mind that the same algorithm, vectorized, would run so much faster. It felt almost like a magic trick to me. In this video, let's figure out how this magic trick really works. Let's take a deeper look at how a vectorized implementation may work on your computer behind the scenes. Let's look at this for loop. A for loop like this runs without vectorization. So if j ranges from 0 to, say, 15, this piece of code performs operations one after another. On the first time step, which I'm going to write as time 0 or t0, it first operates on the values at index 0. At the next time step, it calculates values corresponding to index 1 and so on until the fifth theme step, where it computes that. In other words, it calculates these computations one step at a time, one step after another. In contrast, this function in NumPy is implemented in the computer hardware with vectorization. So the computer can get all values of the vectors w and x, and in a single step, it multiplies each pair of w and x with each other all at the same time in parallel. Then after that, the computer takes these 16 numbers and uses specialized hardware to add them all together very efficiently, rather than needing to carry out distinct additions one after another to add up these 16 numbers. This means that code with vectorization can perform calculations in much less time than code without vectorization. And this matters more when you're running learning algorithms on large data sets or trying to train large models, which is often the case with machine learning. So that's why being able to write vectorized implementations of learning algorithms has been a key step to getting learning algorithms to run efficiently and therefore scale well to the large data sets that many modern machine learning algorithms now have to operate on. Now let's take a look at a concrete example of how this helps with implementing multiple linear regression, that is linear regression with multiple input features. Say you have a problem with 16 features and 16 parameters w1 through w16 in addition to the parameter b. You calculated 16 derivative terms for these 16 weights and in code, maybe you stored the values of w and d in two non-pyrotries with d storing the values of the derivatives. For this example, I'm just going to ignore the parameter b. Now you want to compute and update for each of these 16 parameters. So wj is updated to wj minus the learning rate, say 0.1 times dj for j from 1 through
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2.3 Linear Regression with Multiple Variables | Vectorization part 2 --[Machine Learning |Andrew Ng]
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https://www.youtube.com/watch?v=nRGG50GDNAA
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16. In code, without vectorization, you would be doing something like this, update w1 to be w1 minus the learning rate 0.1 times d1, next update w2 similarly and so on through w16, updated as w16 minus 0.1 times d16. In code without vectorization, you could use a full loop like this for j in range 0, 16, that again goes from 0 to 15, set wj equals wj minus 0.1 times dj. In contrast, with vectorization, you can imagine the computer's parallel processing hardware like this. It takes all 16 values in the vector w and subtracts in parallel 0.1 times all 16 values in the vector d and assign all 16 calculations back to w all at the same time and all in one step. In code, you can implement this as follows. W is assigned to w minus 0.1 times d. Behind the scenes, the computer takes these numpy arrays w and d and uses parallel processing hardware to carry out all 16 computations efficiently. So using a vectorized implementation, you should get a much more efficient implementation of linear regression. Maybe the speed difference won't be huge if you have 16 features, but if you have thousands of features and perhaps very large training sets, this type of vectorized implementation will make a huge difference in the running time of your learning algorithm. It could be the difference between code finishing in one or two minutes versus taking many, many hours to do the same thing. In the optional lab that follows this video, you see an introduction to one of the most used Python libraries in machine learning, which we've already touched on in this video called numpy. You see how they create vectors in code, and these vectors or lists of numbers are called numpy arrays. And you also see how to take the dot product of two vectors using a numpy function called dot. And you also get to see how vectorized code, such as using the dot function, can run much faster than a for loop. In fact, you get to time this code yourself and hopefully see it run much faster. This optional lab introduces a fair amount of new numpy syntax. So when you read through the optional lab, please don't feel like you have to understand all the code right away. But you can save this notebook and use it as a reference to look at when you're working with data stored in numpy arrays. So congrats on finishing this video on vectorization. You've learned one of the most important and useful techniques in implementing machine learning algorithms. In the next video, we'll put the math of multiple linear regression together with vectorization so that you really implement gradient descent for multiple linear regression with vectorization. Let's go on to the next video.
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2.3 Linear Regression with Multiple Variables | Vectorization part 2 --[Machine Learning |Andrew Ng]: 16. In code, without vectorization, you would be doing something like this, update w1 to be w1 minus the learning rate 0.1 times d1, next update w2 similarly and so on through w16, updated as w16 minus 0.1 times d16. In code without vectorization, you could use a full loop like this for j in range 0, 16, that again goes from 0 to 15, set wj equals wj minus 0.1 times dj. In contrast, with vectorization, you can imagine the computer's parallel processing hardware like this. It takes all 16 values in the vector w and subtracts in parallel 0.1 times all 16 values in the vector d and assign all 16 calculations back to w all at the same time and all in one step. In code, you can implement this as follows. W is assigned to w minus 0.1 times d. Behind the scenes, the computer takes these numpy arrays w and d and uses parallel processing hardware to carry out all 16 computations efficiently. So using a vectorized implementation, you should get a much more efficient implementation of linear regression. Maybe the speed difference won't be huge if you have 16 features, but if you have thousands of features and perhaps very large training sets, this type of vectorized implementation will make a huge difference in the running time of your learning algorithm. It could be the difference between code finishing in one or two minutes versus taking many, many hours to do the same thing. In the optional lab that follows this video, you see an introduction to one of the most used Python libraries in machine learning, which we've already touched on in this video called numpy. You see how they create vectors in code, and these vectors or lists of numbers are called numpy arrays. And you also see how to take the dot product of two vectors using a numpy function called dot. And you also get to see how vectorized code, such as using the dot function, can run much faster than a for loop. In fact, you get to time this code yourself and hopefully see it run much faster. This optional lab introduces a fair amount of new numpy syntax. So when you read through the optional lab, please don't feel like you have to understand all the code right away. But you can save this notebook and use it as a reference to look at when you're working with data stored in numpy arrays. So congrats on finishing this video on vectorization. You've learned one of the most important and useful techniques in implementing machine learning algorithms. In the next video, we'll put the math of multiple linear regression together with vectorization so that you really implement gradient descent for multiple linear regression with vectorization. Let's go on to the next video.
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2.4 Linear Regression with Multiple Variables|Gradient descent for multiple linear regression ---ML
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https://www.youtube.com/watch?v=odAhNw-e4o0
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So, you've learned about gradient descent, about multiple linear regression, and also vectorization. Let's put it all together to implement gradient descent for multiple linear regression with vectorization. This would be cool. Let's quickly review what multiple linear regression looks like. Using our previous notation, let's see how you can write it more succinctly using vector notation. We have parameters w1 to wn, as well as b, but instead of thinking of w1 to wn as separate numbers, that is, separate parameters, let's instead collect all of the w's into a vector w, so that now w is a vector of length n. So, we're just going to think of the parameters of this model as a vector w, as well as b, where b is still a number, same as before. For us before, we had defined multiple linear regression like this. Now using vector notation, we can write the model as f sub wb of x equals the vector w dot product with the vector x plus b. And remember that this dot here means dot product. Our cost function can be defined as j of w1 through wn comma b, but instead of just thinking of j as a function of these n different parameters wj as well as b, we're going to write j as a function of parameter vector w and the number b. So this w1 through wn is replaced by this vector w, and j now takes this input, a vector w and a number b, and returns a number. Here's what gradient descent looks like. We're going to repeatedly update each parameter wj to be wj minus alpha times the derivative of the cost j, where j has parameters w1 through wn and b, and once again we just write this as j of vector w and number b. Let's see what this will look like when you implement gradient descent, and in particular, let's take a look at the derivative term. You'll see that gradient descent becomes just a little bit different with multiple features compared to just one feature. Here's what we had when we had gradient descent with one feature. We had an update rule for w and a separate update rule for b. So hopefully these look familiar to you. And this term here is the derivative of the cost function j with respect to the parameter w. And similarly, we have an update rule for parameter b with univariate regression, we had only one feature. We call that feature xi without any subscript. Now here's the new notation for when we have n features where n is two or more. We get this update rule for gradient descent. We have a w1 to be w1 minus alpha times this expression here. And this formula is actually the derivative of the cost j with respect to w1. The formula for the derivative of j with respect to w1 on the right looks very similar to the case of
| 500
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2.4 Linear Regression with Multiple Variables|Gradient descent for multiple linear regression ---ML: So, you've learned about gradient descent, about multiple linear regression, and also vectorization. Let's put it all together to implement gradient descent for multiple linear regression with vectorization. This would be cool. Let's quickly review what multiple linear regression looks like. Using our previous notation, let's see how you can write it more succinctly using vector notation. We have parameters w1 to wn, as well as b, but instead of thinking of w1 to wn as separate numbers, that is, separate parameters, let's instead collect all of the w's into a vector w, so that now w is a vector of length n. So, we're just going to think of the parameters of this model as a vector w, as well as b, where b is still a number, same as before. For us before, we had defined multiple linear regression like this. Now using vector notation, we can write the model as f sub wb of x equals the vector w dot product with the vector x plus b. And remember that this dot here means dot product. Our cost function can be defined as j of w1 through wn comma b, but instead of just thinking of j as a function of these n different parameters wj as well as b, we're going to write j as a function of parameter vector w and the number b. So this w1 through wn is replaced by this vector w, and j now takes this input, a vector w and a number b, and returns a number. Here's what gradient descent looks like. We're going to repeatedly update each parameter wj to be wj minus alpha times the derivative of the cost j, where j has parameters w1 through wn and b, and once again we just write this as j of vector w and number b. Let's see what this will look like when you implement gradient descent, and in particular, let's take a look at the derivative term. You'll see that gradient descent becomes just a little bit different with multiple features compared to just one feature. Here's what we had when we had gradient descent with one feature. We had an update rule for w and a separate update rule for b. So hopefully these look familiar to you. And this term here is the derivative of the cost function j with respect to the parameter w. And similarly, we have an update rule for parameter b with univariate regression, we had only one feature. We call that feature xi without any subscript. Now here's the new notation for when we have n features where n is two or more. We get this update rule for gradient descent. We have a w1 to be w1 minus alpha times this expression here. And this formula is actually the derivative of the cost j with respect to w1. The formula for the derivative of j with respect to w1 on the right looks very similar to the case of
|
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2.4 Linear Regression with Multiple Variables|Gradient descent for multiple linear regression ---ML
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https://www.youtube.com/watch?v=odAhNw-e4o0
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one feature on the left. The error term still takes a prediction f of x minus the target y. One difference is that w and x are now vectors. And just as w on the left has now become w1 here on the right, xi here on the left is now instead xi subscript 1 here on the right. And this is just for j equals 1. For multiple linear regression, we have j ranging from 1 through n, and so we'll update the parameters w1, w2 all the way up to wn. And then as before, we'll update b. And if you implement this, you get gradient descent for multiple regression. So that's it for gradient descent for multiple regression. Before moving on from this video, I want to make a quick aside or a quick side note on an alternative way for finding w and b for linear regression. And this method is called the normal equation. Whereas it turns out gradient descent is a great method for minimizing the cost function j to find w and b. There is one other algorithm that works only for linear regression and pretty much none of the other algorithms you see in this specialization for solving for w and b. And this other method does not need an iterative gradient descent algorithm. Called the normal equation method, it turns out to be possible to use an advanced linear algebra library to just solve for w and b all in one go without iterations. Some disadvantages of the normal equation method are, first, unlike gradient descent, this does not generalize to other learning algorithms, such as the logistic regression algorithm that you learn about next week, or the neural networks or other algorithms you see later in this specialization. The normal equation method is also quite slow if the number of features n is large. Almost no machine learning practitioners should implement the normal equation method themselves. But if you're using a mature machine learning library and call linear regression, there's a chance that on the backend, it will be using this to solve for w and b. So if you're ever in a job interview and hear the term normal equation, that's what this refers to. Don't worry about the details of how the normal equation works. Just be aware that some machine learning libraries may use this complicated method in the backend to solve for w and b. But for most learning algorithms, including how you implement linear regression yourself, gradient descent is often a better way to get the job done. In the optional lab that follows this video, you see how to define a multiple regression model in code, and also how to calculate the prediction f of x. You also see how to calculate the cost and implement gradient descent for multiple linear regression model. This will be using Python's NumPy library, so if any of the code looks very new, that's okay. But you should
| 500
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2.4 Linear Regression with Multiple Variables|Gradient descent for multiple linear regression ---ML: one feature on the left. The error term still takes a prediction f of x minus the target y. One difference is that w and x are now vectors. And just as w on the left has now become w1 here on the right, xi here on the left is now instead xi subscript 1 here on the right. And this is just for j equals 1. For multiple linear regression, we have j ranging from 1 through n, and so we'll update the parameters w1, w2 all the way up to wn. And then as before, we'll update b. And if you implement this, you get gradient descent for multiple regression. So that's it for gradient descent for multiple regression. Before moving on from this video, I want to make a quick aside or a quick side note on an alternative way for finding w and b for linear regression. And this method is called the normal equation. Whereas it turns out gradient descent is a great method for minimizing the cost function j to find w and b. There is one other algorithm that works only for linear regression and pretty much none of the other algorithms you see in this specialization for solving for w and b. And this other method does not need an iterative gradient descent algorithm. Called the normal equation method, it turns out to be possible to use an advanced linear algebra library to just solve for w and b all in one go without iterations. Some disadvantages of the normal equation method are, first, unlike gradient descent, this does not generalize to other learning algorithms, such as the logistic regression algorithm that you learn about next week, or the neural networks or other algorithms you see later in this specialization. The normal equation method is also quite slow if the number of features n is large. Almost no machine learning practitioners should implement the normal equation method themselves. But if you're using a mature machine learning library and call linear regression, there's a chance that on the backend, it will be using this to solve for w and b. So if you're ever in a job interview and hear the term normal equation, that's what this refers to. Don't worry about the details of how the normal equation works. Just be aware that some machine learning libraries may use this complicated method in the backend to solve for w and b. But for most learning algorithms, including how you implement linear regression yourself, gradient descent is often a better way to get the job done. In the optional lab that follows this video, you see how to define a multiple regression model in code, and also how to calculate the prediction f of x. You also see how to calculate the cost and implement gradient descent for multiple linear regression model. This will be using Python's NumPy library, so if any of the code looks very new, that's okay. But you should
|
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2.4 Linear Regression with Multiple Variables|Gradient descent for multiple linear regression ---ML
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https://www.youtube.com/watch?v=odAhNw-e4o0
|
feel free also to take a look at the previous optional lab that introduces NumPy and vectorization for a refresher of NumPy functions and how to implement those in code. So that's it. You now know multiple linear regression. This is probably the single most widely used learning algorithm in the world today. But there's more. With just a few tricks such as picking and scaling features appropriately, and also choosing the learning rate alpha appropriately, you will be able to make this work much better. So just a few more videos to go for this week. Let's go on to the next video to see those little tricks that will help you make multiple linear regression work much better.
| 120
|
2.4 Linear Regression with Multiple Variables|Gradient descent for multiple linear regression ---ML: feel free also to take a look at the previous optional lab that introduces NumPy and vectorization for a refresher of NumPy functions and how to implement those in code. So that's it. You now know multiple linear regression. This is probably the single most widely used learning algorithm in the world today. But there's more. With just a few tricks such as picking and scaling features appropriately, and also choosing the learning rate alpha appropriately, you will be able to make this work much better. So just a few more videos to go for this week. Let's go on to the next video to see those little tricks that will help you make multiple linear regression work much better.
|
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2.5 Practical Tips for Linear Regression | Feature scaling part 1-- [Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=AbtWSXHPfS0
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So, welcome back. Let's take a look at some techniques that will make gradient descent work much better. In this video, you see a technique called feature scaling that will enable gradient descent to run much faster. Let's start by taking a look at the relationship between the size of a feature, that is, how big are the numbers for that feature, and the size of its associated parameter. As a concrete example, let's predict the price of a house using two features, x1, the size of the house, and x2, the number of bedrooms. Let's say that x1 typically ranges from 300 to 2000 square feet, and x2, in the dataset, ranges from 0 to 5 bedrooms. So for this example, x1 takes on a relatively large range of values, and x2 takes on a relatively small range of values. Now let's take an example of a house that has a size of 2000 square feet, has 5 bedrooms, and a price of 500k or $500,000. For this one trading example, what do you think are reasonable values for the size of parameters w1 and w2? Well, let's look at one possible set of parameters. Say w1 is 50, and w2 is 0.1, and b is 50 for the purposes of discussion. So in this case, the estimated price in thousands of dollars is 100,000k here, plus 0.5k, plus 50k, which is slightly over $100 million. So that's clearly very far from the actual price of $500,000. And so this is not a very good set of parameter choices for w1 and w2. Now let's take a look at another possibility. Say w1 and w2 were the other way around. w1 is 0.1, and w2 is 50, and b is still also 50. In this choice of w1 and w2, w1 is relatively small and w2 is relatively large. 50 is much bigger than 0.1. So here the predicted price is 0.1 times 2000 plus 50 times 5 plus 50. The first term becomes 200k, the second term becomes 250k, and then plus 50. So this version of the model predicts a price of $500,000, which is a much more reasonable estimate and happens to be the same price as the true price of the house. So hopefully you might notice that when a possible range of values of a feature is large, like the size in square feet, which goes all the way up to 2000, it's more likely that a good model will learn to choose a relatively small parameter value like 0.1. Likewise, when the possible values of a feature are small, like the number of bedrooms, then a reasonable value for its parameters will be relatively large, like 50. So how does this relate to gradient descent? Well, let's take a look at a scatter plot of the features, where the size in square feet is the horizontal axis, x1, and the number of bedrooms, x2, is on the vertical axis. If you plot the
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2.5 Practical Tips for Linear Regression | Feature scaling part 1-- [Machine Learning | Andrew Ng]: So, welcome back. Let's take a look at some techniques that will make gradient descent work much better. In this video, you see a technique called feature scaling that will enable gradient descent to run much faster. Let's start by taking a look at the relationship between the size of a feature, that is, how big are the numbers for that feature, and the size of its associated parameter. As a concrete example, let's predict the price of a house using two features, x1, the size of the house, and x2, the number of bedrooms. Let's say that x1 typically ranges from 300 to 2000 square feet, and x2, in the dataset, ranges from 0 to 5 bedrooms. So for this example, x1 takes on a relatively large range of values, and x2 takes on a relatively small range of values. Now let's take an example of a house that has a size of 2000 square feet, has 5 bedrooms, and a price of 500k or $500,000. For this one trading example, what do you think are reasonable values for the size of parameters w1 and w2? Well, let's look at one possible set of parameters. Say w1 is 50, and w2 is 0.1, and b is 50 for the purposes of discussion. So in this case, the estimated price in thousands of dollars is 100,000k here, plus 0.5k, plus 50k, which is slightly over $100 million. So that's clearly very far from the actual price of $500,000. And so this is not a very good set of parameter choices for w1 and w2. Now let's take a look at another possibility. Say w1 and w2 were the other way around. w1 is 0.1, and w2 is 50, and b is still also 50. In this choice of w1 and w2, w1 is relatively small and w2 is relatively large. 50 is much bigger than 0.1. So here the predicted price is 0.1 times 2000 plus 50 times 5 plus 50. The first term becomes 200k, the second term becomes 250k, and then plus 50. So this version of the model predicts a price of $500,000, which is a much more reasonable estimate and happens to be the same price as the true price of the house. So hopefully you might notice that when a possible range of values of a feature is large, like the size in square feet, which goes all the way up to 2000, it's more likely that a good model will learn to choose a relatively small parameter value like 0.1. Likewise, when the possible values of a feature are small, like the number of bedrooms, then a reasonable value for its parameters will be relatively large, like 50. So how does this relate to gradient descent? Well, let's take a look at a scatter plot of the features, where the size in square feet is the horizontal axis, x1, and the number of bedrooms, x2, is on the vertical axis. If you plot the
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2.5 Practical Tips for Linear Regression | Feature scaling part 1-- [Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=AbtWSXHPfS0
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training data, you notice that the horizontal axis is on a much larger scale or much larger range of values compared to the vertical axis. Next, let's look at how the cost function might look in a contour plot. You might see a contour plot where the horizontal axis has a much narrower range, say between 0 and 1, whereas the vertical axis takes on much larger values, say between 10 and 100. So the contours form ovals or ellipses, and they're shorter on one side and longer on the other. And this is because a very small change to w1 can have a very large impact on the estimated price and thus a very large impact on the cost j, because w1 tends to be multiplied by a very large number, the size in square feet. In contrast, it takes a much larger change in w2 in order to change the predictions much, and thus small changes to w2 don't change the cost function nearly as much. So where does this leave us? This is what might end up happening if you were to run gradient descent, if you were to use your training data as is. Because the contours are so tall and skinny, gradient descent may end up bouncing back and forth for a long time before it can finally find its way to the global minimum. In situations like this, a useful thing to do is to scale the features. This means performing some transformation of your training data so that x1, say, might now range from 0 to 1, and x2 might also range from 0 to 1. So the data points now look more like this, and you might notice that the scale of the plot on the bottom is now quite different than the one on top. The key point is that the rescaled x1 and x2 are both now taking comparable ranges of values to each other. And if you run gradient descent on a cost function defined on this rescaled x1 and x2 using this transformed data, then the contours will look more like this, more like circles and less tall and skinny, and gradient descent can find a much more direct path to the global minimum. So to recap, when you have different features that take on very different ranges of values, you can cause gradient descent to run slowly, but rescaling the different features so they all take on a comparable range of values can speed up gradient descent significantly. How do you actually do this? Let's take a look at that in the next video.
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2.5 Practical Tips for Linear Regression | Feature scaling part 1-- [Machine Learning | Andrew Ng]: training data, you notice that the horizontal axis is on a much larger scale or much larger range of values compared to the vertical axis. Next, let's look at how the cost function might look in a contour plot. You might see a contour plot where the horizontal axis has a much narrower range, say between 0 and 1, whereas the vertical axis takes on much larger values, say between 10 and 100. So the contours form ovals or ellipses, and they're shorter on one side and longer on the other. And this is because a very small change to w1 can have a very large impact on the estimated price and thus a very large impact on the cost j, because w1 tends to be multiplied by a very large number, the size in square feet. In contrast, it takes a much larger change in w2 in order to change the predictions much, and thus small changes to w2 don't change the cost function nearly as much. So where does this leave us? This is what might end up happening if you were to run gradient descent, if you were to use your training data as is. Because the contours are so tall and skinny, gradient descent may end up bouncing back and forth for a long time before it can finally find its way to the global minimum. In situations like this, a useful thing to do is to scale the features. This means performing some transformation of your training data so that x1, say, might now range from 0 to 1, and x2 might also range from 0 to 1. So the data points now look more like this, and you might notice that the scale of the plot on the bottom is now quite different than the one on top. The key point is that the rescaled x1 and x2 are both now taking comparable ranges of values to each other. And if you run gradient descent on a cost function defined on this rescaled x1 and x2 using this transformed data, then the contours will look more like this, more like circles and less tall and skinny, and gradient descent can find a much more direct path to the global minimum. So to recap, when you have different features that take on very different ranges of values, you can cause gradient descent to run slowly, but rescaling the different features so they all take on a comparable range of values can speed up gradient descent significantly. How do you actually do this? Let's take a look at that in the next video.
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2.6 Practical Tips for Linear Regression | Feature scaling part 2-- [Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=g9bkFTnM-7k
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Let's look at how you can implement feature scaling to take features that take on very different ranges of values and scale them to have comparable ranges of value to each other. So how do you actually scale features? Well, if X1 ranges from 3 to 2000, one way to get a scale version of X1 is to take each original X1 value and divide by 2000 the maximum of the range. So the scale X1 will range from 0.15 up to 1. Similarly, since X2 ranges from 0 to 5, you can calculate a scale version of X2 by taking each original X2 and dividing by 5, which is again the maximum. So the scale X2 will now range from 0 to 1. So if you plot the scaled X1 and X2 on a graph, it might look like this. In addition to dividing by the maximum, you can also do what's called mean normalization. So what this looks like is, you start with the original features and then you rescale them so that both of them are centered around 0. So whereas before they only had values greater than 0, now they have both negative and positive values, but maybe usually between negative 1 and plus 1. So to calculate the mean normalization of X1, first find the average, also called the mean of X1 on your training set, and let's call this mean mu1, with this being the Greek alphabet mu. For example, you may find that the average of feature 1, mu1, is 600 square feet. So let's take each X1, subtract the mean mu1, and then let's divide by the difference 2000 minus 300, where 2000 is the maximum and 300 the minimum. And if you do this, you get the normalized X1 to range from negative 0.18 to 0.82. Similarly to mean normalize X2, you can calculate the average of feature 2, and for instance mu2 may be 2.3. Then you can take each X2, subtract mu2, and divide by 5 minus 0, again the max 5 minus the min which is 0. The mean normalized X2 now ranges from negative 0.46 to 0.54, so if you plot the training data using the mean normalized X1 and X2, it might look like this. There's one last common rescaling method called z-score normalization. To implement z-score normalization, you need to calculate something called the standard deviation of each feature. If you don't know what a standard deviation is, don't worry about it, you won't need to know it for this class. Or if you've heard of the normal distribution or the bell-shaped curve, sometimes also called the Gaussian distribution, this is what the standard deviation for the normal distribution looks like. But if you haven't heard of this, you don't need to worry about that either. But if you do know what is the standard deviation, then to implement a z-score normalization, you first calculate the mean mu as well as the standard deviation, which
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2.6 Practical Tips for Linear Regression | Feature scaling part 2-- [Machine Learning | Andrew Ng]: Let's look at how you can implement feature scaling to take features that take on very different ranges of values and scale them to have comparable ranges of value to each other. So how do you actually scale features? Well, if X1 ranges from 3 to 2000, one way to get a scale version of X1 is to take each original X1 value and divide by 2000 the maximum of the range. So the scale X1 will range from 0.15 up to 1. Similarly, since X2 ranges from 0 to 5, you can calculate a scale version of X2 by taking each original X2 and dividing by 5, which is again the maximum. So the scale X2 will now range from 0 to 1. So if you plot the scaled X1 and X2 on a graph, it might look like this. In addition to dividing by the maximum, you can also do what's called mean normalization. So what this looks like is, you start with the original features and then you rescale them so that both of them are centered around 0. So whereas before they only had values greater than 0, now they have both negative and positive values, but maybe usually between negative 1 and plus 1. So to calculate the mean normalization of X1, first find the average, also called the mean of X1 on your training set, and let's call this mean mu1, with this being the Greek alphabet mu. For example, you may find that the average of feature 1, mu1, is 600 square feet. So let's take each X1, subtract the mean mu1, and then let's divide by the difference 2000 minus 300, where 2000 is the maximum and 300 the minimum. And if you do this, you get the normalized X1 to range from negative 0.18 to 0.82. Similarly to mean normalize X2, you can calculate the average of feature 2, and for instance mu2 may be 2.3. Then you can take each X2, subtract mu2, and divide by 5 minus 0, again the max 5 minus the min which is 0. The mean normalized X2 now ranges from negative 0.46 to 0.54, so if you plot the training data using the mean normalized X1 and X2, it might look like this. There's one last common rescaling method called z-score normalization. To implement z-score normalization, you need to calculate something called the standard deviation of each feature. If you don't know what a standard deviation is, don't worry about it, you won't need to know it for this class. Or if you've heard of the normal distribution or the bell-shaped curve, sometimes also called the Gaussian distribution, this is what the standard deviation for the normal distribution looks like. But if you haven't heard of this, you don't need to worry about that either. But if you do know what is the standard deviation, then to implement a z-score normalization, you first calculate the mean mu as well as the standard deviation, which
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2.6 Practical Tips for Linear Regression | Feature scaling part 2-- [Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=g9bkFTnM-7k
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is often denoted by the lowercase Greek alphabet sigma of each feature. So for instance, maybe feature 1 has a standard deviation of 450 and mean 600, then to z-score normalize x1, take each x1, subtract mu 1, and then divide by the standard deviation, which I'm going to denote as sigma 1. And what you might find is that the z-score normalize x1 now ranges from negative 0.67 to 3.1. Similarly, if you calculate the second feature's standard deviation to be 1.4 and mean to be 2.3, then you can compute x2 minus mu 2 divided by sigma 2. And in this case, the z-score normalize by x2 might now range from negative 1.6 to 1.9. So if you plot the training data on the normalized x1 and x2 on a graph, it might look like this. As a rule of thumb, when performing feature scaling, you might want to aim for getting the features to range from maybe anywhere around negative 1 to somewhere around plus 1 for each feature x. But these values, negative 1 and plus 1 can be a little bit loose. So if the features range from negative 3 to plus 3, or negative 0.3 to plus 0.3, all of these are completely okay. So if you have a feature x1 that winds up being between 0 and 3, that's not a problem. And you can rescale it if you want, but if you don't rescale it, it should work okay too. Or if you have a different feature, x2, whose values are between negative 2 and plus 0.5, again, that's okay. No harm rescaling it, but it might be okay if you leave it alone as well. But if another feature like x3 here ranges from negative 100 to plus 100, then this takes on a very different range of values than something from around negative 1 to plus 1. So you're probably better off rescaling this feature x3 so that it ranges from something closer to negative 1 to plus 1. Similarly, if you have a feature x4 that takes on really small values, say between negative 0.001 and plus 0.001, then these values are so small, that means you may want to rescale it as well. Finally, what if your feature x5, such as measurements of a hospital patient's body temperature, ranges from 98.6 to 105 degrees Fahrenheit? In this case, these values are around 100, which is actually pretty large compared to other scale features, and this will actually cause gradient descent to run more slowly. So in this case, feature rescaling will likely help. There's almost never any harm to carrying out feature rescaling, so when in doubt, I encourage you to just carry it out. And that's it for feature scaling. With this little technique, you'll often be able to get gradient descent to run much faster. So that's feature scaling. And with or without feature scaling, when you run gradient descent, how can you know, how can
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2.6 Practical Tips for Linear Regression | Feature scaling part 2-- [Machine Learning | Andrew Ng]: is often denoted by the lowercase Greek alphabet sigma of each feature. So for instance, maybe feature 1 has a standard deviation of 450 and mean 600, then to z-score normalize x1, take each x1, subtract mu 1, and then divide by the standard deviation, which I'm going to denote as sigma 1. And what you might find is that the z-score normalize x1 now ranges from negative 0.67 to 3.1. Similarly, if you calculate the second feature's standard deviation to be 1.4 and mean to be 2.3, then you can compute x2 minus mu 2 divided by sigma 2. And in this case, the z-score normalize by x2 might now range from negative 1.6 to 1.9. So if you plot the training data on the normalized x1 and x2 on a graph, it might look like this. As a rule of thumb, when performing feature scaling, you might want to aim for getting the features to range from maybe anywhere around negative 1 to somewhere around plus 1 for each feature x. But these values, negative 1 and plus 1 can be a little bit loose. So if the features range from negative 3 to plus 3, or negative 0.3 to plus 0.3, all of these are completely okay. So if you have a feature x1 that winds up being between 0 and 3, that's not a problem. And you can rescale it if you want, but if you don't rescale it, it should work okay too. Or if you have a different feature, x2, whose values are between negative 2 and plus 0.5, again, that's okay. No harm rescaling it, but it might be okay if you leave it alone as well. But if another feature like x3 here ranges from negative 100 to plus 100, then this takes on a very different range of values than something from around negative 1 to plus 1. So you're probably better off rescaling this feature x3 so that it ranges from something closer to negative 1 to plus 1. Similarly, if you have a feature x4 that takes on really small values, say between negative 0.001 and plus 0.001, then these values are so small, that means you may want to rescale it as well. Finally, what if your feature x5, such as measurements of a hospital patient's body temperature, ranges from 98.6 to 105 degrees Fahrenheit? In this case, these values are around 100, which is actually pretty large compared to other scale features, and this will actually cause gradient descent to run more slowly. So in this case, feature rescaling will likely help. There's almost never any harm to carrying out feature rescaling, so when in doubt, I encourage you to just carry it out. And that's it for feature scaling. With this little technique, you'll often be able to get gradient descent to run much faster. So that's feature scaling. And with or without feature scaling, when you run gradient descent, how can you know, how can
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2.6 Practical Tips for Linear Regression | Feature scaling part 2-- [Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=g9bkFTnM-7k
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you check if gradient descent is really working, if it is finding you the global minimum or something close to it? In the next video, let's take a look at how to recognize if gradient descent is converging. And then in the video after that, this will lead to discussion of how to choose a good learning rate for gradient descent.
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2.6 Practical Tips for Linear Regression | Feature scaling part 2-- [Machine Learning | Andrew Ng]: you check if gradient descent is really working, if it is finding you the global minimum or something close to it? In the next video, let's take a look at how to recognize if gradient descent is converging. And then in the video after that, this will lead to discussion of how to choose a good learning rate for gradient descent.
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2.7 Practical Tips for Linear Regression | Checking gradient descent for convergence -- ML Andrew Ng
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https://www.youtube.com/watch?v=9w3HRwTRVUE
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When running gradient descent, how can you tell if it is converging? That is, whether it's helping you to find parameters close to the global minimum of the cost function. By learning to recognize what a well-running implementation of gradient descent looks like, we will also, in a later video, be better able to choose a good learning rate alpha. Let's take a look. As a reminder, here's the gradient descent rule, and one of the key choices is the choice of the learning rate alpha. Here's something that I often do to make sure that gradient descent is working well. Recall that the job of gradient descent is to find parameters w and b that hopefully minimize the cost function j. So what I'll often do is plot the cost function j, which is calculated on the training set, and I'll plot the value of j at each iteration of gradient descent. Remember that each iteration means after each simultaneous update of the parameters w and b, so in this plot, the horizontal axis is the number of iterations of gradient descent that you've run so far. And so you may get a curve that looks like this. Notice that the horizontal axis is the number of iterations of gradient descent, and not a parameter like w or b. This differs from previous graphs you've seen, where the vertical axis was cost j, and the horizontal axis was a single parameter like w or b. This curve is also called a learning curve. Note that there are a few different types of learning curves used in machine learning, and you'll see some of the types later in this course as well. Secondly, if you look at this point on the curve, this means that after you've run gradient descent for 100 iterations, meaning 100 simultaneous updates of the parameters, you have some learned values for w and b. And if you compute the cost, jwb, for those values of w and b, the ones you got after 100 iterations, you get this value for the cost j, that is, this point on the vertical axis. And this point here corresponds to the value of j for the parameters that you got after 200 iterations of gradient descent. So looking at this graph helps you to see how your cost j changes after each iteration of gradient descent. If gradient descent is working properly, then the cost j should decrease after every single iteration. If j ever increases after one iteration, that means either alpha is chosen poorly, and it usually means alpha is too large, or there could be a bug in the code. Another useful thing that this plot can tell you is that if you look at this curve, by the time you reach maybe 300 iterations or so, the cost j is leveling off and is no longer decreasing much. And by 400 iterations, it looks like the curve has flattened out. So this means that
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2.7 Practical Tips for Linear Regression | Checking gradient descent for convergence -- ML Andrew Ng: When running gradient descent, how can you tell if it is converging? That is, whether it's helping you to find parameters close to the global minimum of the cost function. By learning to recognize what a well-running implementation of gradient descent looks like, we will also, in a later video, be better able to choose a good learning rate alpha. Let's take a look. As a reminder, here's the gradient descent rule, and one of the key choices is the choice of the learning rate alpha. Here's something that I often do to make sure that gradient descent is working well. Recall that the job of gradient descent is to find parameters w and b that hopefully minimize the cost function j. So what I'll often do is plot the cost function j, which is calculated on the training set, and I'll plot the value of j at each iteration of gradient descent. Remember that each iteration means after each simultaneous update of the parameters w and b, so in this plot, the horizontal axis is the number of iterations of gradient descent that you've run so far. And so you may get a curve that looks like this. Notice that the horizontal axis is the number of iterations of gradient descent, and not a parameter like w or b. This differs from previous graphs you've seen, where the vertical axis was cost j, and the horizontal axis was a single parameter like w or b. This curve is also called a learning curve. Note that there are a few different types of learning curves used in machine learning, and you'll see some of the types later in this course as well. Secondly, if you look at this point on the curve, this means that after you've run gradient descent for 100 iterations, meaning 100 simultaneous updates of the parameters, you have some learned values for w and b. And if you compute the cost, jwb, for those values of w and b, the ones you got after 100 iterations, you get this value for the cost j, that is, this point on the vertical axis. And this point here corresponds to the value of j for the parameters that you got after 200 iterations of gradient descent. So looking at this graph helps you to see how your cost j changes after each iteration of gradient descent. If gradient descent is working properly, then the cost j should decrease after every single iteration. If j ever increases after one iteration, that means either alpha is chosen poorly, and it usually means alpha is too large, or there could be a bug in the code. Another useful thing that this plot can tell you is that if you look at this curve, by the time you reach maybe 300 iterations or so, the cost j is leveling off and is no longer decreasing much. And by 400 iterations, it looks like the curve has flattened out. So this means that
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2.7 Practical Tips for Linear Regression | Checking gradient descent for convergence -- ML Andrew Ng
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https://www.youtube.com/watch?v=9w3HRwTRVUE
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gradient descent has more or less converged because the curve is no longer decreasing. So looking at this learning curve, you can try to spot whether or not gradient descent is converging. By the way, the number of iterations that gradient descent takes to converge can vary a lot between different applications. In one application, it may converge after just 30 iterations. For a different application, it could take a thousand or a hundred thousand iterations. It turns out to be very difficult to tell in advance how many iterations gradient descent needs to converge, which is why you can create a graph like this, a learning curve, to try to find out when you can stop training your particular model. Another way to decide when your model is done training is with an automatic convergence test. So let's let epsilon, this here is the Greek alphabet epsilon, let's let epsilon be a variable representing a small number, such as 0.001 or 10 to the power of negative three. If the cost J decreases by less than this number epsilon on one iteration, then you're likely on this flattened part of the curve that you see on the left and you can declare convergence. Remember, convergence hopefully indicates that you found parameters W and B that are close to the minimum possible value of J. I usually find that choosing the right threshold epsilon is pretty difficult. So I actually tend to look at graphs like this one on the left, rather than rely on automatic convergence tests. Looking at this other figure can tell you, I'll give you some advanced warning if maybe gradient descent is not working correctly as well. So you've now seen what the learning curve should look like when gradient descent is running well. Let's take these insights and in the next video, take a look at how to choose an appropriate learning rate.
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2.7 Practical Tips for Linear Regression | Checking gradient descent for convergence -- ML Andrew Ng: gradient descent has more or less converged because the curve is no longer decreasing. So looking at this learning curve, you can try to spot whether or not gradient descent is converging. By the way, the number of iterations that gradient descent takes to converge can vary a lot between different applications. In one application, it may converge after just 30 iterations. For a different application, it could take a thousand or a hundred thousand iterations. It turns out to be very difficult to tell in advance how many iterations gradient descent needs to converge, which is why you can create a graph like this, a learning curve, to try to find out when you can stop training your particular model. Another way to decide when your model is done training is with an automatic convergence test. So let's let epsilon, this here is the Greek alphabet epsilon, let's let epsilon be a variable representing a small number, such as 0.001 or 10 to the power of negative three. If the cost J decreases by less than this number epsilon on one iteration, then you're likely on this flattened part of the curve that you see on the left and you can declare convergence. Remember, convergence hopefully indicates that you found parameters W and B that are close to the minimum possible value of J. I usually find that choosing the right threshold epsilon is pretty difficult. So I actually tend to look at graphs like this one on the left, rather than rely on automatic convergence tests. Looking at this other figure can tell you, I'll give you some advanced warning if maybe gradient descent is not working correctly as well. So you've now seen what the learning curve should look like when gradient descent is running well. Let's take these insights and in the next video, take a look at how to choose an appropriate learning rate.
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2.8 Practical Tips for Linear Regression | | Choosing the learning rate-[MachineLearning|Andrew Ng]
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https://www.youtube.com/watch?v=j4uDxRNjYlA
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Your learning algorithm will run much better with an appropriate choice of learning rate. If it's too small, it will run very slowly, and if it's too large, it may not even converge. Let's take a look at how you can choose a good learning rate for your model. Concretely, if you plot the cost for a number of iterations and notice that the cost sometimes goes up and sometimes goes down, you should take that as a clear sign that gradient descent is not working properly. This could mean that there's a bug in the code, or sometimes it could mean that your learning rate is too large. So here's an illustration of what might be happening. Here the vertical axis is a cost function J, and the horizontal axis represents a parameter like maybe W1, and if the learning rate is too big, then if you start off here, your update step may overshoot the minimum and end up here, and in the next update step here, you're again overshooting, so you end up here, and so on. And that's why the cost can sometimes go up instead of decreasing. To fix this, you can use a smaller learning rate. So then your updates may start here and go down a little bit and down a bit, and will hopefully consistently decrease until it reaches the global minimum. Sometimes you may see that the cost consistently increases after each iteration, like this curve here. This is also likely due to a learning rate that is too large, and it could be addressed by choosing a smaller learning rate. But learning rates like this could also be a sign of a possible bug in the code. For example, if I wrote my code so that W1 gets updated as W1 plus alpha times this derivative term, this could result in the cost consistently increasing at each iteration. And this is because adding the derivative term moves your cost j further from the global minimum instead of close up. So remember, you want to use a minus sign, so the code should be updated W1, updated by W1 minus alpha times the derivative term. One debugging tip for a correct implementation of gradient descent is that with a small enough learning rate, the cost function should decrease on every single iteration. So if gradient descent isn't working, one thing I will often do, and I hope you find this tip useful too, one thing I'll often do is just set alpha to be a very, very small number and see if that causes the cost to decrease on every iteration. If even with alpha set to a very small number, j doesn't decrease on every single iteration, but instead sometimes increases, then that usually means there's a bug somewhere in the code. Note that setting alpha to be really, really small is meant here as a debugging step, and a very, very small value of alpha is not going to
| 500
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2.8 Practical Tips for Linear Regression | | Choosing the learning rate-[MachineLearning|Andrew Ng]: Your learning algorithm will run much better with an appropriate choice of learning rate. If it's too small, it will run very slowly, and if it's too large, it may not even converge. Let's take a look at how you can choose a good learning rate for your model. Concretely, if you plot the cost for a number of iterations and notice that the cost sometimes goes up and sometimes goes down, you should take that as a clear sign that gradient descent is not working properly. This could mean that there's a bug in the code, or sometimes it could mean that your learning rate is too large. So here's an illustration of what might be happening. Here the vertical axis is a cost function J, and the horizontal axis represents a parameter like maybe W1, and if the learning rate is too big, then if you start off here, your update step may overshoot the minimum and end up here, and in the next update step here, you're again overshooting, so you end up here, and so on. And that's why the cost can sometimes go up instead of decreasing. To fix this, you can use a smaller learning rate. So then your updates may start here and go down a little bit and down a bit, and will hopefully consistently decrease until it reaches the global minimum. Sometimes you may see that the cost consistently increases after each iteration, like this curve here. This is also likely due to a learning rate that is too large, and it could be addressed by choosing a smaller learning rate. But learning rates like this could also be a sign of a possible bug in the code. For example, if I wrote my code so that W1 gets updated as W1 plus alpha times this derivative term, this could result in the cost consistently increasing at each iteration. And this is because adding the derivative term moves your cost j further from the global minimum instead of close up. So remember, you want to use a minus sign, so the code should be updated W1, updated by W1 minus alpha times the derivative term. One debugging tip for a correct implementation of gradient descent is that with a small enough learning rate, the cost function should decrease on every single iteration. So if gradient descent isn't working, one thing I will often do, and I hope you find this tip useful too, one thing I'll often do is just set alpha to be a very, very small number and see if that causes the cost to decrease on every iteration. If even with alpha set to a very small number, j doesn't decrease on every single iteration, but instead sometimes increases, then that usually means there's a bug somewhere in the code. Note that setting alpha to be really, really small is meant here as a debugging step, and a very, very small value of alpha is not going to
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2.8 Practical Tips for Linear Regression | | Choosing the learning rate-[MachineLearning|Andrew Ng]
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https://www.youtube.com/watch?v=j4uDxRNjYlA
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be the most efficient choice for actually training your learning algorithm. One important trade off is that if your learning rate is too small, then gradient descent can take a lot of iterations to converge. So when I am running gradient descent, I will usually try a range of values for the learning rate alpha. So I might start by trying a learning rate of 0.001, and I might also try a learning rate that's 10 times as large, say 0.01 and 0.1 and so on. And for each choice of alpha, you might run gradient descent just for a handful of iterations and plot the cost function j as a function of the number of iterations. And after trying a few different values, you might then pick the value of alpha that seems to decrease the learning rate rapidly, but also consistently. In fact, what I actually do is try a range of values like this. After trying 0.001, I'll then increase the learning rate threefold to 0.003. And after that, I'll try 0.01, which is again about three times as large as 0.003. So these are roughly trying out gradient descent with each value of alpha being roughly three times bigger than the previous value. So what I'll do is try a range of values until I found a value that's too small. And then also make sure I found a value that's too large. And I'll slowly try to pick the largest possible learning rate or just something slightly smaller than the largest reasonable value that I found. And when I do that, it usually gives me a good learning rate for my model. So I hope this technique too will be useful for you to choose a good learning rate for your implementation of gradient descent. In the upcoming optional lab, you can also take a look at how feature scaling is done in code and also see how different choices of the learning rate alpha can lead to either better or worse training of your model. I hope you have fun playing with the value of alpha and seeing the outcomes of different choices of alpha. So please take a look and run the code in the optional lab to gain a deeper intuition about feature scaling as well as the learning rate alpha. Training learning rates is an important part of training many learning algorithms. And I hope that this video gives you intuition about different choices and how to pick a good value for alpha. Now there are a couple more ideas they can use to make multiple linear regression much more powerful. And that is choosing custom features, which will also allow you to fit curves, not just a straight line to your data. Let's take a look at that in the next video.
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2.8 Practical Tips for Linear Regression | | Choosing the learning rate-[MachineLearning|Andrew Ng]: be the most efficient choice for actually training your learning algorithm. One important trade off is that if your learning rate is too small, then gradient descent can take a lot of iterations to converge. So when I am running gradient descent, I will usually try a range of values for the learning rate alpha. So I might start by trying a learning rate of 0.001, and I might also try a learning rate that's 10 times as large, say 0.01 and 0.1 and so on. And for each choice of alpha, you might run gradient descent just for a handful of iterations and plot the cost function j as a function of the number of iterations. And after trying a few different values, you might then pick the value of alpha that seems to decrease the learning rate rapidly, but also consistently. In fact, what I actually do is try a range of values like this. After trying 0.001, I'll then increase the learning rate threefold to 0.003. And after that, I'll try 0.01, which is again about three times as large as 0.003. So these are roughly trying out gradient descent with each value of alpha being roughly three times bigger than the previous value. So what I'll do is try a range of values until I found a value that's too small. And then also make sure I found a value that's too large. And I'll slowly try to pick the largest possible learning rate or just something slightly smaller than the largest reasonable value that I found. And when I do that, it usually gives me a good learning rate for my model. So I hope this technique too will be useful for you to choose a good learning rate for your implementation of gradient descent. In the upcoming optional lab, you can also take a look at how feature scaling is done in code and also see how different choices of the learning rate alpha can lead to either better or worse training of your model. I hope you have fun playing with the value of alpha and seeing the outcomes of different choices of alpha. So please take a look and run the code in the optional lab to gain a deeper intuition about feature scaling as well as the learning rate alpha. Training learning rates is an important part of training many learning algorithms. And I hope that this video gives you intuition about different choices and how to pick a good value for alpha. Now there are a couple more ideas they can use to make multiple linear regression much more powerful. And that is choosing custom features, which will also allow you to fit curves, not just a straight line to your data. Let's take a look at that in the next video.
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2.9 Practical Tips for Linear Regression | Feature engineering --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=sO_JcsxQRz8
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The choice of features can have a huge impact on your learning algorithm's performance. In fact, for many practical applications, choosing or entering the right features is a critical step to making the algorithm work well. In this video, let's take a look at how you can choose or engineer the most appropriate features for your learning algorithm. Let's take a look at feature engineering by revisiting the example of predicting the price of a house. Say you have two features for each house. X1 is the width of the lot size of the plot of land that the house is built on. This in real estate is also called the frontage of the lot. And the second feature, X2, is the depth of the lot size of, let's assume, the rectangular plot of land that the house was built on. Given these two features, X1 and X2, you might build a model like this, where f of x is w1 x1 plus w2 x2 plus b, where x1 is the frontage or width and x2 is the depth. And this model might work okay. But here's another option for how you might choose a different way to use these features in the model that could be even more effective. You might notice that the area of the land can be calculated as the frontage or width times the depth. And you may have an intuition that the area of the land is more predictive of the price than the frontage and depth as separate features. So you might define a new feature, X3, as X1 times X2. So this new feature X3 is equal to the area of the plot of land. For this feature, you can then have a model fwb of x equals w1 x1 plus w2 x2 plus w3 x3 plus b, so that the model can now choose parameters w1, w2, and w3, depending on whether the data shows that the frontage or the depth or the area x3 of the lot turns out to be the most important thing for predicting the price of the house. What we just did, creating a new feature, is an example of what's called feature engineering, in which you might use your knowledge or intuition about the problem to design new features, usually by transforming or combining the original features of the problem in order to make it easier for the learning algorithm to make accurate predictions. So depending on what insights you may have into the application, rather than just taking the features that you happen to have started off with, sometimes by defining new features, you might be able to get a much better model. So that's feature engineering. And it turns out that there's one flavor of feature engineering that will allow you to fit not just straight lines, but curves, nonlinear functions to your data. Let's take a look in the next video at how you can do that.
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2.9 Practical Tips for Linear Regression | Feature engineering --[Machine Learning | Andrew Ng]: The choice of features can have a huge impact on your learning algorithm's performance. In fact, for many practical applications, choosing or entering the right features is a critical step to making the algorithm work well. In this video, let's take a look at how you can choose or engineer the most appropriate features for your learning algorithm. Let's take a look at feature engineering by revisiting the example of predicting the price of a house. Say you have two features for each house. X1 is the width of the lot size of the plot of land that the house is built on. This in real estate is also called the frontage of the lot. And the second feature, X2, is the depth of the lot size of, let's assume, the rectangular plot of land that the house was built on. Given these two features, X1 and X2, you might build a model like this, where f of x is w1 x1 plus w2 x2 plus b, where x1 is the frontage or width and x2 is the depth. And this model might work okay. But here's another option for how you might choose a different way to use these features in the model that could be even more effective. You might notice that the area of the land can be calculated as the frontage or width times the depth. And you may have an intuition that the area of the land is more predictive of the price than the frontage and depth as separate features. So you might define a new feature, X3, as X1 times X2. So this new feature X3 is equal to the area of the plot of land. For this feature, you can then have a model fwb of x equals w1 x1 plus w2 x2 plus w3 x3 plus b, so that the model can now choose parameters w1, w2, and w3, depending on whether the data shows that the frontage or the depth or the area x3 of the lot turns out to be the most important thing for predicting the price of the house. What we just did, creating a new feature, is an example of what's called feature engineering, in which you might use your knowledge or intuition about the problem to design new features, usually by transforming or combining the original features of the problem in order to make it easier for the learning algorithm to make accurate predictions. So depending on what insights you may have into the application, rather than just taking the features that you happen to have started off with, sometimes by defining new features, you might be able to get a much better model. So that's feature engineering. And it turns out that there's one flavor of feature engineering that will allow you to fit not just straight lines, but curves, nonlinear functions to your data. Let's take a look in the next video at how you can do that.
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2.10 Practical Tips for Linear Regression | Polynomial regression --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=UobsU2oQwBA
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So far, we've just been fitting straight lines to our data. Let's take the ideas of multiple linear regression and feature engineering to come up with a new algorithm called polynomial regression, which will let you fit curves, nonlinear functions, to your data. Let's say you have a housing data set that looks like this, where feature x is the size in square feet. It doesn't look like a straight line fits this data set very well, so maybe you want to fit a curve, maybe a quadratic function, to the data like this, which includes a size x and also x squared, which is the size raised to the power of 2. And maybe that will give you a better fit to the data. But then you may decide that your quadratic model doesn't really make sense, because a quadratic function eventually comes back down, and well, we wouldn't really expect housing prices to go down when the size increases, right? Big houses seem like they should usually cost more. So then you may choose a cubic function, where we now have not only x squared, but x cubed. So maybe this model produces this curve here, which is a somewhat better fit to the data, because the size does eventually come back up as the size increases. These are both examples of polynomial regression, because you took your optional feature x and raised it to the power of 2, or 3, or any other power. And in the case of the cubic function, the first feature is the size, the second feature is the size squared, and the third feature is the size cubed. I just want to point out one more thing, which is that if you create features that are these powers, like the square of the original features like this, then feature scaling becomes increasingly important. So if the size of the house ranges from say, 1 to 1000 square feet, then the second feature, which is the size squared, would range from 1 to a million, and the third feature, which is size cubed, ranges from 1 to a billion. So these two features, x squared and x cubed, take on very different ranges of values compared to the original feature x. And if you're using gradient descent, it's important to apply feature scaling to get your features into comparable ranges of values. Finally, here's one last example of how you really have a wide range of choices of features to use. Another reasonable alternative to taking the size squared and size cubed is to say use the square root of x. So your model may look like w1 times x plus w2 times the square root of x plus b. The square root function looks like this, and it becomes a bit less steep as x increases, but it doesn't ever completely flatten out, and it certainly never ever comes back down. So this would be another choice of features that might work
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2.10 Practical Tips for Linear Regression | Polynomial regression --[Machine Learning | Andrew Ng]: So far, we've just been fitting straight lines to our data. Let's take the ideas of multiple linear regression and feature engineering to come up with a new algorithm called polynomial regression, which will let you fit curves, nonlinear functions, to your data. Let's say you have a housing data set that looks like this, where feature x is the size in square feet. It doesn't look like a straight line fits this data set very well, so maybe you want to fit a curve, maybe a quadratic function, to the data like this, which includes a size x and also x squared, which is the size raised to the power of 2. And maybe that will give you a better fit to the data. But then you may decide that your quadratic model doesn't really make sense, because a quadratic function eventually comes back down, and well, we wouldn't really expect housing prices to go down when the size increases, right? Big houses seem like they should usually cost more. So then you may choose a cubic function, where we now have not only x squared, but x cubed. So maybe this model produces this curve here, which is a somewhat better fit to the data, because the size does eventually come back up as the size increases. These are both examples of polynomial regression, because you took your optional feature x and raised it to the power of 2, or 3, or any other power. And in the case of the cubic function, the first feature is the size, the second feature is the size squared, and the third feature is the size cubed. I just want to point out one more thing, which is that if you create features that are these powers, like the square of the original features like this, then feature scaling becomes increasingly important. So if the size of the house ranges from say, 1 to 1000 square feet, then the second feature, which is the size squared, would range from 1 to a million, and the third feature, which is size cubed, ranges from 1 to a billion. So these two features, x squared and x cubed, take on very different ranges of values compared to the original feature x. And if you're using gradient descent, it's important to apply feature scaling to get your features into comparable ranges of values. Finally, here's one last example of how you really have a wide range of choices of features to use. Another reasonable alternative to taking the size squared and size cubed is to say use the square root of x. So your model may look like w1 times x plus w2 times the square root of x plus b. The square root function looks like this, and it becomes a bit less steep as x increases, but it doesn't ever completely flatten out, and it certainly never ever comes back down. So this would be another choice of features that might work
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2.10 Practical Tips for Linear Regression | Polynomial regression --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=UobsU2oQwBA
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well for this dataset as well. So you may ask yourself, how do I decide what features to use? Later, in the second course in this specialization, you see how you can choose different features and different models that include or don't include these features, and you have a process for measuring how well these different models perform to help you decide which features to include or not include. For now, I just want you to be aware that you have a choice in what features you use, and by using feature engineering and polynomial functions, you can potentially get a much better model for your data. In the optional lab that follows this video, you will see some code that implements polynomial regression using features like x, x squared, and x cubed. So please take a look and run the code and see how it works. There's also another optional lab after that one that shows how to use a popular open source toolkit that implements linear regression. Scikit-learn is a very widely used open source machine learning library that is used by many practitioners in many of the top AI, internet, machine learning companies in the world. So if either now or in the future you're using machine learning in your job, there's a very good chance you'll be using tools like Scikit-learn to train your models. And so working through that optional lab will give you a chance to not only better understand linear regression, but also see how this can be done in just a few lines of code using a library like Scikit-learn. For you to have a solid understanding of these algorithms and be able to apply them, I do think it's important that you know how to implement linear regression yourself, and not just call some Scikit-learn function that is a black box. But Scikit-learn also has an important role in the way machine learning is done in practice today. So, we're just about at the end of this week. Congratulations on finishing all of this week's videos. Please do take a look at the practice quizzes and also the practice lab, which I hope will let you try out and practice ideas that we've discussed. In this week's practice lab, you implement linear regression. I hope you have a lot of fun getting this learning algorithm to work for yourself. Best of luck with that, and I also look forward to seeing you in next week's videos, where we'll go beyond regression, that is predicting numbers, to talk about our first classification algorithm, which can predict categories. I'll see you next week.
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2.10 Practical Tips for Linear Regression | Polynomial regression --[Machine Learning | Andrew Ng]: well for this dataset as well. So you may ask yourself, how do I decide what features to use? Later, in the second course in this specialization, you see how you can choose different features and different models that include or don't include these features, and you have a process for measuring how well these different models perform to help you decide which features to include or not include. For now, I just want you to be aware that you have a choice in what features you use, and by using feature engineering and polynomial functions, you can potentially get a much better model for your data. In the optional lab that follows this video, you will see some code that implements polynomial regression using features like x, x squared, and x cubed. So please take a look and run the code and see how it works. There's also another optional lab after that one that shows how to use a popular open source toolkit that implements linear regression. Scikit-learn is a very widely used open source machine learning library that is used by many practitioners in many of the top AI, internet, machine learning companies in the world. So if either now or in the future you're using machine learning in your job, there's a very good chance you'll be using tools like Scikit-learn to train your models. And so working through that optional lab will give you a chance to not only better understand linear regression, but also see how this can be done in just a few lines of code using a library like Scikit-learn. For you to have a solid understanding of these algorithms and be able to apply them, I do think it's important that you know how to implement linear regression yourself, and not just call some Scikit-learn function that is a black box. But Scikit-learn also has an important role in the way machine learning is done in practice today. So, we're just about at the end of this week. Congratulations on finishing all of this week's videos. Please do take a look at the practice quizzes and also the practice lab, which I hope will let you try out and practice ideas that we've discussed. In this week's practice lab, you implement linear regression. I hope you have a lot of fun getting this learning algorithm to work for yourself. Best of luck with that, and I also look forward to seeing you in next week's videos, where we'll go beyond regression, that is predicting numbers, to talk about our first classification algorithm, which can predict categories. I'll see you next week.
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3.1 Classification | Motivations --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=wul3SmrpAeY
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Welcome to the third week of this course. By the end of this week, you have completed the first course of this specialization. So let's jump in. Last week, you learned about linear regression, which predicts a number. This week, you learned about classification, where your output variable y can take on only one of a small handful of possible values, instead of any number in an infinite range of numbers. It turns out that linear regression is not a good algorithm for classification problems. Let's take a look at why, and this will lead us into a different algorithm called logistic regression, which is one of the most popular and most widely used learning algorithms today. Here are some examples of classification problems. Recall the example of trying to figure out whether an email is spam. So the answer you want to output is going to be either a no or a yes. Another example would be figuring out if an online financial transaction is fraudulent. Fighting online financial fraud is something I once worked on, and it was strangely exhilarating because I knew there were forces out there trying to steal money, and my team's job was to stop them. So the problem is, given a financial transaction, can your learning algorithm figure out is this transaction fraudulent, such as was this credit card stolen? Another example we've touched on before was trying to classify a tumor as malignant versus not. In each of these problems, the variable that you want to predict can only be one of two possible values, no or yes. This type of classification problem, where there are only two possible outputs, is called binary classification, where the word binary refers to there being only two possible classes, or two possible categories. In these problems, I will use the terms class and category relatively interchangeably. They mean basically the same thing. By convention, we can refer to these two classes or categories in a few common ways. We often designate classes as no or yes, or sometimes equivalently, false or true, or very commonly using the numbers 0 or 1, following the common convention in computer science with 0 denoting false and 1 denoting true. I'm usually going to use the numbers 0 and 1 to represent the answer why, because that will fit in most easily with the types of learning algorithms we want to implement. But when we talk about it, we'll often say no or yes, or false or true, as well. One of the terminology is commonly used is to call the false or zero class the negative class and the true or the one class the positive class. For example, for spam classification, an email that is not spam may be referred to as a negative example because the output to the question of is it spam? The output is no or zero. In contrast, an email that is spam might be referred to as a positive training example
| 500
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3.1 Classification | Motivations --[Machine Learning | Andrew Ng]: Welcome to the third week of this course. By the end of this week, you have completed the first course of this specialization. So let's jump in. Last week, you learned about linear regression, which predicts a number. This week, you learned about classification, where your output variable y can take on only one of a small handful of possible values, instead of any number in an infinite range of numbers. It turns out that linear regression is not a good algorithm for classification problems. Let's take a look at why, and this will lead us into a different algorithm called logistic regression, which is one of the most popular and most widely used learning algorithms today. Here are some examples of classification problems. Recall the example of trying to figure out whether an email is spam. So the answer you want to output is going to be either a no or a yes. Another example would be figuring out if an online financial transaction is fraudulent. Fighting online financial fraud is something I once worked on, and it was strangely exhilarating because I knew there were forces out there trying to steal money, and my team's job was to stop them. So the problem is, given a financial transaction, can your learning algorithm figure out is this transaction fraudulent, such as was this credit card stolen? Another example we've touched on before was trying to classify a tumor as malignant versus not. In each of these problems, the variable that you want to predict can only be one of two possible values, no or yes. This type of classification problem, where there are only two possible outputs, is called binary classification, where the word binary refers to there being only two possible classes, or two possible categories. In these problems, I will use the terms class and category relatively interchangeably. They mean basically the same thing. By convention, we can refer to these two classes or categories in a few common ways. We often designate classes as no or yes, or sometimes equivalently, false or true, or very commonly using the numbers 0 or 1, following the common convention in computer science with 0 denoting false and 1 denoting true. I'm usually going to use the numbers 0 and 1 to represent the answer why, because that will fit in most easily with the types of learning algorithms we want to implement. But when we talk about it, we'll often say no or yes, or false or true, as well. One of the terminology is commonly used is to call the false or zero class the negative class and the true or the one class the positive class. For example, for spam classification, an email that is not spam may be referred to as a negative example because the output to the question of is it spam? The output is no or zero. In contrast, an email that is spam might be referred to as a positive training example
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3.1 Classification | Motivations --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=wul3SmrpAeY
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because the answer to is it spam is yes or true or one. To be clear, negative and positive do not necessarily mean bad versus good or evil versus good. It's just that negative and positive examples are used to convey the concepts of absence or zero or false versus the presence or true or one of something you might be looking for, such as the absence or presence of the spam units or the spam property of an email or the absence of presence of fraudulent activity or absence of presence of malignancy in a tumor. Between non-spam and spam emails, which one you call false or zero and which one you call true or one is a little bit arbitrary. Often either choice could work. So a different engineer might actually swap it around and have the positive class be the presence of a good email or the positive class be the presence of a real financial transaction or a healthy patient. So how do you build a classification algorithm? Here's the example of a training set for classifying if a tumor is malignant. A class one positive class, yes class or benign class zero or negative class. I plotted both the tumor size on the horizontal axis as well as the label Y on the vertical axis. By the way, in week one, when we first talked about classification, this is how we previously visualized it on the number line, except that now we're calling the classes zero and one and plotting them on the vertical axis. Now one thing you could try on this training set is to apply the algorithm you already know linear regression and try to fit a straight line to the data. If you do that, maybe the straight line looks like this, right? And that's your f of X. Linear regression predicts not just the values zero and one, but all numbers between zero and one or even less than zero or greater than one. But here we want to predict categories. One thing you could try is to pick a threshold of say 0.5 so that if the model outputs a value below 0.5, then you predict Y equals zero or not malignant. And if the model outputs a number equal to or greater than 0.5, then predict Y equals one or malignant. Notice that this threshold value 0.5 intersects the best fit straight line at this point. So if you draw this vertical line here, everything to the left ends up with a prediction of Y equals zero and everything on the right ends up with a prediction of Y equals one. Now for this particular data set, it looks like linear regression could do something reasonable. But now let's see what happens if your data set has one more training example. This one way over here on the right. Let's also extend the horizontal axis. Notice that this training example shouldn't really change how you classify the data
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3.1 Classification | Motivations --[Machine Learning | Andrew Ng]: because the answer to is it spam is yes or true or one. To be clear, negative and positive do not necessarily mean bad versus good or evil versus good. It's just that negative and positive examples are used to convey the concepts of absence or zero or false versus the presence or true or one of something you might be looking for, such as the absence or presence of the spam units or the spam property of an email or the absence of presence of fraudulent activity or absence of presence of malignancy in a tumor. Between non-spam and spam emails, which one you call false or zero and which one you call true or one is a little bit arbitrary. Often either choice could work. So a different engineer might actually swap it around and have the positive class be the presence of a good email or the positive class be the presence of a real financial transaction or a healthy patient. So how do you build a classification algorithm? Here's the example of a training set for classifying if a tumor is malignant. A class one positive class, yes class or benign class zero or negative class. I plotted both the tumor size on the horizontal axis as well as the label Y on the vertical axis. By the way, in week one, when we first talked about classification, this is how we previously visualized it on the number line, except that now we're calling the classes zero and one and plotting them on the vertical axis. Now one thing you could try on this training set is to apply the algorithm you already know linear regression and try to fit a straight line to the data. If you do that, maybe the straight line looks like this, right? And that's your f of X. Linear regression predicts not just the values zero and one, but all numbers between zero and one or even less than zero or greater than one. But here we want to predict categories. One thing you could try is to pick a threshold of say 0.5 so that if the model outputs a value below 0.5, then you predict Y equals zero or not malignant. And if the model outputs a number equal to or greater than 0.5, then predict Y equals one or malignant. Notice that this threshold value 0.5 intersects the best fit straight line at this point. So if you draw this vertical line here, everything to the left ends up with a prediction of Y equals zero and everything on the right ends up with a prediction of Y equals one. Now for this particular data set, it looks like linear regression could do something reasonable. But now let's see what happens if your data set has one more training example. This one way over here on the right. Let's also extend the horizontal axis. Notice that this training example shouldn't really change how you classify the data
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3.1 Classification | Motivations --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=wul3SmrpAeY
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points. This vertical dividing line that we drew just now still makes sense as the cutoff where two meters smaller than this should be classified as zero and two meters greater than this should be classified as one. But once you've added this extra training example on the right, the best fit line for linear regression will shift over like this. And if you continue using the threshold of 0.5, you now notice that everything to the left of this point is predicted as zero, non-malignant, and everything to the right of this point is predicted to be one or malignant. This isn't what we want, because adding that example way to the right shouldn't change any of our conclusions about how to classify malignant versus benign tumors. But if you try to do this with linear regression, adding this one example, which feels like it shouldn't be changing anything, it ends up with us learning a much worse function for this classification problem. Clearly, when a tumor is large, we want the algorithm to classify it as malignant. So what we just saw was linear regression causes the best fit line when we added one more example to the right to shift over, and thus the dividing line also called the decision boundary to shift over to the right. You learn more about the decision boundary in the next video. You also learn about an algorithm called logistic regression, where the output value of the outcome will always be between 0 and 1, and the algorithm will avoid these problems that we're seeing on the slide. By the way, one thing confusing about the name logistic regression is that even though it has the word regression in it, it's actually used for classification. Don't be confused by the name, which was given for historical reasons. It's actually used to solve binary classification problems where the output label y is either 0 or 1. In the upcoming optional lab, you also get to take a look at what happens when you try to use linear regression for classification. Sometimes you get lucky and it may work, but often it will not work well, which is why I don't use linear regression myself for classification. In the optional lab, you see an interactive plot that attempts to classify between two categories, and you hopefully notice how this often doesn't work very well, which is okay, because that motivates the need for a different model to do classification tasks. So please check out this optional lab, and after that, we'll go on to the next video to look at logistic regression for classification.
| 441
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3.1 Classification | Motivations --[Machine Learning | Andrew Ng]: points. This vertical dividing line that we drew just now still makes sense as the cutoff where two meters smaller than this should be classified as zero and two meters greater than this should be classified as one. But once you've added this extra training example on the right, the best fit line for linear regression will shift over like this. And if you continue using the threshold of 0.5, you now notice that everything to the left of this point is predicted as zero, non-malignant, and everything to the right of this point is predicted to be one or malignant. This isn't what we want, because adding that example way to the right shouldn't change any of our conclusions about how to classify malignant versus benign tumors. But if you try to do this with linear regression, adding this one example, which feels like it shouldn't be changing anything, it ends up with us learning a much worse function for this classification problem. Clearly, when a tumor is large, we want the algorithm to classify it as malignant. So what we just saw was linear regression causes the best fit line when we added one more example to the right to shift over, and thus the dividing line also called the decision boundary to shift over to the right. You learn more about the decision boundary in the next video. You also learn about an algorithm called logistic regression, where the output value of the outcome will always be between 0 and 1, and the algorithm will avoid these problems that we're seeing on the slide. By the way, one thing confusing about the name logistic regression is that even though it has the word regression in it, it's actually used for classification. Don't be confused by the name, which was given for historical reasons. It's actually used to solve binary classification problems where the output label y is either 0 or 1. In the upcoming optional lab, you also get to take a look at what happens when you try to use linear regression for classification. Sometimes you get lucky and it may work, but often it will not work well, which is why I don't use linear regression myself for classification. In the optional lab, you see an interactive plot that attempts to classify between two categories, and you hopefully notice how this often doesn't work very well, which is okay, because that motivates the need for a different model to do classification tasks. So please check out this optional lab, and after that, we'll go on to the next video to look at logistic regression for classification.
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3.2 Classification | Logistic regression --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=Pm8mRCZmYiU
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Let's talk about logistic regression, which is probably the single most widely used classification algorithm in the world. This is something that I use all the time in my work. Let's continue with the example of classifying whether a tumor is malignant, where as before we're going to use the label 1 or yes the positive class to represent malignant tumors and 0 or no negative examples to represent benign tumors. Here's a graph of the dataset where the horizontal axis is the tumor size and the vertical axis takes on only values of 0 and 1 because it's a classification problem. You saw in the last video that linear regression is not a good algorithm for this problem. In contrast, what logistic regression will end up doing is fit a curve that looks like this, a sort of S-shaped curve to this dataset. And so for this example, if a patient comes in with a tumor of this size, which I'm showing on the x-axis, then the algorithm will output 0.7, suggesting that it's closer or maybe more likely to be malignant and benign. We'll say more later what 0.7 actually means in this context, but the output label y is never 0.7, is only ever 0 or 1. To build up to the logistic regression algorithm, there's an important mathematical function I'd like to describe which is called the sigmoid function, sometimes also referred to as the logistic function. The sigmoid function looks like this. Notice that the x-axis of the graphs on the left and right are different. In the graph to the left, on the x-axis is the tumor size, so it's all positive numbers. Whereas in the graph on the right, you have 0 down here, and the horizontal axis takes on both negative and positive values, and I've labeled the horizontal axis z. And I'm showing here just a range of negative 3 to plus 3. So the sigmoid function outputs values between 0 and 1. And if I use g of z to denote this function, then the formula of g of z is equal to 1 over 1 plus e to the negative z, where here e is a mathematical constant that takes on a value of about 2.7, and so e to the negative z is that mathematical constant to the power of negative z. Notice if z were really big, say 100, e to the negative z is e to the negative 100, which is a tiny, tiny number. So this ends up being 1 over 1 plus a tiny little number, and so the denominator will be basically very, very close to 1, which is why when z is large, g of z, that is the sigmoid function of z, is going to be very close to 1. And conversely, you can also check for yourself that when z is a very large negative number, then g of z becomes 1 over a giant number, which is why g
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3.2 Classification | Logistic regression --[Machine Learning | Andrew Ng]: Let's talk about logistic regression, which is probably the single most widely used classification algorithm in the world. This is something that I use all the time in my work. Let's continue with the example of classifying whether a tumor is malignant, where as before we're going to use the label 1 or yes the positive class to represent malignant tumors and 0 or no negative examples to represent benign tumors. Here's a graph of the dataset where the horizontal axis is the tumor size and the vertical axis takes on only values of 0 and 1 because it's a classification problem. You saw in the last video that linear regression is not a good algorithm for this problem. In contrast, what logistic regression will end up doing is fit a curve that looks like this, a sort of S-shaped curve to this dataset. And so for this example, if a patient comes in with a tumor of this size, which I'm showing on the x-axis, then the algorithm will output 0.7, suggesting that it's closer or maybe more likely to be malignant and benign. We'll say more later what 0.7 actually means in this context, but the output label y is never 0.7, is only ever 0 or 1. To build up to the logistic regression algorithm, there's an important mathematical function I'd like to describe which is called the sigmoid function, sometimes also referred to as the logistic function. The sigmoid function looks like this. Notice that the x-axis of the graphs on the left and right are different. In the graph to the left, on the x-axis is the tumor size, so it's all positive numbers. Whereas in the graph on the right, you have 0 down here, and the horizontal axis takes on both negative and positive values, and I've labeled the horizontal axis z. And I'm showing here just a range of negative 3 to plus 3. So the sigmoid function outputs values between 0 and 1. And if I use g of z to denote this function, then the formula of g of z is equal to 1 over 1 plus e to the negative z, where here e is a mathematical constant that takes on a value of about 2.7, and so e to the negative z is that mathematical constant to the power of negative z. Notice if z were really big, say 100, e to the negative z is e to the negative 100, which is a tiny, tiny number. So this ends up being 1 over 1 plus a tiny little number, and so the denominator will be basically very, very close to 1, which is why when z is large, g of z, that is the sigmoid function of z, is going to be very close to 1. And conversely, you can also check for yourself that when z is a very large negative number, then g of z becomes 1 over a giant number, which is why g
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3.2 Classification | Logistic regression --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=Pm8mRCZmYiU
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of z is very close to 0. So that's why the sigmoid function has this shape, where it starts very close to 0 and slowly builds up, or grows, to the value of 1. Also, in the sigmoid function, when z is equal to 0, then e to the negative z is e to the negative 0, which is equal to 1, and so g of z is equal to 1 over 1 plus 1, which is 0.5. So that's why it passes the vertical axis at 0.5. Now let's use this to build up to the logistic regression algorithm. We're going to do this in two steps. In the first step, I hope you remember that a straight line function, like a linear regression function, can be defined as W dot product of x plus b. So let's store this value in a variable, which I'm going to call z. And this will turn out to be the same z as the one you saw on the previous slide, but we'll get to that in a minute. The next step then is to take this value of z and pass it to the sigmoid function, also called the logistic function g. So now g of z then outputs a value computed by this formula, 1 over 1 plus e to the negative z, that's going to be between 0 and 1. When you take these two equations and put them together, they then give you the logistic regression model f of x, which is equal to g of w x plus b, or equivalently g of z, which is equal to this formula over here. So this is the logistic regression model. And what it does is it inputs a feature or set of features x, and it outputs a number between 0 and 1. Next, let's take a look at how to interpret the output of logistic regression. We'll return to the tumor classification example. The way I'd encourage you to think of logistic regression's output is to think of it as outputting the probability that the class or the label y will be equal to 1, given a certain input x. So, for example, in this application where x is the tumor size and y is either 0 or 1, if you have a patient come in and she has a tumor of a certain size x, and if based on this input x, the model outputs 0.7, then what that means is that the model is predicting or the model thinks there's a 70% chance that the true label y will be equal to 1 for this patient. In other words, the model is telling us that it thinks the patient has a 70% chance of their tumor turning out to be malignant. Now, let me ask you a question. See if you can get this right. We know that y has to be either 0 or 1. So if y has a 70% chance of
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3.2 Classification | Logistic regression --[Machine Learning | Andrew Ng]: of z is very close to 0. So that's why the sigmoid function has this shape, where it starts very close to 0 and slowly builds up, or grows, to the value of 1. Also, in the sigmoid function, when z is equal to 0, then e to the negative z is e to the negative 0, which is equal to 1, and so g of z is equal to 1 over 1 plus 1, which is 0.5. So that's why it passes the vertical axis at 0.5. Now let's use this to build up to the logistic regression algorithm. We're going to do this in two steps. In the first step, I hope you remember that a straight line function, like a linear regression function, can be defined as W dot product of x plus b. So let's store this value in a variable, which I'm going to call z. And this will turn out to be the same z as the one you saw on the previous slide, but we'll get to that in a minute. The next step then is to take this value of z and pass it to the sigmoid function, also called the logistic function g. So now g of z then outputs a value computed by this formula, 1 over 1 plus e to the negative z, that's going to be between 0 and 1. When you take these two equations and put them together, they then give you the logistic regression model f of x, which is equal to g of w x plus b, or equivalently g of z, which is equal to this formula over here. So this is the logistic regression model. And what it does is it inputs a feature or set of features x, and it outputs a number between 0 and 1. Next, let's take a look at how to interpret the output of logistic regression. We'll return to the tumor classification example. The way I'd encourage you to think of logistic regression's output is to think of it as outputting the probability that the class or the label y will be equal to 1, given a certain input x. So, for example, in this application where x is the tumor size and y is either 0 or 1, if you have a patient come in and she has a tumor of a certain size x, and if based on this input x, the model outputs 0.7, then what that means is that the model is predicting or the model thinks there's a 70% chance that the true label y will be equal to 1 for this patient. In other words, the model is telling us that it thinks the patient has a 70% chance of their tumor turning out to be malignant. Now, let me ask you a question. See if you can get this right. We know that y has to be either 0 or 1. So if y has a 70% chance of
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3.2 Classification | Logistic regression --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=Pm8mRCZmYiU
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being 1, what is the chance that it is 0? So y has got to be either 0 or 1, and thus the probability of it being 0 or 1, these two numbers have to add up to 1 or to 100% chance. So that's why if the chance of y being 1 is 0.7 or 70% chance, then the chance of it being 0 has got to be 0.3 or 30% chance. If someday you read research papers or blog posts about logistic regression, sometimes you see this notation that f of x is equal to p of y equals 1, given the input features x and with parameters w and b. What the semicolon here is used to denote is just that w and b are parameters that affect this computation of what is the probability of y being equal to 1, given the input feature x. For the purpose of this class, don't worry too much about what this vertical line and what the semicolon mean. You don't need to remember or follow any of this mathematical notation for this class. I'm mentioning this only because you may see this in other places. In the optional lab that follows this video, you also get to see how the sigmoid function is implemented in code. You can see a plot that uses the sigmoid function so as to do better on the classification task that you saw in the previous optional lab. Remember that the code will be provided to you, so you just have to run it. I hope you take a look and get familiar with the code. So congrats on getting here. You now know what is the logistic regression model as well as the mathematical formula that defines logistic regression. For a long time, a lot of internet advertising was actually driven by basically a slight variation of logistic regression. This was very lucrative for some large companies, and this was basically the algorithm that decided what ad was shown to you and many others on some large websites. Now there's even more to learn about this algorithm. In the next video, we'll take a look at the details of logistic regression. We'll look at some visualizations and also examine something called the decision boundary. This will give you a few different ways to map the numbers that this model outputs, such as 0.3 or 0.7 or 0.65, to a prediction of whether y is actually 0 or 1. So let's go on to the next video to learn more about logistic regression.
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3.2 Classification | Logistic regression --[Machine Learning | Andrew Ng]: being 1, what is the chance that it is 0? So y has got to be either 0 or 1, and thus the probability of it being 0 or 1, these two numbers have to add up to 1 or to 100% chance. So that's why if the chance of y being 1 is 0.7 or 70% chance, then the chance of it being 0 has got to be 0.3 or 30% chance. If someday you read research papers or blog posts about logistic regression, sometimes you see this notation that f of x is equal to p of y equals 1, given the input features x and with parameters w and b. What the semicolon here is used to denote is just that w and b are parameters that affect this computation of what is the probability of y being equal to 1, given the input feature x. For the purpose of this class, don't worry too much about what this vertical line and what the semicolon mean. You don't need to remember or follow any of this mathematical notation for this class. I'm mentioning this only because you may see this in other places. In the optional lab that follows this video, you also get to see how the sigmoid function is implemented in code. You can see a plot that uses the sigmoid function so as to do better on the classification task that you saw in the previous optional lab. Remember that the code will be provided to you, so you just have to run it. I hope you take a look and get familiar with the code. So congrats on getting here. You now know what is the logistic regression model as well as the mathematical formula that defines logistic regression. For a long time, a lot of internet advertising was actually driven by basically a slight variation of logistic regression. This was very lucrative for some large companies, and this was basically the algorithm that decided what ad was shown to you and many others on some large websites. Now there's even more to learn about this algorithm. In the next video, we'll take a look at the details of logistic regression. We'll look at some visualizations and also examine something called the decision boundary. This will give you a few different ways to map the numbers that this model outputs, such as 0.3 or 0.7 or 0.65, to a prediction of whether y is actually 0 or 1. So let's go on to the next video to learn more about logistic regression.
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3.3 Classification | Decision boundary --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=QJdIpRcL_4U
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In the last video, you learned about the logistic regression model. Now, let's take a look at the decision boundary to get a better sense of how logistic regression is computing its predictions. To recap, here's how the logistic regression model's outputs are computed in two steps. In the first step, you compute z as w dot x plus b. Then you apply the sigmoid function g to this value z, and here again is the formula for the sigmoid function. Another way to write this is we can say f of x is equal to g, the sigmoid function, also called the logistic function, applied to w dot x plus b, where this is of course the value of z. And if you take the definition of the sigmoid function and plug in the definition of z, then you find that f of x is equal to this formula over here, 1 over 1 plus e to the negative z, where z is w x plus b. And you may remember we said in the previous video that we interpret this as the probability that y is equal to 1, given x, and with parameters w and b. And so this is going to be a number, like maybe 0.7 or 0.3. Now what if you want the learning algorithm to predict, is the value of y going to be 0 or 1? Well one thing you might do is set a threshold above which you predict y is 1, or you set y hat, the prediction, to be equal to 1, and below which you might say y hat, my prediction, is going to be equal to 0. So a common choice would be to pick a threshold of 0.5, so that if f of x is greater than or equal to 0.5, then predict y is 1, and we write that prediction as y hat equals 1, or if f of x is less than 0.5, then predict y is 0, or in other words, the prediction y hat is equal to 0. So now let's dive deeper into when the model would predict 1. In other words, when is f of x greater than or equal to 0.5? Well recall that f of x is just equal to g of z, and so f is greater than or equal to 0.5 whenever g of z is greater than or equal to 0.5. But when is g of z greater than or equal to 0.5? Well here's the sigmoid function over here, and so g of z is greater than or equal to 0.5 whenever z is greater than or equal to 0, right? That is, whenever z is on the right half of this axis. And finally, when is z greater than or equal to 0? Well z is equal to w dot x plus b, and so z is greater than or equal to 0 whenever w dot x plus b is greater than or
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3.3 Classification | Decision boundary --[Machine Learning | Andrew Ng]: In the last video, you learned about the logistic regression model. Now, let's take a look at the decision boundary to get a better sense of how logistic regression is computing its predictions. To recap, here's how the logistic regression model's outputs are computed in two steps. In the first step, you compute z as w dot x plus b. Then you apply the sigmoid function g to this value z, and here again is the formula for the sigmoid function. Another way to write this is we can say f of x is equal to g, the sigmoid function, also called the logistic function, applied to w dot x plus b, where this is of course the value of z. And if you take the definition of the sigmoid function and plug in the definition of z, then you find that f of x is equal to this formula over here, 1 over 1 plus e to the negative z, where z is w x plus b. And you may remember we said in the previous video that we interpret this as the probability that y is equal to 1, given x, and with parameters w and b. And so this is going to be a number, like maybe 0.7 or 0.3. Now what if you want the learning algorithm to predict, is the value of y going to be 0 or 1? Well one thing you might do is set a threshold above which you predict y is 1, or you set y hat, the prediction, to be equal to 1, and below which you might say y hat, my prediction, is going to be equal to 0. So a common choice would be to pick a threshold of 0.5, so that if f of x is greater than or equal to 0.5, then predict y is 1, and we write that prediction as y hat equals 1, or if f of x is less than 0.5, then predict y is 0, or in other words, the prediction y hat is equal to 0. So now let's dive deeper into when the model would predict 1. In other words, when is f of x greater than or equal to 0.5? Well recall that f of x is just equal to g of z, and so f is greater than or equal to 0.5 whenever g of z is greater than or equal to 0.5. But when is g of z greater than or equal to 0.5? Well here's the sigmoid function over here, and so g of z is greater than or equal to 0.5 whenever z is greater than or equal to 0, right? That is, whenever z is on the right half of this axis. And finally, when is z greater than or equal to 0? Well z is equal to w dot x plus b, and so z is greater than or equal to 0 whenever w dot x plus b is greater than or
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3.3 Classification | Decision boundary --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=QJdIpRcL_4U
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equal to 0. So to recap, what you've seen here is that the model predicts 1 whenever w dot x plus b is greater than or equal to 0. And conversely, when w dot x plus b is less than 0, the algorithm predicts y is 0. So given this, let's now visualize how the model makes predictions. I'm going to take an example of a classification problem where you have two features, x1 and x2 instead of just one feature. Here's a training set where the little red crosses denote the positive examples, and the little blue circles denote negative examples. So the red crosses corresponds to y equals 1, and the blue circles correspond to y equals 0. So the logistic regression model will make predictions using this function, f of x equals g of z, where z is now this expression over here, w1 x1 plus w2 x2 plus b, because we have two features, x1 and x2. And let's just say for this example that the value of the parameters are w1 equals 1, w2 equals 1, and b equals negative 3. And let's now take a look at how logistic regression makes predictions. In particular, let's figure out when wx plus b is greater than or equal to 0, and when wx plus b is less than 0. To figure that out, there's a very interesting line to look at, which is when wx plus b is exactly equal to 0. It turns out that this line is also called the decision boundary, because that's the line where you're just almost neutral about whether y is 0 or y is 1. Now for the values of the parameters w1, w2, and b that we had written down above, this decision boundary is just x1 plus x2 minus 3. And so when is x1 plus x2 minus 3 equal to 0? Well that will correspond to the line x1 plus x2 equals 3. And that is this line shown over here. And so this line turns out to be the decision boundary, where if the features x are to the right of this line, logistic regression would predict 1, and to the left of this line, logistic regression would predict 0. In other words, what we have just visualized is the decision boundary for logistic regression when the parameters w1, w2, and b are 1, 1, and negative 3. Of course, if you had a different choice of the parameters, the decision boundary would be a different line. Now let's look at a more complex example where the decision boundary is no longer a straight line. As before, crosses denote the class y equals 1, and the little circles denote the class y equals 0. Earlier last week, you saw how to use polynomials in linear regression, and you can do the same in logistic regression. So let's set z to be w1 x1 squared plus w2 x2 squared plus b. With this choice of features,
| 500
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3.3 Classification | Decision boundary --[Machine Learning | Andrew Ng]: equal to 0. So to recap, what you've seen here is that the model predicts 1 whenever w dot x plus b is greater than or equal to 0. And conversely, when w dot x plus b is less than 0, the algorithm predicts y is 0. So given this, let's now visualize how the model makes predictions. I'm going to take an example of a classification problem where you have two features, x1 and x2 instead of just one feature. Here's a training set where the little red crosses denote the positive examples, and the little blue circles denote negative examples. So the red crosses corresponds to y equals 1, and the blue circles correspond to y equals 0. So the logistic regression model will make predictions using this function, f of x equals g of z, where z is now this expression over here, w1 x1 plus w2 x2 plus b, because we have two features, x1 and x2. And let's just say for this example that the value of the parameters are w1 equals 1, w2 equals 1, and b equals negative 3. And let's now take a look at how logistic regression makes predictions. In particular, let's figure out when wx plus b is greater than or equal to 0, and when wx plus b is less than 0. To figure that out, there's a very interesting line to look at, which is when wx plus b is exactly equal to 0. It turns out that this line is also called the decision boundary, because that's the line where you're just almost neutral about whether y is 0 or y is 1. Now for the values of the parameters w1, w2, and b that we had written down above, this decision boundary is just x1 plus x2 minus 3. And so when is x1 plus x2 minus 3 equal to 0? Well that will correspond to the line x1 plus x2 equals 3. And that is this line shown over here. And so this line turns out to be the decision boundary, where if the features x are to the right of this line, logistic regression would predict 1, and to the left of this line, logistic regression would predict 0. In other words, what we have just visualized is the decision boundary for logistic regression when the parameters w1, w2, and b are 1, 1, and negative 3. Of course, if you had a different choice of the parameters, the decision boundary would be a different line. Now let's look at a more complex example where the decision boundary is no longer a straight line. As before, crosses denote the class y equals 1, and the little circles denote the class y equals 0. Earlier last week, you saw how to use polynomials in linear regression, and you can do the same in logistic regression. So let's set z to be w1 x1 squared plus w2 x2 squared plus b. With this choice of features,
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3.3 Classification | Decision boundary --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=QJdIpRcL_4U
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polynomial features into logistic regression, so f of x, which equals g of z, is now g of this expression over here. And let's say that we end up choosing w1 and w2 to be 1, and b to be negative 1. So z is equal to 1 times x1 squared plus 1 times x2 squared minus 1. And the decision boundary as before will correspond to when z is equal to 0, and so this expression will be equal to 0 when x1 squared plus x2 squared is equal to 1. And if you plot on the diagram on the left, the curve corresponding to x1 squared plus x2 squared equals 1, this turns out to be this circle. When x1 squared plus x2 squared is greater than or equal to 1, that's this area outside the circle, and that's when you predict y to be 1. Conversely, when x1 squared plus x2 squared is less than 1, that's this area inside the circle, and that's when you would predict y to be 0. So can we come up with even more complex decision boundaries than these? Yes you can. You can do so by having even higher order polynomial terms. Say z is w1 x1 plus w2 x2 plus w3 x1 squared plus w4 x1 x2 plus w5 x2 squared. Then it's possible you can get even more complex decision boundaries. The model can define decision boundaries such as this example, an ellipse that's like this, or with a different choice of the parameters. You can even get more complex decision boundaries which can look like functions that maybe look like that. So this is an example of an even more complex decision boundary than the ones we've seen previously, and this implementation of logistic regression will predict y equals 1 inside this shape, and outside the shape it will predict y equals 0. So with these polynomial features you can get very complex decision boundaries. In other words logistic regression can learn to fit pretty complex data. Although if you were to not include any of these higher order polynomials, so if the only features you use are x1 x2 x3 and so on, then the decision boundary for logistic regression will always be linear, will always be a straight line. In the upcoming optional lab you also get to see the code implementation of the decision boundary. In the example in the lab there will be two features so you can see the decision boundary as a line. So with these visualizations I hope that you now have a sense of the range of possible models you can get with logistic regression. Now that you've seen what f of x can potentially compute, let's take a look at how you can actually train a logistic regression model. We'll start by looking at the cost function for logistic regression and after that figure out how to apply gradient descent to it. Let's go on to the next
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3.3 Classification | Decision boundary --[Machine Learning | Andrew Ng]: polynomial features into logistic regression, so f of x, which equals g of z, is now g of this expression over here. And let's say that we end up choosing w1 and w2 to be 1, and b to be negative 1. So z is equal to 1 times x1 squared plus 1 times x2 squared minus 1. And the decision boundary as before will correspond to when z is equal to 0, and so this expression will be equal to 0 when x1 squared plus x2 squared is equal to 1. And if you plot on the diagram on the left, the curve corresponding to x1 squared plus x2 squared equals 1, this turns out to be this circle. When x1 squared plus x2 squared is greater than or equal to 1, that's this area outside the circle, and that's when you predict y to be 1. Conversely, when x1 squared plus x2 squared is less than 1, that's this area inside the circle, and that's when you would predict y to be 0. So can we come up with even more complex decision boundaries than these? Yes you can. You can do so by having even higher order polynomial terms. Say z is w1 x1 plus w2 x2 plus w3 x1 squared plus w4 x1 x2 plus w5 x2 squared. Then it's possible you can get even more complex decision boundaries. The model can define decision boundaries such as this example, an ellipse that's like this, or with a different choice of the parameters. You can even get more complex decision boundaries which can look like functions that maybe look like that. So this is an example of an even more complex decision boundary than the ones we've seen previously, and this implementation of logistic regression will predict y equals 1 inside this shape, and outside the shape it will predict y equals 0. So with these polynomial features you can get very complex decision boundaries. In other words logistic regression can learn to fit pretty complex data. Although if you were to not include any of these higher order polynomials, so if the only features you use are x1 x2 x3 and so on, then the decision boundary for logistic regression will always be linear, will always be a straight line. In the upcoming optional lab you also get to see the code implementation of the decision boundary. In the example in the lab there will be two features so you can see the decision boundary as a line. So with these visualizations I hope that you now have a sense of the range of possible models you can get with logistic regression. Now that you've seen what f of x can potentially compute, let's take a look at how you can actually train a logistic regression model. We'll start by looking at the cost function for logistic regression and after that figure out how to apply gradient descent to it. Let's go on to the next
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3.4 Cost function | Cost function for logistic regression --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=E-ZcnndnY2I
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Remember that the cost function gives you a way to measure how well a specific set of parameters fits the training data, and it thereby gives you a way to try to choose better parameters. In this video, we'll look at how the squared error cost function is not an ideal cost function for logistic regression, and we'll take a look at a different cost function that can help us choose better parameters for logistic regression. Notice what the training set for a logistic regression model might look like, where here each row might correspond to a patient that was paying a visit to the doctor and wound up with some sort of diagnosis. As before, we'll use m to denote the number of training examples. Each training example has one or more features, such as the tumor size, the patient's age, and so on, for a total of n features. And so let's call the features x1 through xn. And since this is a binary classification task, the target label y takes on only two values, either 0 or 1. And finally, the logistic regression model is defined by this equation. Okay, so the question you want to answer is, given this training set, how can you choose parameters w and b? Recall for linear regression, this is the squared error cost function. The only thing I've changed is that I put the one half inside the summation instead of outside the summation. And you might remember that in the case of linear regression, where f of x is the linear function w dot x plus b, the cost function looks like this is a convex function or a bow shape or a hammock shape. And so gradient descent will look like this, where you take one step, one step, one step, and so on to converge at the global minimum. Now, you could try to use the same cost function for logistic regression, but it turns out that if I were to write f of x equals 1 over 1 plus e to the negative w x plus b, and plot the cost function using this value of f of x, then the cost will look like this. This becomes what's called a non-convex cost function, it's not convex. And what this means is that if you were to try to use gradient descent, there are lots of local minima that you can get stuck in. So it turns out that for logistic regression, this squared error cost function is not a good choice. Instead, there will be a different cost function that can make the cost function convex again, so that gradient descent can be guaranteed to converge to the global minimum. The only thing I've changed is that I put the one half inside the summation instead of outside the summation. This will make the math you see later on the slide a little bit simpler. In order to build a new cost function, one that
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3.4 Cost function | Cost function for logistic regression --[Machine Learning | Andrew Ng]: Remember that the cost function gives you a way to measure how well a specific set of parameters fits the training data, and it thereby gives you a way to try to choose better parameters. In this video, we'll look at how the squared error cost function is not an ideal cost function for logistic regression, and we'll take a look at a different cost function that can help us choose better parameters for logistic regression. Notice what the training set for a logistic regression model might look like, where here each row might correspond to a patient that was paying a visit to the doctor and wound up with some sort of diagnosis. As before, we'll use m to denote the number of training examples. Each training example has one or more features, such as the tumor size, the patient's age, and so on, for a total of n features. And so let's call the features x1 through xn. And since this is a binary classification task, the target label y takes on only two values, either 0 or 1. And finally, the logistic regression model is defined by this equation. Okay, so the question you want to answer is, given this training set, how can you choose parameters w and b? Recall for linear regression, this is the squared error cost function. The only thing I've changed is that I put the one half inside the summation instead of outside the summation. And you might remember that in the case of linear regression, where f of x is the linear function w dot x plus b, the cost function looks like this is a convex function or a bow shape or a hammock shape. And so gradient descent will look like this, where you take one step, one step, one step, and so on to converge at the global minimum. Now, you could try to use the same cost function for logistic regression, but it turns out that if I were to write f of x equals 1 over 1 plus e to the negative w x plus b, and plot the cost function using this value of f of x, then the cost will look like this. This becomes what's called a non-convex cost function, it's not convex. And what this means is that if you were to try to use gradient descent, there are lots of local minima that you can get stuck in. So it turns out that for logistic regression, this squared error cost function is not a good choice. Instead, there will be a different cost function that can make the cost function convex again, so that gradient descent can be guaranteed to converge to the global minimum. The only thing I've changed is that I put the one half inside the summation instead of outside the summation. This will make the math you see later on the slide a little bit simpler. In order to build a new cost function, one that
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3.4 Cost function | Cost function for logistic regression --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=E-ZcnndnY2I
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we'll use for logistic regression, I'm going to change a little bit the definition of the cost function j of w and b. In particular, if you look inside the summation, let's call this term inside the loss on a single training example. And I'm going to denote the loss via this capital L, and is a function of the prediction of the learning algorithm f of x, as well as of the true label y. And so the loss, given the predicted f of x and the true label y, is equal in this case to one half of the squared difference. We'll see shortly that by choosing a different form for this loss function, we'll be able to keep the overall cost function, which is one over m times the sum of these loss functions, to be a convex function. Now the loss function inputs f of x and the true label y, and tells us how well we're doing on that example. I'm going to just write down here the definition of the loss function we'll use for logistic regression. If the label y is equal to one, then the loss is negative log of f of x. And if the label y is equal to zero, then the loss is negative log of one minus f of x. Let's take a look at why this loss function hopefully makes sense. Let's first consider the case of y equals one, and plot what this function looks like to gain some intuition about what this loss function is doing. And remember, the loss function measures how well you're doing on one training example, and is by summing up the losses on all of the training examples that you then get the cost function, which measures how well you're doing on the entire training set. So if you plot log of f, it looks like this curve here, where f here is on the horizontal axis. And so a plot of negative of the log of f looks like this, where we just flip the curve along the horizontal axis. Notice that it intersects the horizontal axis at f equals one, and continues downward from there. Now f is the output of logistic regression. Thus, f is always between zero and one, because the output of logistic regression is always between zero and one. The only part of the function that's relevant is therefore this part over here, corresponding to f between zero and one. So let's zoom in and take a closer look at this part of the graph. If the algorithm predicts a probability close to one, and the true label is one, then the loss is very small, it's pretty much zero, because you're very close to the right answer. Now continuing with the example of the true label y being one, so say it really is a malignant tumor, if the algorithm predicts 0.5, then the loss is at this point here, which is a
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3.4 Cost function | Cost function for logistic regression --[Machine Learning | Andrew Ng]: we'll use for logistic regression, I'm going to change a little bit the definition of the cost function j of w and b. In particular, if you look inside the summation, let's call this term inside the loss on a single training example. And I'm going to denote the loss via this capital L, and is a function of the prediction of the learning algorithm f of x, as well as of the true label y. And so the loss, given the predicted f of x and the true label y, is equal in this case to one half of the squared difference. We'll see shortly that by choosing a different form for this loss function, we'll be able to keep the overall cost function, which is one over m times the sum of these loss functions, to be a convex function. Now the loss function inputs f of x and the true label y, and tells us how well we're doing on that example. I'm going to just write down here the definition of the loss function we'll use for logistic regression. If the label y is equal to one, then the loss is negative log of f of x. And if the label y is equal to zero, then the loss is negative log of one minus f of x. Let's take a look at why this loss function hopefully makes sense. Let's first consider the case of y equals one, and plot what this function looks like to gain some intuition about what this loss function is doing. And remember, the loss function measures how well you're doing on one training example, and is by summing up the losses on all of the training examples that you then get the cost function, which measures how well you're doing on the entire training set. So if you plot log of f, it looks like this curve here, where f here is on the horizontal axis. And so a plot of negative of the log of f looks like this, where we just flip the curve along the horizontal axis. Notice that it intersects the horizontal axis at f equals one, and continues downward from there. Now f is the output of logistic regression. Thus, f is always between zero and one, because the output of logistic regression is always between zero and one. The only part of the function that's relevant is therefore this part over here, corresponding to f between zero and one. So let's zoom in and take a closer look at this part of the graph. If the algorithm predicts a probability close to one, and the true label is one, then the loss is very small, it's pretty much zero, because you're very close to the right answer. Now continuing with the example of the true label y being one, so say it really is a malignant tumor, if the algorithm predicts 0.5, then the loss is at this point here, which is a
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3.4 Cost function | Cost function for logistic regression --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=E-ZcnndnY2I
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bit higher, but not that high. Whereas in contrast, if the algorithm were to have output 0.1, if it thinks that there's only a 10% chance of the tumor being malignant, but y really is one, it really is malignant, then the loss is this much higher value over here. So when y is equal to one, the loss function incentivizes, or nudges, or helps push the algorithm to make more accurate predictions, because the loss is lowest when it predicts values close to one. Now on this slide, we've been looking at what the loss is when y is equal to one. On this slide, let's look at the second part of the loss function corresponding to when y is equal to zero. In this case, the loss is negative log of one minus f of x. When this function is plotted, it actually looks like this. The range of f is limited to zero to one, because logistic regression only outputs values between zero and one, and if we zoom in, this is what it looks like. So in this plot, corresponding to y equals zero, the vertical axis shows the value of the loss for different values of f of x. So when f is zero or very close to zero, the loss is also going to be very small, which means that if the true label is zero and the model's prediction is very close to zero, well, you nearly got it right. So the loss is appropriately very close to zero. And the larger the value of f of x gets, the bigger the loss, because the prediction is further from the true label zero. And in fact, as that prediction approaches one, the loss actually approaches infinity. Going back to the tumor prediction example, this is if a model predicts that the patient's tumor is almost certain to be malignant, say 99.9% chance of malignancy, but it turns out to actually not be malignant. So y equals zero, then we penalize the model with a very high loss. So in this case of y equals zero, similar to the case of y equals one on the previous slide, the further the prediction f of x is away from the true value of y, the higher the loss. And in fact, if f of x approaches zero, the loss here actually goes really, really large and in fact approaches infinity. So when the true label is one, the algorithm is strongly incentivized not to predict something too close to zero. So in this video, you saw why the squared error cost function doesn't work well for logistic regression. We also defined the loss for a single training example and came up with a new definition for the loss function for logistic regression. It turns out that with this choice of loss function, the overall cost function will be convex and thus you can reliably use gradient descent to take you to the global minimum. Proving
| 500
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3.4 Cost function | Cost function for logistic regression --[Machine Learning | Andrew Ng]: bit higher, but not that high. Whereas in contrast, if the algorithm were to have output 0.1, if it thinks that there's only a 10% chance of the tumor being malignant, but y really is one, it really is malignant, then the loss is this much higher value over here. So when y is equal to one, the loss function incentivizes, or nudges, or helps push the algorithm to make more accurate predictions, because the loss is lowest when it predicts values close to one. Now on this slide, we've been looking at what the loss is when y is equal to one. On this slide, let's look at the second part of the loss function corresponding to when y is equal to zero. In this case, the loss is negative log of one minus f of x. When this function is plotted, it actually looks like this. The range of f is limited to zero to one, because logistic regression only outputs values between zero and one, and if we zoom in, this is what it looks like. So in this plot, corresponding to y equals zero, the vertical axis shows the value of the loss for different values of f of x. So when f is zero or very close to zero, the loss is also going to be very small, which means that if the true label is zero and the model's prediction is very close to zero, well, you nearly got it right. So the loss is appropriately very close to zero. And the larger the value of f of x gets, the bigger the loss, because the prediction is further from the true label zero. And in fact, as that prediction approaches one, the loss actually approaches infinity. Going back to the tumor prediction example, this is if a model predicts that the patient's tumor is almost certain to be malignant, say 99.9% chance of malignancy, but it turns out to actually not be malignant. So y equals zero, then we penalize the model with a very high loss. So in this case of y equals zero, similar to the case of y equals one on the previous slide, the further the prediction f of x is away from the true value of y, the higher the loss. And in fact, if f of x approaches zero, the loss here actually goes really, really large and in fact approaches infinity. So when the true label is one, the algorithm is strongly incentivized not to predict something too close to zero. So in this video, you saw why the squared error cost function doesn't work well for logistic regression. We also defined the loss for a single training example and came up with a new definition for the loss function for logistic regression. It turns out that with this choice of loss function, the overall cost function will be convex and thus you can reliably use gradient descent to take you to the global minimum. Proving
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3.4 Cost function | Cost function for logistic regression --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=E-ZcnndnY2I
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that this function is convex is beyond the scope of this course. You may remember that the cost function is a function of the entire training set and is therefore the average or one over m times the sum of the loss function on the individual training examples. So the cost on a certain set of parameters w and b is equal to one over m times the sum over all the training examples of the loss on the training examples. And if you can find the value of the parameters w and b that minimizes this, then you'd have a pretty good set of values for the parameters w and b for logistic regression. In the upcoming optional lab, you get to take a look at how the squared error cost function doesn't work very well for classification because you see that the surface plot results in a very wiggly cost surface with many local minima. Then you take a look at the new logistic loss function and as you can see here, this produces a nice and smooth convex surface plot that does not have all those local minima. So please take a look at the code and the plots after this video. Alright so we've seen a lot in this video. In the next video, let's go back and take the loss function for a single training example and use that to define the overall cost function for the entire training set. And we'll also figure out a simpler way to write out the cost function, which will then later allow us to run gradient descent to find good parameters for logistic regression. Let's go on to the next video.
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3.4 Cost function | Cost function for logistic regression --[Machine Learning | Andrew Ng]: that this function is convex is beyond the scope of this course. You may remember that the cost function is a function of the entire training set and is therefore the average or one over m times the sum of the loss function on the individual training examples. So the cost on a certain set of parameters w and b is equal to one over m times the sum over all the training examples of the loss on the training examples. And if you can find the value of the parameters w and b that minimizes this, then you'd have a pretty good set of values for the parameters w and b for logistic regression. In the upcoming optional lab, you get to take a look at how the squared error cost function doesn't work very well for classification because you see that the surface plot results in a very wiggly cost surface with many local minima. Then you take a look at the new logistic loss function and as you can see here, this produces a nice and smooth convex surface plot that does not have all those local minima. So please take a look at the code and the plots after this video. Alright so we've seen a lot in this video. In the next video, let's go back and take the loss function for a single training example and use that to define the overall cost function for the entire training set. And we'll also figure out a simpler way to write out the cost function, which will then later allow us to run gradient descent to find good parameters for logistic regression. Let's go on to the next video.
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3.5 Cost Function | Simplified Cost Function for Logistic Regression --[Machine Learning|Andrew Ng]
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https://www.youtube.com/watch?v=wB08Jlmhi24
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In the last video, you saw the loss function and the cost function for logistic regression. In this video, you'll see a slightly simpler way to write out the loss and cost functions so that the implementation can be a bit simpler when we get to gradient descent for fitting the parameters of a logistic regression model. Let's take a look. As a reminder, here's the loss function that we had defined in the previous video for logistic regression. Now, because we're still working on the binary classification problem, y is either 0 or 1. Because y is either 0 or 1 and cannot take on any value other than 0 or 1, we'll be able to come up with a simpler way to write this loss function. You can write the loss function as follows. Given the prediction f of x and the target label y, the loss equals negative y times log of f minus 1 minus y times log of 1 minus f. And it turns out this equation, which we just wrote in one line, is completely equivalent to this more complex formula up here. Let's see why this is the case. Now remember, y can only take on the values of either 1 or 0. In the first case, let's say y equals 1. This first y over here is 1, and this 1 minus y is 1 minus 1, which is therefore equal to 0. And so the loss becomes negative 1 times log of f of x minus 0 times a bunch of stuff that becomes 0 and goes away. And so when y is equal to 1, the loss is indeed the first term on top, negative log of f of x. Let's look at the second case, when y is equal to 0. In this case, this y here is equal to 0, so this first term goes away. And the second term is 1 minus 0 times that logarithmic term. So the loss becomes this negative 1 times log of 1 minus f of x. And that's just equal to this second term up here. And so in the case of y equals 0, we also get back the original loss function as defined above. So what you see is that whether y is 1 or 0, the single expression here is equivalent to the more complex expression up here, which is why this gives us a simpler way to write the loss with just one equation without separating out these two cases like we did on top. Using this simplified loss function, let's go back and write out the cost function for logistic regression. So here again is the simplified loss function. And recall that the cost j is just the average loss, average across the entire training set of m examples. So it's 1 over m times the sum of the loss from i equals 1 to m. Now if you plug in a definition for the simplified
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3.5 Cost Function | Simplified Cost Function for Logistic Regression --[Machine Learning|Andrew Ng]: In the last video, you saw the loss function and the cost function for logistic regression. In this video, you'll see a slightly simpler way to write out the loss and cost functions so that the implementation can be a bit simpler when we get to gradient descent for fitting the parameters of a logistic regression model. Let's take a look. As a reminder, here's the loss function that we had defined in the previous video for logistic regression. Now, because we're still working on the binary classification problem, y is either 0 or 1. Because y is either 0 or 1 and cannot take on any value other than 0 or 1, we'll be able to come up with a simpler way to write this loss function. You can write the loss function as follows. Given the prediction f of x and the target label y, the loss equals negative y times log of f minus 1 minus y times log of 1 minus f. And it turns out this equation, which we just wrote in one line, is completely equivalent to this more complex formula up here. Let's see why this is the case. Now remember, y can only take on the values of either 1 or 0. In the first case, let's say y equals 1. This first y over here is 1, and this 1 minus y is 1 minus 1, which is therefore equal to 0. And so the loss becomes negative 1 times log of f of x minus 0 times a bunch of stuff that becomes 0 and goes away. And so when y is equal to 1, the loss is indeed the first term on top, negative log of f of x. Let's look at the second case, when y is equal to 0. In this case, this y here is equal to 0, so this first term goes away. And the second term is 1 minus 0 times that logarithmic term. So the loss becomes this negative 1 times log of 1 minus f of x. And that's just equal to this second term up here. And so in the case of y equals 0, we also get back the original loss function as defined above. So what you see is that whether y is 1 or 0, the single expression here is equivalent to the more complex expression up here, which is why this gives us a simpler way to write the loss with just one equation without separating out these two cases like we did on top. Using this simplified loss function, let's go back and write out the cost function for logistic regression. So here again is the simplified loss function. And recall that the cost j is just the average loss, average across the entire training set of m examples. So it's 1 over m times the sum of the loss from i equals 1 to m. Now if you plug in a definition for the simplified
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3.5 Cost Function | Simplified Cost Function for Logistic Regression --[Machine Learning|Andrew Ng]
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https://www.youtube.com/watch?v=wB08Jlmhi24
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loss from above, then it looks like this. 1 over m times the sum of this term above. And if you bring the negative signs and move them outside, then you end up with this expression over here. And this is the cost function. The cost function that pretty much everyone uses to train logistic regression. Now you might be wondering, why do we choose this particular function when there could be tons of other cost functions we could have chosen? Although we won't have time to go into great detail on this in this class, I'd just like to mention that this particular cost function is derived from statistics using a statistical principle called maximum likelihood estimation, which is an idea from statistics on how to efficiently find parameters for different models. And this cost function has the nice property that it is convex. But don't worry about learning the details of maximum likelihood. This puts a deeper rationale and justification behind this particular cost function. The upcoming optional lab will show you how the logistic cost function is implemented in code. I recommend taking a look at it because you implement this later in the practice lab at the end of the week. This upcoming optional lab also shows you how two different choices of the parameters will lead to different cost calculations. So you can see in the plot that the better fitting blue decision boundary has a lower cost relative to the magenta decision boundary. So with the simplified cost function, we're now ready to jump into applying gradient descent to logistic regression. Let's go see that in the next video.
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3.5 Cost Function | Simplified Cost Function for Logistic Regression --[Machine Learning|Andrew Ng]: loss from above, then it looks like this. 1 over m times the sum of this term above. And if you bring the negative signs and move them outside, then you end up with this expression over here. And this is the cost function. The cost function that pretty much everyone uses to train logistic regression. Now you might be wondering, why do we choose this particular function when there could be tons of other cost functions we could have chosen? Although we won't have time to go into great detail on this in this class, I'd just like to mention that this particular cost function is derived from statistics using a statistical principle called maximum likelihood estimation, which is an idea from statistics on how to efficiently find parameters for different models. And this cost function has the nice property that it is convex. But don't worry about learning the details of maximum likelihood. This puts a deeper rationale and justification behind this particular cost function. The upcoming optional lab will show you how the logistic cost function is implemented in code. I recommend taking a look at it because you implement this later in the practice lab at the end of the week. This upcoming optional lab also shows you how two different choices of the parameters will lead to different cost calculations. So you can see in the plot that the better fitting blue decision boundary has a lower cost relative to the magenta decision boundary. So with the simplified cost function, we're now ready to jump into applying gradient descent to logistic regression. Let's go see that in the next video.
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3.6 Gradient Descent | Gradient Descent Implementation --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=D59jK8T9dfI
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To fit the parameters of a logistic regression model, we're going to try to find the values of the parameters w and b that minimise the cost function j of w and b. And we're going to apply gradient descent to do this. Let's take a look at how. In this video, we'll focus on how to find a good choice of the parameters w and b. After you've done so, if you give the model a new input x, say a new patient at the hospital with a certain tumour size and age that needs a diagnosis, the model can then make a prediction or it can try to estimate the probability that the label y is 1. The algorithm you can use to minimise the cost function is gradient descent. Here again is the cost function, and so if you want to minimise the cost j as a function of w and b, well, here's the usual gradient descent algorithm where you repeatedly update each parameter as the O value minus alpha, the learning rate, times this derivative term. Let's take a look at the derivative of j with respect to wj, this term up on top here, where as usual j goes from 1 through n, where n is the number of features. If someone were to apply the rules of calculus, you can show that the derivative with respect to wj of the cost function capital J is equal to this expression over here. Is 1 over m times the sum from 1 through m of this error term, that is f minus the label y times xj. Here this xij is the j feature of training example i. Now let's also look at the derivative of j with respect to the parameter b. It turns out to be this expression over here, and it's quite similar to the expression above, except that it is not multiplied by this x super strip i subscript j at the end. So just as a reminder, similar to what you saw for linear regression, the way to carry out these updates is to use simultaneous updates, meaning that you would first compute the right hand side for all of these updates, and then simultaneously overwrite all the values on the left at the same time. So let me take these derivative expressions here and plug them into these terms here. This gives you gradient descent for logistic regression. Now one funny thing you might be wondering is, huh, that's weird. These two equations look exactly like the algorithm we had come up with previously for linear regression. So you might be wondering, is linear regression actually secretly the same as logistic regression? Well, even though these equations look the same, the reason that this is not linear regression is because the definition for the function f of x has changed. In linear regression, f of x is this is w x plus b, but in logistic regression, f of
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3.6 Gradient Descent | Gradient Descent Implementation --[Machine Learning | Andrew Ng]: To fit the parameters of a logistic regression model, we're going to try to find the values of the parameters w and b that minimise the cost function j of w and b. And we're going to apply gradient descent to do this. Let's take a look at how. In this video, we'll focus on how to find a good choice of the parameters w and b. After you've done so, if you give the model a new input x, say a new patient at the hospital with a certain tumour size and age that needs a diagnosis, the model can then make a prediction or it can try to estimate the probability that the label y is 1. The algorithm you can use to minimise the cost function is gradient descent. Here again is the cost function, and so if you want to minimise the cost j as a function of w and b, well, here's the usual gradient descent algorithm where you repeatedly update each parameter as the O value minus alpha, the learning rate, times this derivative term. Let's take a look at the derivative of j with respect to wj, this term up on top here, where as usual j goes from 1 through n, where n is the number of features. If someone were to apply the rules of calculus, you can show that the derivative with respect to wj of the cost function capital J is equal to this expression over here. Is 1 over m times the sum from 1 through m of this error term, that is f minus the label y times xj. Here this xij is the j feature of training example i. Now let's also look at the derivative of j with respect to the parameter b. It turns out to be this expression over here, and it's quite similar to the expression above, except that it is not multiplied by this x super strip i subscript j at the end. So just as a reminder, similar to what you saw for linear regression, the way to carry out these updates is to use simultaneous updates, meaning that you would first compute the right hand side for all of these updates, and then simultaneously overwrite all the values on the left at the same time. So let me take these derivative expressions here and plug them into these terms here. This gives you gradient descent for logistic regression. Now one funny thing you might be wondering is, huh, that's weird. These two equations look exactly like the algorithm we had come up with previously for linear regression. So you might be wondering, is linear regression actually secretly the same as logistic regression? Well, even though these equations look the same, the reason that this is not linear regression is because the definition for the function f of x has changed. In linear regression, f of x is this is w x plus b, but in logistic regression, f of
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3.6 Gradient Descent | Gradient Descent Implementation --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=D59jK8T9dfI
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x is defined to be the sigmoid function applied to w x plus b. So although the algorithm written looked the same for both linear regression and logistic regression, actually they're two very different algorithms because the definition for f of x is not the same. When we talked about gradient descent for linear regression previously, you saw how you can monitor gradient descent to make sure it converges. You can just apply the same method for logistic regression to make sure it also converges. I've written out these updates as if you're updating the parameters w j one parameter at a time. Similar to the discussion on vectorized implementations of linear regression, you can also use vectorization to make gradient descent run faster for logistic regression. I won't dive into the details of the vectorized implementation in this video, but you can also learn more about it and see the code in the optional labs. So now you know how to implement gradient descent for logistic regression. You might also remember feature scaling when we were using linear regression, where you saw how feature scaling that is scaling all the features to take on similar ranges of values, say between negative one and plus one, how that can help gradient descent to converge faster. Feature scaling applied the same way to scale the different features to take on similar ranges of values can also speed up gradient descent for logistic regression. In the upcoming optional lab, you also see how the gradient for the regression can be calculated in code. This will be useful to look at because you also implement this in a practice lab at the end of this week. After you run gradient descent in this lab, there'll be a nice set of animated plots that show gradient descent in action. You see the sigmoid function, the control plot of the cost, the 3d surface plot to the cost and the learning curve all evolve as gradient descent runs. There will be another optional lab after that, which is short and sweet, but also very useful because they'll show you how to use the popular scikit-learn library to train the logistic regression model for classification. Many machine learning practitioners in many companies today use scikit-learn regularly as part of their job. And so I hope you check out the scikit-learn function as well and take a look at how that is used. So that's it. You now know how to implement logistic regression. This is a very powerful, very widely used learning algorithm and you now know how to get it to work yourself. Congratulations.
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3.6 Gradient Descent | Gradient Descent Implementation --[Machine Learning | Andrew Ng]: x is defined to be the sigmoid function applied to w x plus b. So although the algorithm written looked the same for both linear regression and logistic regression, actually they're two very different algorithms because the definition for f of x is not the same. When we talked about gradient descent for linear regression previously, you saw how you can monitor gradient descent to make sure it converges. You can just apply the same method for logistic regression to make sure it also converges. I've written out these updates as if you're updating the parameters w j one parameter at a time. Similar to the discussion on vectorized implementations of linear regression, you can also use vectorization to make gradient descent run faster for logistic regression. I won't dive into the details of the vectorized implementation in this video, but you can also learn more about it and see the code in the optional labs. So now you know how to implement gradient descent for logistic regression. You might also remember feature scaling when we were using linear regression, where you saw how feature scaling that is scaling all the features to take on similar ranges of values, say between negative one and plus one, how that can help gradient descent to converge faster. Feature scaling applied the same way to scale the different features to take on similar ranges of values can also speed up gradient descent for logistic regression. In the upcoming optional lab, you also see how the gradient for the regression can be calculated in code. This will be useful to look at because you also implement this in a practice lab at the end of this week. After you run gradient descent in this lab, there'll be a nice set of animated plots that show gradient descent in action. You see the sigmoid function, the control plot of the cost, the 3d surface plot to the cost and the learning curve all evolve as gradient descent runs. There will be another optional lab after that, which is short and sweet, but also very useful because they'll show you how to use the popular scikit-learn library to train the logistic regression model for classification. Many machine learning practitioners in many companies today use scikit-learn regularly as part of their job. And so I hope you check out the scikit-learn function as well and take a look at how that is used. So that's it. You now know how to implement logistic regression. This is a very powerful, very widely used learning algorithm and you now know how to get it to work yourself. Congratulations.
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3.7 Regularization to Reduce Overfitting | The problem of Overfitting -[Machine Learning|Andrew Ng]
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https://www.youtube.com/watch?v=wBx3NZ0ucgc
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Now you've seen a couple of different learning algorithms, linear regression and logistic regression. They work well for many tasks, but sometimes in an application the algorithm could run into a problem called overfitting, which can cause it to perform poorly. What I'd like to do in this video is to show you what is overfitting, as well as a closely related almost opposite problem called underfitting. And in the next videos after this, I'll share with you some techniques for addressing overfitting. In particular, there's a method called regularization. Very useful technique, I use it all the time, but regularization will help you minimize this overfitting problem and get your learning algorithms to work much better. So let's take a look at what is overfitting. To help us understand what is overfitting, let's take a look at a few examples. Let's go back to our original example of predicting housing prices with linear regression, where you want to predict the price as a function of the size of a house. To help us understand what is overfitting, let's take a look at a linear regression example. And I'm going to go back to our original running example of predicting housing prices with linear regression. Suppose your data set looks like this, with the input feature x being the size of the house and the value y they're trying to predict the price of the house. One thing you could do is fit a linear function to this data. And if you do that, you get a straight line fit to the data that maybe looks like this. But this isn't a very good model. Looking at the data, it seems pretty clear that as the size of the house increases, the housing prices kind of flatten out. So this algorithm does not fit the training data very well. The technical term for this is the model is underfitting the training data. Another term is the algorithm has high bias. You may have read in the news about some learning algorithms, really unfortunately, demonstrating bias against certain ethnicities or certain genders. In machine learning, the term bias has multiple meanings. Checking learning algorithms for bias based on characteristics such as gender or ethnicity is absolutely critical. But the term bias has a second technical meaning as well, which is the one I'm using here, which is if the algorithm has underfit the data, meaning that it's just not even able to fit the training set that well, that there's a clear pattern in the training data that the algorithm is just unable to capture. Another way to think of this form of bias is as if the learning algorithm has a very strong preconception or we say a very strong bias that the housing prices are going to be a completely linear function of the size, despite data to the contrary. So this preconception that the data is linear causes it to fit a straight line that fits the data
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3.7 Regularization to Reduce Overfitting | The problem of Overfitting -[Machine Learning|Andrew Ng]: Now you've seen a couple of different learning algorithms, linear regression and logistic regression. They work well for many tasks, but sometimes in an application the algorithm could run into a problem called overfitting, which can cause it to perform poorly. What I'd like to do in this video is to show you what is overfitting, as well as a closely related almost opposite problem called underfitting. And in the next videos after this, I'll share with you some techniques for addressing overfitting. In particular, there's a method called regularization. Very useful technique, I use it all the time, but regularization will help you minimize this overfitting problem and get your learning algorithms to work much better. So let's take a look at what is overfitting. To help us understand what is overfitting, let's take a look at a few examples. Let's go back to our original example of predicting housing prices with linear regression, where you want to predict the price as a function of the size of a house. To help us understand what is overfitting, let's take a look at a linear regression example. And I'm going to go back to our original running example of predicting housing prices with linear regression. Suppose your data set looks like this, with the input feature x being the size of the house and the value y they're trying to predict the price of the house. One thing you could do is fit a linear function to this data. And if you do that, you get a straight line fit to the data that maybe looks like this. But this isn't a very good model. Looking at the data, it seems pretty clear that as the size of the house increases, the housing prices kind of flatten out. So this algorithm does not fit the training data very well. The technical term for this is the model is underfitting the training data. Another term is the algorithm has high bias. You may have read in the news about some learning algorithms, really unfortunately, demonstrating bias against certain ethnicities or certain genders. In machine learning, the term bias has multiple meanings. Checking learning algorithms for bias based on characteristics such as gender or ethnicity is absolutely critical. But the term bias has a second technical meaning as well, which is the one I'm using here, which is if the algorithm has underfit the data, meaning that it's just not even able to fit the training set that well, that there's a clear pattern in the training data that the algorithm is just unable to capture. Another way to think of this form of bias is as if the learning algorithm has a very strong preconception or we say a very strong bias that the housing prices are going to be a completely linear function of the size, despite data to the contrary. So this preconception that the data is linear causes it to fit a straight line that fits the data
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3.7 Regularization to Reduce Overfitting | The problem of Overfitting -[Machine Learning|Andrew Ng]
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https://www.youtube.com/watch?v=wBx3NZ0ucgc
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poorly, leading it to underfit the data. Now let's look at a second variation of a model, which is if you instead fit a quadratic function to the data with two features, x and x squared, then when you fit the parameters w1 and w2, you can get a curve that fits the data somewhat better. Maybe it looks like this. So if you were to get a new house that's not in this set of five training examples, this model would probably do quite well on that new house. So if you're a real estate agent, the idea that you want your learning algorithm to do well, even on examples that are not on the training set, that's called generalization. Technically, we say that you want your learning algorithm to generalize well, which means to make good predictions even on brand new examples that it has never seen before. So this quadratic model seems to fit the training set not perfectly, but pretty well. And I think it will generalize well to new examples. Now let's look at the other extreme. What if you were to fit a fourth order polynomial to the data? So you have x, x squared, x cubed, and x to the fourth all as features. With this fourth order polynomial, you can actually fit the curve that passes through all five of the training examples exactly. And you might get a curve that looks like this. This on one hand seems to do an extremely good job fitting the training data, because it passes through all of the training data perfectly. In fact, you'll be able to choose parameters that will result in the cost function being exactly equal to zero, because the errors are zero on all five training examples. But this is a very wiggly curve. It's going up and down all over the place. And if you have this house size right here, the model would predict that this house is cheaper than houses that are smaller than it. So we don't think that this is a particularly good model for predicting housing prices. The technical term is that we'll say this model has overfit the data, or this model has an overfitting problem. Because even though it fits the training set very well, it has fit the data almost too well, hence is overfit. And it does not look like this model will generalize to new examples that has never seen before. Another term for this is that the algorithm has high variance. In machine learning, many people will use the terms overfit and high variance almost interchangeably, and we use the terms underfit and high bias almost interchangeably. The intuition behind overfitting or high variance is that the algorithm is trying very, very hard to fit every single training example. And it turns out that if your training set were just even a little bit different, say one house was priced just a little bit more, a little bit
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3.7 Regularization to Reduce Overfitting | The problem of Overfitting -[Machine Learning|Andrew Ng]: poorly, leading it to underfit the data. Now let's look at a second variation of a model, which is if you instead fit a quadratic function to the data with two features, x and x squared, then when you fit the parameters w1 and w2, you can get a curve that fits the data somewhat better. Maybe it looks like this. So if you were to get a new house that's not in this set of five training examples, this model would probably do quite well on that new house. So if you're a real estate agent, the idea that you want your learning algorithm to do well, even on examples that are not on the training set, that's called generalization. Technically, we say that you want your learning algorithm to generalize well, which means to make good predictions even on brand new examples that it has never seen before. So this quadratic model seems to fit the training set not perfectly, but pretty well. And I think it will generalize well to new examples. Now let's look at the other extreme. What if you were to fit a fourth order polynomial to the data? So you have x, x squared, x cubed, and x to the fourth all as features. With this fourth order polynomial, you can actually fit the curve that passes through all five of the training examples exactly. And you might get a curve that looks like this. This on one hand seems to do an extremely good job fitting the training data, because it passes through all of the training data perfectly. In fact, you'll be able to choose parameters that will result in the cost function being exactly equal to zero, because the errors are zero on all five training examples. But this is a very wiggly curve. It's going up and down all over the place. And if you have this house size right here, the model would predict that this house is cheaper than houses that are smaller than it. So we don't think that this is a particularly good model for predicting housing prices. The technical term is that we'll say this model has overfit the data, or this model has an overfitting problem. Because even though it fits the training set very well, it has fit the data almost too well, hence is overfit. And it does not look like this model will generalize to new examples that has never seen before. Another term for this is that the algorithm has high variance. In machine learning, many people will use the terms overfit and high variance almost interchangeably, and we use the terms underfit and high bias almost interchangeably. The intuition behind overfitting or high variance is that the algorithm is trying very, very hard to fit every single training example. And it turns out that if your training set were just even a little bit different, say one house was priced just a little bit more, a little bit
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3.7 Regularization to Reduce Overfitting | The problem of Overfitting -[Machine Learning|Andrew Ng]
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https://www.youtube.com/watch?v=wBx3NZ0ucgc
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less, then the function that the algorithm fits could end up being totally different. So if two different machine learning engineers were to fit this fourth order polynomial model to just slightly different data sets, they could end up with totally different predictions or highly variable predictions. And that's why we say the algorithm has high variance. Using this rightmost model with the one in the middle for the same house, it seems the middle model gives the much more reasonable prediction for price. There isn't really a name for this case in the middle, but I'm just going to call this just right because it is neither underfit nor overfit. So we can say that the goal of machine learning is to find a model that hopefully is neither underfitting nor overfitting. In other words, hopefully a model that has neither high bias nor high variance. When I think about underfitting and overfitting, high bias and high variance, I'm sometimes reminded of the children's story of Goldilocks and the three bears. In this children's tale, a girl called Goldilocks visits the home of a bear family. There's a bowl of porridge that's too cold to taste and so that's no good. There's also a bowl of porridge that's too hot to eat. So that's no good either. But there's a bowl of porridge that is neither too cold nor too hot. The temperature is in the middle, which is just right to eat. So to recap, if you have too many features, like the full water polynomial on the right, then the model may fit the training set well, but almost too well or overfit in the high variance. On the flip side, if you have too few features, then in this example, like the one on the left, it underfits and has high bias. And in this example, using quadratic features x and x squared, that seems to be just right. So far, we've looked at underfitting and overfitting for linear regression model. Similarly, overfitting applies to classification as well. Here's a classification example with two features x1 and x2, where x1 is maybe the tumor size and x2 is the age of patient. And we're trying to classify if a tumor is malignant or benign, as denoted by these crosses and circles. One thing you can do is fit a logistic regression model, just a simple model like this, where as usual, g is the sigmoid function, and this term here inside is z. So if you do that, you end up with a straight line as the decision boundary. This is the line where z is equal to zero that separates the positive and negative examples. This straight line doesn't look terrible. It looks kind of okay, but it doesn't look like a very good fit to the data either. So this is an example of underfitting or of high bias. Let's look at another example. If you were to add to your features these quadratic
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3.7 Regularization to Reduce Overfitting | The problem of Overfitting -[Machine Learning|Andrew Ng]: less, then the function that the algorithm fits could end up being totally different. So if two different machine learning engineers were to fit this fourth order polynomial model to just slightly different data sets, they could end up with totally different predictions or highly variable predictions. And that's why we say the algorithm has high variance. Using this rightmost model with the one in the middle for the same house, it seems the middle model gives the much more reasonable prediction for price. There isn't really a name for this case in the middle, but I'm just going to call this just right because it is neither underfit nor overfit. So we can say that the goal of machine learning is to find a model that hopefully is neither underfitting nor overfitting. In other words, hopefully a model that has neither high bias nor high variance. When I think about underfitting and overfitting, high bias and high variance, I'm sometimes reminded of the children's story of Goldilocks and the three bears. In this children's tale, a girl called Goldilocks visits the home of a bear family. There's a bowl of porridge that's too cold to taste and so that's no good. There's also a bowl of porridge that's too hot to eat. So that's no good either. But there's a bowl of porridge that is neither too cold nor too hot. The temperature is in the middle, which is just right to eat. So to recap, if you have too many features, like the full water polynomial on the right, then the model may fit the training set well, but almost too well or overfit in the high variance. On the flip side, if you have too few features, then in this example, like the one on the left, it underfits and has high bias. And in this example, using quadratic features x and x squared, that seems to be just right. So far, we've looked at underfitting and overfitting for linear regression model. Similarly, overfitting applies to classification as well. Here's a classification example with two features x1 and x2, where x1 is maybe the tumor size and x2 is the age of patient. And we're trying to classify if a tumor is malignant or benign, as denoted by these crosses and circles. One thing you can do is fit a logistic regression model, just a simple model like this, where as usual, g is the sigmoid function, and this term here inside is z. So if you do that, you end up with a straight line as the decision boundary. This is the line where z is equal to zero that separates the positive and negative examples. This straight line doesn't look terrible. It looks kind of okay, but it doesn't look like a very good fit to the data either. So this is an example of underfitting or of high bias. Let's look at another example. If you were to add to your features these quadratic
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3.7 Regularization to Reduce Overfitting | The problem of Overfitting -[Machine Learning|Andrew Ng]
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https://www.youtube.com/watch?v=wBx3NZ0ucgc
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terms, then z becomes this new term in the middle and the decision boundary, that is where z equals zero, can look more like this, more like an ellipse or part of an ellipse. And this is a pretty good fit to the data, even though it does not perfectly classify every single training sample in the training set. Notice how some of these crosses get classified among the circles. But this model looks pretty good. I'm going to call it just right, and it looks like this will generalize pretty well to new patients. And finally, at the other extreme, if you were to fit a very high order polynomial with many, many features like these, then the model may try really hard and can't hold or twist itself to find a decision boundary that fits your training data perfectly. Having all these higher order polynomial features allows the algorithm to choose this really overly complex decision boundary. If the features are tumor size and age, and you're trying to classify tumors as malignant or benign, then this doesn't really look like a very good model for making predictions. So once again, this is an instance of overfitting and high variance, because this model, despite doing very well in the training set, doesn't look like it will generalize well to new examples. So now you've seen how an algorithm can underfit or have high bias or overfit and have high variance. You may want to know how you can get a model that is just right. In the next video, we'll look at some ways you can address the issue of overfitting. We'll also touch on some ideas relevant for accessing underfitting. Let's go on to the next video.
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3.7 Regularization to Reduce Overfitting | The problem of Overfitting -[Machine Learning|Andrew Ng]: terms, then z becomes this new term in the middle and the decision boundary, that is where z equals zero, can look more like this, more like an ellipse or part of an ellipse. And this is a pretty good fit to the data, even though it does not perfectly classify every single training sample in the training set. Notice how some of these crosses get classified among the circles. But this model looks pretty good. I'm going to call it just right, and it looks like this will generalize pretty well to new patients. And finally, at the other extreme, if you were to fit a very high order polynomial with many, many features like these, then the model may try really hard and can't hold or twist itself to find a decision boundary that fits your training data perfectly. Having all these higher order polynomial features allows the algorithm to choose this really overly complex decision boundary. If the features are tumor size and age, and you're trying to classify tumors as malignant or benign, then this doesn't really look like a very good model for making predictions. So once again, this is an instance of overfitting and high variance, because this model, despite doing very well in the training set, doesn't look like it will generalize well to new examples. So now you've seen how an algorithm can underfit or have high bias or overfit and have high variance. You may want to know how you can get a model that is just right. In the next video, we'll look at some ways you can address the issue of overfitting. We'll also touch on some ideas relevant for accessing underfitting. Let's go on to the next video.
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3.8 Regularization to Reduce Overfitting | Addressing Overfitting-[Machine Learning|Andrew Ng]
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https://www.youtube.com/watch?v=ce4CPW8AFE4
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Later in this specialization, we'll talk about debugging and diagnosing things that can go wrong with learning algorithms. You also learn about specific tools to recognize when overfitting and underfitting may be occurring. But for now, when you think overfitting has occurred, let's talk about what you can do to address it. Let's say you fit a model and it has high variance as overfit. Here's our overfit house price prediction model. One way to address this problem is to collect more training data. So that's one option. If you're able to get more data, that is more training examples on sizes and prices of houses, then with the larger training set, the learning algorithm will learn to fit a function that is less wiggly. So you can continue to fit a high order polynomial or some other function with a lot of features. And if you have enough training examples, it will still do okay. So to summarize, the number one tool you can use against overfitting is to get more training data. Now, getting more data isn't always an option. Maybe only so many houses have been sold in this location. So maybe there just isn't more data to be had. So when the data is available, this can work really well. A second option for addressing overfitting is to see if you can use fewer features. In the previous video, our model's features included the size x as well as the size squared, that is x squared, and x cubed and x to the 4 and so on. These were a lot of polynomial features. So in that case, one way to reduce overfitting is to just not use so many of these polynomial features. But now let's look at a different example. Maybe you have a lot of different features of a house with which to try to predict its price, ranging from the size, number of bedrooms, number of floors, the age, average income of the neighborhood, and so on and so forth to the distance to the nearest coffee shop. It turns out that if you have a lot of features like these, but don't have enough training data, then your learning algorithm may also overfit to your training set. Now instead of using all 100 features, if we were to pick just a subset of the most useful ones, maybe size, bedrooms, and the age of the house, if you think those are the most relevant features, then using just that smaller subset of features, you may find that your model no longer overfits as badly. Choosing the most appropriate set of features to use is sometimes also called feature selection. One way you could do so is to use your intuition to choose what you think is the best set of features. What's most relevant for predicting the price. Now one disadvantage of feature selection is that by using only a subset of the features, the algorithm is throwing away some of
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3.8 Regularization to Reduce Overfitting | Addressing Overfitting-[Machine Learning|Andrew Ng]: Later in this specialization, we'll talk about debugging and diagnosing things that can go wrong with learning algorithms. You also learn about specific tools to recognize when overfitting and underfitting may be occurring. But for now, when you think overfitting has occurred, let's talk about what you can do to address it. Let's say you fit a model and it has high variance as overfit. Here's our overfit house price prediction model. One way to address this problem is to collect more training data. So that's one option. If you're able to get more data, that is more training examples on sizes and prices of houses, then with the larger training set, the learning algorithm will learn to fit a function that is less wiggly. So you can continue to fit a high order polynomial or some other function with a lot of features. And if you have enough training examples, it will still do okay. So to summarize, the number one tool you can use against overfitting is to get more training data. Now, getting more data isn't always an option. Maybe only so many houses have been sold in this location. So maybe there just isn't more data to be had. So when the data is available, this can work really well. A second option for addressing overfitting is to see if you can use fewer features. In the previous video, our model's features included the size x as well as the size squared, that is x squared, and x cubed and x to the 4 and so on. These were a lot of polynomial features. So in that case, one way to reduce overfitting is to just not use so many of these polynomial features. But now let's look at a different example. Maybe you have a lot of different features of a house with which to try to predict its price, ranging from the size, number of bedrooms, number of floors, the age, average income of the neighborhood, and so on and so forth to the distance to the nearest coffee shop. It turns out that if you have a lot of features like these, but don't have enough training data, then your learning algorithm may also overfit to your training set. Now instead of using all 100 features, if we were to pick just a subset of the most useful ones, maybe size, bedrooms, and the age of the house, if you think those are the most relevant features, then using just that smaller subset of features, you may find that your model no longer overfits as badly. Choosing the most appropriate set of features to use is sometimes also called feature selection. One way you could do so is to use your intuition to choose what you think is the best set of features. What's most relevant for predicting the price. Now one disadvantage of feature selection is that by using only a subset of the features, the algorithm is throwing away some of
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3.8 Regularization to Reduce Overfitting | Addressing Overfitting-[Machine Learning|Andrew Ng]
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https://www.youtube.com/watch?v=ce4CPW8AFE4
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the information that you have about the houses. For example, maybe all of these features, or 100 of them, are actually useful for predicting the price of a house. So maybe you don't want to throw away some of the information by throwing away some of the features. Later in course two, you also see some algorithms for automatically choosing the most appropriate set of features to use for prediction tasks. Now this takes us to the third option for reducing overfitting. This technique, which we'll look at in even greater depth in the next video, is called regularization. If you look at an overfit model, here's a model using polynomial features x, x squared, x cubed, and so on. You find that the parameters are often relatively large. Now if you were to eliminate some of these features, say if you were to eliminate the feature x4, that corresponds to setting this parameter to zero. So setting a parameter to zero is equivalent to eliminating a feature, which is what we saw on the previous slide. It turns out that regularization is a way to more gently reduce the impacts of some of the features without doing something as harsh as eliminating it outright. What regularization does is encourage the learning algorithm to shrink the values of the parameters without necessarily demanding that the parameter is set to exactly zero. And it turns out that even if you fit a higher order polynomial like this, so long as you can get the algorithm to use smaller parameter values, w1, w2, w3, w4, you end up with a curve that ends up fitting the training data much better. So what regularization does is it lets you keep all of your features, but it just prevents the features from having an overly large effect, which is what sometimes can cause overfitting. By the way, by convention, we normally just reduce the size of the wj parameters, that is w1 through wn. It kind of doesn't make a huge difference whether you regularize the parameter b as well. You could do so if you want or not if you don't. I usually don't and it's just fine to regularize w1, w2 all the way to wn, but not really encourage b to become smaller. In practice, it should make very little difference whether you also regularize b or not. So to recap, these are the three ways you saw in this video for addressing overfitting. One, collect more data. If you can get more data, this can really help reduce overfitting. Sometimes that's not possible, in which case some of the options are, two, try selecting and using only a subset of the features. You learn more about feature selection in course two. Three would be to reduce the size of the parameters using regularization. This will be the subject of the next video as well. As for myself, I use regularization all the time. So this is a very useful technique
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3.8 Regularization to Reduce Overfitting | Addressing Overfitting-[Machine Learning|Andrew Ng]: the information that you have about the houses. For example, maybe all of these features, or 100 of them, are actually useful for predicting the price of a house. So maybe you don't want to throw away some of the information by throwing away some of the features. Later in course two, you also see some algorithms for automatically choosing the most appropriate set of features to use for prediction tasks. Now this takes us to the third option for reducing overfitting. This technique, which we'll look at in even greater depth in the next video, is called regularization. If you look at an overfit model, here's a model using polynomial features x, x squared, x cubed, and so on. You find that the parameters are often relatively large. Now if you were to eliminate some of these features, say if you were to eliminate the feature x4, that corresponds to setting this parameter to zero. So setting a parameter to zero is equivalent to eliminating a feature, which is what we saw on the previous slide. It turns out that regularization is a way to more gently reduce the impacts of some of the features without doing something as harsh as eliminating it outright. What regularization does is encourage the learning algorithm to shrink the values of the parameters without necessarily demanding that the parameter is set to exactly zero. And it turns out that even if you fit a higher order polynomial like this, so long as you can get the algorithm to use smaller parameter values, w1, w2, w3, w4, you end up with a curve that ends up fitting the training data much better. So what regularization does is it lets you keep all of your features, but it just prevents the features from having an overly large effect, which is what sometimes can cause overfitting. By the way, by convention, we normally just reduce the size of the wj parameters, that is w1 through wn. It kind of doesn't make a huge difference whether you regularize the parameter b as well. You could do so if you want or not if you don't. I usually don't and it's just fine to regularize w1, w2 all the way to wn, but not really encourage b to become smaller. In practice, it should make very little difference whether you also regularize b or not. So to recap, these are the three ways you saw in this video for addressing overfitting. One, collect more data. If you can get more data, this can really help reduce overfitting. Sometimes that's not possible, in which case some of the options are, two, try selecting and using only a subset of the features. You learn more about feature selection in course two. Three would be to reduce the size of the parameters using regularization. This will be the subject of the next video as well. As for myself, I use regularization all the time. So this is a very useful technique
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3.8 Regularization to Reduce Overfitting | Addressing Overfitting-[Machine Learning|Andrew Ng]
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https://www.youtube.com/watch?v=ce4CPW8AFE4
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for training learning algorithms, including neural networks specifically, which you see later in the specialization as well. I hope you also check out the optional lab on overfitting. In the lab, you will see different examples of overfitting and adjust those examples by clicking on options in the plots. You also be able to add your own data points by clicking on the plot and see how that changes the curve that is fit. You can also try examples for both regression and classification, and you really change the degree of the polynomial to be x, x squared, x cubed, and so on. The lab also lets you play with two different options for addressing overfitting. You can add additional training data to reduce overfitting, and you can also select which features to include or to exclude as another way to try to reduce overfitting. So please take a look at the lab, which I hope will help you build your intuition about overfitting as well as some methods for addressing it. In this video, you also saw the idea of regularization at a relatively high level. I realize that all of these details on regularization may not fully make sense to you yet, but in the next video, we'll start to formulate exactly how to apply regularization and exactly what regularization means. And then we'll start to figure out how to make this work with our learning algorithms to make linear regression and literacy regression and in the future, other algorithms as well, avoid overfitting. Let's take a look at that in the next video.
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3.8 Regularization to Reduce Overfitting | Addressing Overfitting-[Machine Learning|Andrew Ng]: for training learning algorithms, including neural networks specifically, which you see later in the specialization as well. I hope you also check out the optional lab on overfitting. In the lab, you will see different examples of overfitting and adjust those examples by clicking on options in the plots. You also be able to add your own data points by clicking on the plot and see how that changes the curve that is fit. You can also try examples for both regression and classification, and you really change the degree of the polynomial to be x, x squared, x cubed, and so on. The lab also lets you play with two different options for addressing overfitting. You can add additional training data to reduce overfitting, and you can also select which features to include or to exclude as another way to try to reduce overfitting. So please take a look at the lab, which I hope will help you build your intuition about overfitting as well as some methods for addressing it. In this video, you also saw the idea of regularization at a relatively high level. I realize that all of these details on regularization may not fully make sense to you yet, but in the next video, we'll start to formulate exactly how to apply regularization and exactly what regularization means. And then we'll start to figure out how to make this work with our learning algorithms to make linear regression and literacy regression and in the future, other algorithms as well, avoid overfitting. Let's take a look at that in the next video.
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3.9 Regularization to Reduce Overfitting | Cost function with regularization- [ML | Andrew Ng]
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https://www.youtube.com/watch?v=SCj3h47dKL0
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In the last video, we saw that regularization tries to make the parameter values w1 through wn small to reduce overfitting. In this video, we'll build on that intuition and develop a modified cost function for your learning algorithm they can use to actually apply regularization. Let's jump in. Recall this example from the previous video in which we saw that if you fit a quadratic function to this data, it gives a pretty good fit. But if you fit a very high order polynomial, you end up with a curve that overfits the data. But now consider the following. Suppose that you had a way to make the parameters w3 and w4 really, really small, say close to zero. Here's what I mean. Let's say instead of minimizing this objective function, this is a cost function for linear regression. Let's say you were to modify the cost function and add to it 1000 times w3 squared plus 1000 times w4 squared. And here I'm just choosing 1000 because it's a big number, but any other really large number would be okay. So with this modified cost function, you'd in fact be penalizing the model if w3 and w4 are large. Because if you want to minimize this function, the only way to make this new cost function small is if w3 and w4 are both small, right? Because otherwise this 1000 times w3 squared and 1000 times w4 squared terms are going to be really, really big. So when you minimize this function, you're going to end up with w3 close to zero and w4 close to zero. So we're effectively nearly canceling out the effect of the features x cubed and x to the power of four and getting rid of these two terms over here. And if we do that, then we end up with a fit to the data that's much closer to the quadratic function, including maybe just tiny contributions from the features x cubed and x to the fourth. And this is good because it's a much better fit to the data compared to if all the parameters could be large and you end up with this wiggly quadratic function. More generally, here's the idea behind regularization. The idea is that if there are smaller values for the parameters, then that's a bit like having a simpler model, maybe one with fewer features, which is therefore less prone to overfitting. On the last slide, we penalize or we say we regularize only w3 and w4. But more generally, the way that regularization tends to be implemented is if you have a lot of features, say 100 features, you may not know which are the most important features and which ones to penalize. So the way regularization is typically implemented is to penalize all of the features or more precisely, you penalize all the wj parameters. And it's possible to show that this will usually result in fitting a smoother, simpler, less wiggly function that's
| 500
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3.9 Regularization to Reduce Overfitting | Cost function with regularization- [ML | Andrew Ng]: In the last video, we saw that regularization tries to make the parameter values w1 through wn small to reduce overfitting. In this video, we'll build on that intuition and develop a modified cost function for your learning algorithm they can use to actually apply regularization. Let's jump in. Recall this example from the previous video in which we saw that if you fit a quadratic function to this data, it gives a pretty good fit. But if you fit a very high order polynomial, you end up with a curve that overfits the data. But now consider the following. Suppose that you had a way to make the parameters w3 and w4 really, really small, say close to zero. Here's what I mean. Let's say instead of minimizing this objective function, this is a cost function for linear regression. Let's say you were to modify the cost function and add to it 1000 times w3 squared plus 1000 times w4 squared. And here I'm just choosing 1000 because it's a big number, but any other really large number would be okay. So with this modified cost function, you'd in fact be penalizing the model if w3 and w4 are large. Because if you want to minimize this function, the only way to make this new cost function small is if w3 and w4 are both small, right? Because otherwise this 1000 times w3 squared and 1000 times w4 squared terms are going to be really, really big. So when you minimize this function, you're going to end up with w3 close to zero and w4 close to zero. So we're effectively nearly canceling out the effect of the features x cubed and x to the power of four and getting rid of these two terms over here. And if we do that, then we end up with a fit to the data that's much closer to the quadratic function, including maybe just tiny contributions from the features x cubed and x to the fourth. And this is good because it's a much better fit to the data compared to if all the parameters could be large and you end up with this wiggly quadratic function. More generally, here's the idea behind regularization. The idea is that if there are smaller values for the parameters, then that's a bit like having a simpler model, maybe one with fewer features, which is therefore less prone to overfitting. On the last slide, we penalize or we say we regularize only w3 and w4. But more generally, the way that regularization tends to be implemented is if you have a lot of features, say 100 features, you may not know which are the most important features and which ones to penalize. So the way regularization is typically implemented is to penalize all of the features or more precisely, you penalize all the wj parameters. And it's possible to show that this will usually result in fitting a smoother, simpler, less wiggly function that's
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3.9 Regularization to Reduce Overfitting | Cost function with regularization- [ML | Andrew Ng]
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https://www.youtube.com/watch?v=SCj3h47dKL0
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less prone to overfitting. So for this example, if you have data with 100 features for each house, it may be hard to pick in advance which features to include and which ones to exclude. So let's build a model that uses all 100 features. So you have these 100 parameters, w1 through w100, as well as the 101st parameter b. Because we don't know which of these parameters are going to be the important ones, let's penalize all of them a bit and shrink all of them by adding this new term lambda times the sum from j equals 1 through n, where n is 100, the number of features of wj squared. This value lambda here is the Greek alphabet lambda, and it's also called a regularization parameter. So similar to picking a learning rate alpha, you now also have to choose a number for lambda. A couple things I would like to point out, by convention, instead of using lambda times the sum of wj squared, we also divide lambda by 2m, so that both the first and second terms here are scaled by 1 over 2m. It turns out that by scaling both terms the same way, it becomes a little bit easier to choose a good value for lambda, and in particular, you find that even if your training set size grows, say you find more training examples, so m, the training set size, is now bigger, the same value of lambda that you picked previously is now also more likely to continue to work if you have this extra scaling by 2m. Also by the way, by convention, we're not going to penalize the parameter b for being large. In practice, it makes very little difference whether you do or not, and some machine learning engineers, and actually some learning algorithm implementations, will also include lambda over 2m times the b squared term. But this makes very little difference in practice, and the more common convention which we'll use in this course is to regularize only the parameters w rather than the parameter b. So to summarize, in this modified cost function, we want to minimize the original cost, which is the mean squared error cost, plus additionally the second term, which is called the regularization term. And so this new cost function trades off two goals that you might have. Trying to minimize this first term encourages the algorithm to fit the training data well by minimizing the squared differences of the predictions and the actual values, and trying to minimize the second term, the algorithm also tries to keep the parameters wj small, which will tend to reduce overfitting. The value of lambda that you choose specifies the relative importance or the relative tradeoff or how you balance between these two goals. Let's take a look at what different values of lambda will cause your learning algorithm to do. Let's use the housing price prediction example using linear regression. So f of
| 500
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3.9 Regularization to Reduce Overfitting | Cost function with regularization- [ML | Andrew Ng]: less prone to overfitting. So for this example, if you have data with 100 features for each house, it may be hard to pick in advance which features to include and which ones to exclude. So let's build a model that uses all 100 features. So you have these 100 parameters, w1 through w100, as well as the 101st parameter b. Because we don't know which of these parameters are going to be the important ones, let's penalize all of them a bit and shrink all of them by adding this new term lambda times the sum from j equals 1 through n, where n is 100, the number of features of wj squared. This value lambda here is the Greek alphabet lambda, and it's also called a regularization parameter. So similar to picking a learning rate alpha, you now also have to choose a number for lambda. A couple things I would like to point out, by convention, instead of using lambda times the sum of wj squared, we also divide lambda by 2m, so that both the first and second terms here are scaled by 1 over 2m. It turns out that by scaling both terms the same way, it becomes a little bit easier to choose a good value for lambda, and in particular, you find that even if your training set size grows, say you find more training examples, so m, the training set size, is now bigger, the same value of lambda that you picked previously is now also more likely to continue to work if you have this extra scaling by 2m. Also by the way, by convention, we're not going to penalize the parameter b for being large. In practice, it makes very little difference whether you do or not, and some machine learning engineers, and actually some learning algorithm implementations, will also include lambda over 2m times the b squared term. But this makes very little difference in practice, and the more common convention which we'll use in this course is to regularize only the parameters w rather than the parameter b. So to summarize, in this modified cost function, we want to minimize the original cost, which is the mean squared error cost, plus additionally the second term, which is called the regularization term. And so this new cost function trades off two goals that you might have. Trying to minimize this first term encourages the algorithm to fit the training data well by minimizing the squared differences of the predictions and the actual values, and trying to minimize the second term, the algorithm also tries to keep the parameters wj small, which will tend to reduce overfitting. The value of lambda that you choose specifies the relative importance or the relative tradeoff or how you balance between these two goals. Let's take a look at what different values of lambda will cause your learning algorithm to do. Let's use the housing price prediction example using linear regression. So f of
|
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3.9 Regularization to Reduce Overfitting | Cost function with regularization- [ML | Andrew Ng]
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https://www.youtube.com/watch?v=SCj3h47dKL0
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x is the linear regression model. If lambda was set to be zero, then you're not using the regularization term at all, because the regularization term is multiplied by zero. And so if lambda was zero, you end up fitting this overly wiggly, overly complex curve, and it overfits. So that was one extreme of if lambda was zero. Let's now look at the other extreme. If you set lambda to be a really, really, really large number, say lambda equals 10 to the power of 10, then you're placing a very heavy weight on this regularization term on the right. And the only way to minimize this is to be sure that all the values of w are pretty much very close to zero. So if lambda is very, very large, the learning algorithm will choose w1, w2, w3, and w4 to be extremely close to zero. And thus, f of x is basically equal to b, and so the learning algorithm fits a horizontal straight line and underfits. To recap, if lambda is zero, this model will overfit. If lambda is enormous, like 10 to the power of 10, this model will underfit. And so what you want is some value of lambda that is in between that more appropriately balances these first and second terms of trading off, minimizing the mean squared error and keeping the parameters small. And when the value of lambda is not too small and not too large, but just right, then hopefully you end up able to fit a fourth order polynomial keeping all of these features, but with a function that looks like this. So that's how regularization works. When we talk about model selection later in the specialization, we'll also see a variety of ways to choose good values for lambda. In the next two videos, we'll flesh out how to apply regularization to linear regression and logistic regression, and how to train these models with gradient descent. With that, you'll be able to avoid overfitting with both of these algorithms.
| 341
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3.9 Regularization to Reduce Overfitting | Cost function with regularization- [ML | Andrew Ng]: x is the linear regression model. If lambda was set to be zero, then you're not using the regularization term at all, because the regularization term is multiplied by zero. And so if lambda was zero, you end up fitting this overly wiggly, overly complex curve, and it overfits. So that was one extreme of if lambda was zero. Let's now look at the other extreme. If you set lambda to be a really, really, really large number, say lambda equals 10 to the power of 10, then you're placing a very heavy weight on this regularization term on the right. And the only way to minimize this is to be sure that all the values of w are pretty much very close to zero. So if lambda is very, very large, the learning algorithm will choose w1, w2, w3, and w4 to be extremely close to zero. And thus, f of x is basically equal to b, and so the learning algorithm fits a horizontal straight line and underfits. To recap, if lambda is zero, this model will overfit. If lambda is enormous, like 10 to the power of 10, this model will underfit. And so what you want is some value of lambda that is in between that more appropriately balances these first and second terms of trading off, minimizing the mean squared error and keeping the parameters small. And when the value of lambda is not too small and not too large, but just right, then hopefully you end up able to fit a fourth order polynomial keeping all of these features, but with a function that looks like this. So that's how regularization works. When we talk about model selection later in the specialization, we'll also see a variety of ways to choose good values for lambda. In the next two videos, we'll flesh out how to apply regularization to linear regression and logistic regression, and how to train these models with gradient descent. With that, you'll be able to avoid overfitting with both of these algorithms.
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3.10 Regularization to Reduce Overfitting | Regularized linear regression-- [ML | Andrew Ng]
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https://www.youtube.com/watch?v=yRSKygmsvSI
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In this video, we'll figure out how to get gradient descent to work with regularized linear regression. Let's jump in. Here's the cost function we've come up with in the last video for regularized linear regression. The first part is the usual squared error cost function. And now you have this additional regularization term, where lambda is the regularization parameter. And you'd like to find parameters w and b that minimize the regularized cost function. Previously, we were using gradient descent for the original cost function, just the first term, before we added that second regularization term. And previously, we had the following gradient descent algorithm, which is that we repeatedly update the parameters wj and b for j equals 1 through n according to this formula, and b is also updated similarly. Again, alpha is a very small positive number called the learning rate. In fact, the updates for regularized linear regression look exactly the same, except that now the cost j is defined a bit differently. Previously, the derivative of j with respect to wj was given by this expression over here, and the derivative with respect to b was given by this expression over here. Now that we've added this additional regularization term, the only thing that changes is that the expression for the derivative with respect to wj ends up with one additional term. This plus lambda over m times wj. And in particular, for the new definition of the cost function j, these two expressions over here, these are the new derivatives of j with respect to wj, and the derivative of j with respect to b. Recall that we don't regularize b, so we're not trying to shrink b. That's why the update to b remains the same as before, whereas the update to w changes because the regularization term causes us to try to shrink wj. So let's take these definitions for the derivatives and put them back into the expression on the left to write out the gradient descent algorithm for regularized linear regression. So to implement gradient descent for regularized linear regression, this is what you would have your code do. Here's the update for wj for j equals one through n, and here's the update for b. And as usual, please remember to carry out simultaneous updates for all of these parameters. Now, in order for you to get this algorithm to work, this is all you need to know. But what I'd like to do in the remainder of this video is to go over some optional material to convey a slightly deeper intuition about what this formula is actually doing, as well as chat briefly about how these derivatives are derived. The rest of this video is completely optional. It's completely okay if you skip the rest of this video. But if you have a strong interest in math, then stick with me. It's always nice to hang out with you here. And through these equations, perhaps you
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3.10 Regularization to Reduce Overfitting | Regularized linear regression-- [ML | Andrew Ng]: In this video, we'll figure out how to get gradient descent to work with regularized linear regression. Let's jump in. Here's the cost function we've come up with in the last video for regularized linear regression. The first part is the usual squared error cost function. And now you have this additional regularization term, where lambda is the regularization parameter. And you'd like to find parameters w and b that minimize the regularized cost function. Previously, we were using gradient descent for the original cost function, just the first term, before we added that second regularization term. And previously, we had the following gradient descent algorithm, which is that we repeatedly update the parameters wj and b for j equals 1 through n according to this formula, and b is also updated similarly. Again, alpha is a very small positive number called the learning rate. In fact, the updates for regularized linear regression look exactly the same, except that now the cost j is defined a bit differently. Previously, the derivative of j with respect to wj was given by this expression over here, and the derivative with respect to b was given by this expression over here. Now that we've added this additional regularization term, the only thing that changes is that the expression for the derivative with respect to wj ends up with one additional term. This plus lambda over m times wj. And in particular, for the new definition of the cost function j, these two expressions over here, these are the new derivatives of j with respect to wj, and the derivative of j with respect to b. Recall that we don't regularize b, so we're not trying to shrink b. That's why the update to b remains the same as before, whereas the update to w changes because the regularization term causes us to try to shrink wj. So let's take these definitions for the derivatives and put them back into the expression on the left to write out the gradient descent algorithm for regularized linear regression. So to implement gradient descent for regularized linear regression, this is what you would have your code do. Here's the update for wj for j equals one through n, and here's the update for b. And as usual, please remember to carry out simultaneous updates for all of these parameters. Now, in order for you to get this algorithm to work, this is all you need to know. But what I'd like to do in the remainder of this video is to go over some optional material to convey a slightly deeper intuition about what this formula is actually doing, as well as chat briefly about how these derivatives are derived. The rest of this video is completely optional. It's completely okay if you skip the rest of this video. But if you have a strong interest in math, then stick with me. It's always nice to hang out with you here. And through these equations, perhaps you
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3.10 Regularization to Reduce Overfitting | Regularized linear regression-- [ML | Andrew Ng]
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https://www.youtube.com/watch?v=yRSKygmsvSI
|
can build a deeper intuition about what the math and what the derivatives are doing as well. So let's take a look. Let's look at the update rule for wj and rewrite it in another way. We're updating wj as one times wj minus alpha times lambda over m times wj. So I've moved the term from the end to the front here. And then minus alpha times one over m and then the rest of that term over there. So we just rearrange the terms a little bit. And if we simplify, then we're seeing that wj is updated as wj times one minus alpha times lambda over m minus alpha times this other term over here. And you might recognize this second term as the usual gradient descent update for unregularized linear regression. This is the update for linear regression before we had regularization. And this is the term we saw in week two of this course. And so the only change when you have regularization is that instead of wj being set to be equal to wj minus alpha times this term is now w times this number minus the usual update. So this is what we had in week one of this course. So what is this first term over here? Well alpha is a very small positive number, say 0.01. Lambda is usually a small number, say one or maybe 10. Let's say lambda is one for this example, and m is the training set size, say 50. And so when you multiply alpha lambda over m, say 0.01 times one divided by 50, this term ends up being a small positive number, say 0.0002. And thus, one minus alpha lambda over m is going to be a number just slightly less than one. So this is wj's 0.9998. And so the effect of this term is that on every single iteration of gradient descent, you're taking wj and multiplying it by 0.9998. That is by some numbers slightly less than one before carrying out the usual update. So what regularization is doing on every single iteration is you're multiplying w by a number slightly less than one, and that has the effect of shrinking the value of wj just a little bit. So this gives us another view on why regularization has the effect of shrinking the parameters wj a little bit on every iteration. And so that's how regularization works. If you're curious about how these derivative terms were computed, I have just one last optional slide that goes through just a little bit of the calculation of the derivative term. Again, this slide and the rest of this video are completely optional, meaning you won't need any of this to do the practice labs and the quizzes. So let's step through quickly the derivative calculation. The derivative of j with respect to wj looks like this. Recall that f of x for linear regression is defined as w dot x plus b
| 500
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3.10 Regularization to Reduce Overfitting | Regularized linear regression-- [ML | Andrew Ng]: can build a deeper intuition about what the math and what the derivatives are doing as well. So let's take a look. Let's look at the update rule for wj and rewrite it in another way. We're updating wj as one times wj minus alpha times lambda over m times wj. So I've moved the term from the end to the front here. And then minus alpha times one over m and then the rest of that term over there. So we just rearrange the terms a little bit. And if we simplify, then we're seeing that wj is updated as wj times one minus alpha times lambda over m minus alpha times this other term over here. And you might recognize this second term as the usual gradient descent update for unregularized linear regression. This is the update for linear regression before we had regularization. And this is the term we saw in week two of this course. And so the only change when you have regularization is that instead of wj being set to be equal to wj minus alpha times this term is now w times this number minus the usual update. So this is what we had in week one of this course. So what is this first term over here? Well alpha is a very small positive number, say 0.01. Lambda is usually a small number, say one or maybe 10. Let's say lambda is one for this example, and m is the training set size, say 50. And so when you multiply alpha lambda over m, say 0.01 times one divided by 50, this term ends up being a small positive number, say 0.0002. And thus, one minus alpha lambda over m is going to be a number just slightly less than one. So this is wj's 0.9998. And so the effect of this term is that on every single iteration of gradient descent, you're taking wj and multiplying it by 0.9998. That is by some numbers slightly less than one before carrying out the usual update. So what regularization is doing on every single iteration is you're multiplying w by a number slightly less than one, and that has the effect of shrinking the value of wj just a little bit. So this gives us another view on why regularization has the effect of shrinking the parameters wj a little bit on every iteration. And so that's how regularization works. If you're curious about how these derivative terms were computed, I have just one last optional slide that goes through just a little bit of the calculation of the derivative term. Again, this slide and the rest of this video are completely optional, meaning you won't need any of this to do the practice labs and the quizzes. So let's step through quickly the derivative calculation. The derivative of j with respect to wj looks like this. Recall that f of x for linear regression is defined as w dot x plus b
|
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