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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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or w dot product x plus b. And it turns out that by the rules of calculus, the derivatives look like this is one over two m times the sum i equals one through m of w dot x plus b minus y times two xj plus the derivative of the regularization term, which is lambda over two m times two wj. Notice that the second term does not have the summation term from j equals one through n anymore. The twos cancel out here and here and also here and here. And so it simplifies to this expression over here. And finally, remember that wx plus b is f of x. And so you can rewrite it as this expression down here. So this is why this expression is used to compute the gradient in regularized linear regression. So you now know how to implement regularized linear regression. Using this, you will reduce overfitting when you have a lot of features and relatively small training set. And this should let you get linear regression to work much better on many problems. In the next video, we'll take this regularization idea and apply it to logistic regression to avoid overfitting for logistic regression as well. Let's take a look at that in the next video.
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3.10 Regularization to Reduce Overfitting | Regularized linear regression-- [ML | Andrew Ng]: or w dot product x plus b. And it turns out that by the rules of calculus, the derivatives look like this is one over two m times the sum i equals one through m of w dot x plus b minus y times two xj plus the derivative of the regularization term, which is lambda over two m times two wj. Notice that the second term does not have the summation term from j equals one through n anymore. The twos cancel out here and here and also here and here. And so it simplifies to this expression over here. And finally, remember that wx plus b is f of x. And so you can rewrite it as this expression down here. So this is why this expression is used to compute the gradient in regularized linear regression. So you now know how to implement regularized linear regression. Using this, you will reduce overfitting when you have a lot of features and relatively small training set. And this should let you get linear regression to work much better on many problems. In the next video, we'll take this regularization idea and apply it to logistic regression to avoid overfitting for logistic regression as well. Let's take a look at that in the next video.
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3.11 Regularization to Reduce Overfitting | Regularized logistic regression-- [ML | Andrew Ng]
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https://www.youtube.com/watch?v=MFp4uQMQ1rk
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In this video, you see how to implement regularized logistic regression. Just as the gradient update for logistic regression has seemed surprisingly similar to the gradient update for linear regression, you find that the gradient descent update for regularized logistic regression will also look similar to the update for regularized linear regression. Let's take a look. Here's the idea. We saw earlier that logistic regression can be prone to overfitting if you fit it with very high-order polynomial features like this. Here z is a high-order polynomial that gets passed into the sigmoid function, like so, to compute f. And in particular, you can end up with a decision boundary that is overly complex and overfits this training set. More generally, when you train logistic regression with a lot of features, whether polynomial features or some other features, there can be a higher risk of overfitting. This was the cost function for logistic regression. If you want to modify it to use regularization, all you need to do is add to it the following term. Let's add lambda, the regularization parameter, over 2m, times the sum from j equals 1 through n, where n is the number of features as usual, of wj squared. So when you minimize this cost function as a function of w and b, it has the effect of penalizing parameters w1, w2, through wn and preventing them from being too large. And if you do this, then even though you're fitting a high-order polynomial with a lot of parameters, you still get a decision boundary that looks like this, something that looks more reasonable for separating positive and negative examples while also generalizing, hopefully, to new examples not in the training set. So when using regularization, even when you have a lot of features, how can you actually implement this? How can you actually minimize this cost function j of wb that includes the regularization term? Well, let's use gradient descent as before. So here's the cost function that you want to minimize. And to implement gradient descent, as before, we'll carry out the following simultaneous updates over wj and b. These are the usual update rules for gradient descent. And just like regularized linear regression, when you compute what are these derivative terms, the only thing that changes now is that the derivative with respect to wj gets this additional term, lambda over m times wj added here at the end. And again, it looks a lot like the update for regularized linear regression. In fact, it's the exact same equation, except for the fact that the definition of f is now no longer the linear function, it is the logistic function applied to z. And similar to linear regression, we will regularize only the parameters wj, but not the parameter b, which is why there's no change to the update you would make for b. In the final optional lab of this week, you revisit overfitting. And in the interactive plot in the
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3.11 Regularization to Reduce Overfitting | Regularized logistic regression-- [ML | Andrew Ng]: In this video, you see how to implement regularized logistic regression. Just as the gradient update for logistic regression has seemed surprisingly similar to the gradient update for linear regression, you find that the gradient descent update for regularized logistic regression will also look similar to the update for regularized linear regression. Let's take a look. Here's the idea. We saw earlier that logistic regression can be prone to overfitting if you fit it with very high-order polynomial features like this. Here z is a high-order polynomial that gets passed into the sigmoid function, like so, to compute f. And in particular, you can end up with a decision boundary that is overly complex and overfits this training set. More generally, when you train logistic regression with a lot of features, whether polynomial features or some other features, there can be a higher risk of overfitting. This was the cost function for logistic regression. If you want to modify it to use regularization, all you need to do is add to it the following term. Let's add lambda, the regularization parameter, over 2m, times the sum from j equals 1 through n, where n is the number of features as usual, of wj squared. So when you minimize this cost function as a function of w and b, it has the effect of penalizing parameters w1, w2, through wn and preventing them from being too large. And if you do this, then even though you're fitting a high-order polynomial with a lot of parameters, you still get a decision boundary that looks like this, something that looks more reasonable for separating positive and negative examples while also generalizing, hopefully, to new examples not in the training set. So when using regularization, even when you have a lot of features, how can you actually implement this? How can you actually minimize this cost function j of wb that includes the regularization term? Well, let's use gradient descent as before. So here's the cost function that you want to minimize. And to implement gradient descent, as before, we'll carry out the following simultaneous updates over wj and b. These are the usual update rules for gradient descent. And just like regularized linear regression, when you compute what are these derivative terms, the only thing that changes now is that the derivative with respect to wj gets this additional term, lambda over m times wj added here at the end. And again, it looks a lot like the update for regularized linear regression. In fact, it's the exact same equation, except for the fact that the definition of f is now no longer the linear function, it is the logistic function applied to z. And similar to linear regression, we will regularize only the parameters wj, but not the parameter b, which is why there's no change to the update you would make for b. In the final optional lab of this week, you revisit overfitting. And in the interactive plot in the
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3.11 Regularization to Reduce Overfitting | Regularized logistic regression-- [ML | Andrew Ng]
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https://www.youtube.com/watch?v=MFp4uQMQ1rk
|
optional lab, you can now choose to regularize your models, both regression and classification, by enabling regularization during gradient descent, by selecting a value for lambda. Please take a look at the code for implementing regularized logistic regression in particular, because you implement this in a practice lab yourself at the end of this week. So, now you know how to implement regularized logistic regression. When I walk around Silicon Valley, there are many engineers using machine learning to create a ton of value, sometimes making a lot of money for the companies. And I know you've only been studying this stuff for a few weeks. But if you understand and can apply linear regression and logistic regression, that's actually all you need to create some very valuable applications. While the specific learning algorithms you use are important, knowing things like when and how to reduce overfitting turns out to be one of the very valuable skills in the real world as well. So, I want to say, congratulations on how far you've come. And I want to say great job for getting through all the way to the end of this video. I hope you also work through the practice labs and quizzes. Having said that, there's still many more exciting things to learn. In the second course of this specialization, you learn about neural networks, also called deep learning algorithms. Neural networks are responsible for many of the latest breakthroughs in AI today, from practical speech recognition, to computers accurately recognizing objects and images, to self-driving cars. The way a neural network gets built actually uses a lot of what you've already learned, like cost functions and gradient descent and sigmoid functions. So again, congratulations on reaching the end of this third and final week of course one. I hope you have fun in the labs and I will see you in next week's material on neural networks.
| 320
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3.11 Regularization to Reduce Overfitting | Regularized logistic regression-- [ML | Andrew Ng]: optional lab, you can now choose to regularize your models, both regression and classification, by enabling regularization during gradient descent, by selecting a value for lambda. Please take a look at the code for implementing regularized logistic regression in particular, because you implement this in a practice lab yourself at the end of this week. So, now you know how to implement regularized logistic regression. When I walk around Silicon Valley, there are many engineers using machine learning to create a ton of value, sometimes making a lot of money for the companies. And I know you've only been studying this stuff for a few weeks. But if you understand and can apply linear regression and logistic regression, that's actually all you need to create some very valuable applications. While the specific learning algorithms you use are important, knowing things like when and how to reduce overfitting turns out to be one of the very valuable skills in the real world as well. So, I want to say, congratulations on how far you've come. And I want to say great job for getting through all the way to the end of this video. I hope you also work through the practice labs and quizzes. Having said that, there's still many more exciting things to learn. In the second course of this specialization, you learn about neural networks, also called deep learning algorithms. Neural networks are responsible for many of the latest breakthroughs in AI today, from practical speech recognition, to computers accurately recognizing objects and images, to self-driving cars. The way a neural network gets built actually uses a lot of what you've already learned, like cost functions and gradient descent and sigmoid functions. So again, congratulations on reaching the end of this third and final week of course one. I hope you have fun in the labs and I will see you in next week's material on neural networks.
|
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4.1 Advanced Learning Algorithms | Welcome! --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=cuU8pCflXCo
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Welcome to course two of this machine learning specialization. In this course, you learn about neural networks, also called deep learning algorithms, as well as decision trees. These are some of the most powerful and widely used machine learning algorithms, and you get to implement them and get them to work for yourself. One of the things you see also in this course is practical advice on how to build machine learning systems. This part of the material is quite unique to this course. When you're building a practical machine learning system, there are a lot of decisions you have to make, such as should you spend more time collecting data, or should you buy a much bigger GPU to build a much bigger neural network. Even today, when I visit a leading tech company and talk to the team working there on a machine learning application, unfortunately, sometimes I look at what they've been doing for the last six months and go, gee, someone could have told you maybe even six months ago that that approach wasn't going to work that well. With some of the tips that you learn in this course, I hope that you'll be one of the ones to not waste those six months, but instead be able to make more systematic and better decisions about how to build practical working machine learning applications. So with that, let's dive in. In detail, this is what you see in the four weeks of this course. In week one, we'll go over neural networks and how to carry out inference or prediction. So if you were to go to the internet and download the parameters of a neural network that someone else had trained and whose parameters had posted on the internet, then to use that neural network to make predictions would be called inference. And you learn how neural networks work and how to do inference in week one in this week. Next week, you learn how to train your own neural network. In particular, if you have a trading set of labeled examples, X and Y, how do you train the parameters of a neural network for yourself? In the third week, we'll then go into practical advice for building machine learning systems. And I'll share with you some tips that I think even highly paid engineers building machine learning systems very successfully today don't really always manage to consistently apply. And I think that will help you build systems yourself efficiently and quickly. And then in the final week of this course, you learn about decision trees. While decision trees don't get as much buzz in the media, there's a little bit less hype about decision trees compared to neural networks. They are also one of the widely used and very powerful learning algorithms that I think there's a good chance you end up using yourself if you end up building an application. So with that, let's jump into neural networks, and we're going
| 500
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4.1 Advanced Learning Algorithms | Welcome! --[Machine Learning | Andrew Ng]: Welcome to course two of this machine learning specialization. In this course, you learn about neural networks, also called deep learning algorithms, as well as decision trees. These are some of the most powerful and widely used machine learning algorithms, and you get to implement them and get them to work for yourself. One of the things you see also in this course is practical advice on how to build machine learning systems. This part of the material is quite unique to this course. When you're building a practical machine learning system, there are a lot of decisions you have to make, such as should you spend more time collecting data, or should you buy a much bigger GPU to build a much bigger neural network. Even today, when I visit a leading tech company and talk to the team working there on a machine learning application, unfortunately, sometimes I look at what they've been doing for the last six months and go, gee, someone could have told you maybe even six months ago that that approach wasn't going to work that well. With some of the tips that you learn in this course, I hope that you'll be one of the ones to not waste those six months, but instead be able to make more systematic and better decisions about how to build practical working machine learning applications. So with that, let's dive in. In detail, this is what you see in the four weeks of this course. In week one, we'll go over neural networks and how to carry out inference or prediction. So if you were to go to the internet and download the parameters of a neural network that someone else had trained and whose parameters had posted on the internet, then to use that neural network to make predictions would be called inference. And you learn how neural networks work and how to do inference in week one in this week. Next week, you learn how to train your own neural network. In particular, if you have a trading set of labeled examples, X and Y, how do you train the parameters of a neural network for yourself? In the third week, we'll then go into practical advice for building machine learning systems. And I'll share with you some tips that I think even highly paid engineers building machine learning systems very successfully today don't really always manage to consistently apply. And I think that will help you build systems yourself efficiently and quickly. And then in the final week of this course, you learn about decision trees. While decision trees don't get as much buzz in the media, there's a little bit less hype about decision trees compared to neural networks. They are also one of the widely used and very powerful learning algorithms that I think there's a good chance you end up using yourself if you end up building an application. So with that, let's jump into neural networks, and we're going
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4.2 Neural Networks Intuition |Neurons and the brain --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=QpJ35mMLIOA
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When neural networks were first invented many decades ago, the original motivation was the right software that could mimic how the human brain or how the biological brain learns and thinks. And even though today, neural networks, sometimes also called artificial neural networks, have become very different than how any of us might think about how the brain actually works and learns, some of the biological motivation still remains in the way we think about artificial neural networks or computer neural networks today. So let's start by taking a look at how the brain works and how that relates to neural networks. The human brain, or maybe more generally the biological brain, demonstrates a higher level or more capable level of intelligence than anything else we've been able to build so far. And so neural networks have started with the motivation of trying to build software to mimic the brain. Work in neural networks had started back in the 1950s and then it fell out of favor for a while. Then in the 1980s and early 1990s, they gained in popularity again and showed tremendous traction in some applications like handwritten digit recognition, which were used even back then to read postal codes for routing mail and for reading dollar figures in handwritten checks. But then it fell out of favor again in the late 1990s and it was from about 2005 that it enjoyed a resurgence and also became maybe rebranded a little bit with deep learning. One of the things that surprised me back then was deep learning and neural networks meant very similar things, but I maybe underappreciated at the time that the term deep learning just sounds much better because it's deep and it's learning. And so that turned out to be the brand that took off in the last decade or decade and a half. And since then, neural networks have revolutionized application area after application area. I think the first application area that modern neural networks or deep learning had a huge impact on was probably speech recognition, where we started to see much better speech recognition systems due to modern deep learning and authors such as Lee Dang and Jeff Hinton were instrumental to this. And then it started to make inroads into computer vision. And sometimes people still speak of the ImageNet moment in 2012 and that was maybe a bigger splash where it then caught broader imagination and had a big impact on computer vision. Then the next few years, it made its inroads into text or into natural language processing and so on and so forth. And now neural networks are used in everything from climate change to medical imaging to online advertising to product recommendations and really lots of application areas of machine learning now use neural networks. Even though today's neural networks have almost nothing to do with how the brain learns, there was the early motivation of trying to build software to mimic the brain. So how
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4.2 Neural Networks Intuition |Neurons and the brain --[Machine Learning | Andrew Ng]: When neural networks were first invented many decades ago, the original motivation was the right software that could mimic how the human brain or how the biological brain learns and thinks. And even though today, neural networks, sometimes also called artificial neural networks, have become very different than how any of us might think about how the brain actually works and learns, some of the biological motivation still remains in the way we think about artificial neural networks or computer neural networks today. So let's start by taking a look at how the brain works and how that relates to neural networks. The human brain, or maybe more generally the biological brain, demonstrates a higher level or more capable level of intelligence than anything else we've been able to build so far. And so neural networks have started with the motivation of trying to build software to mimic the brain. Work in neural networks had started back in the 1950s and then it fell out of favor for a while. Then in the 1980s and early 1990s, they gained in popularity again and showed tremendous traction in some applications like handwritten digit recognition, which were used even back then to read postal codes for routing mail and for reading dollar figures in handwritten checks. But then it fell out of favor again in the late 1990s and it was from about 2005 that it enjoyed a resurgence and also became maybe rebranded a little bit with deep learning. One of the things that surprised me back then was deep learning and neural networks meant very similar things, but I maybe underappreciated at the time that the term deep learning just sounds much better because it's deep and it's learning. And so that turned out to be the brand that took off in the last decade or decade and a half. And since then, neural networks have revolutionized application area after application area. I think the first application area that modern neural networks or deep learning had a huge impact on was probably speech recognition, where we started to see much better speech recognition systems due to modern deep learning and authors such as Lee Dang and Jeff Hinton were instrumental to this. And then it started to make inroads into computer vision. And sometimes people still speak of the ImageNet moment in 2012 and that was maybe a bigger splash where it then caught broader imagination and had a big impact on computer vision. Then the next few years, it made its inroads into text or into natural language processing and so on and so forth. And now neural networks are used in everything from climate change to medical imaging to online advertising to product recommendations and really lots of application areas of machine learning now use neural networks. Even though today's neural networks have almost nothing to do with how the brain learns, there was the early motivation of trying to build software to mimic the brain. So how
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4.2 Neural Networks Intuition |Neurons and the brain --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=QpJ35mMLIOA
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does the brain work? Here's a diagram illustrating what neurons in a brain looks like. All of human thought is from neurons like these in your brain and mind sending electrical impulses and sometimes forming new connections with other neurons. Given a neuron like this one, it has a number of inputs where it receives electrical impulses from other neurons. And then this neuron that I've circled carries out some computations and will then send this output to other neurons via these electrical impulses. And this upper neurons output in turn becomes the input to this neuron down below, which again, advocates inputs from multiple other neurons to then maybe send his own output to yet other neurons. And this is the stuff of which human thought is made. Here's a simplified diagram of a biological neuron. A neuron comprises a cell body shown here on the left. And if you have taken a class in biology, you may recognize this to be the nucleus of the neuron. And as we saw on the previous slide, the neuron has different inputs. And in a biological neuron, the input wires are called the dendrites. And it then occasionally sends electrical impulses to other neurons via the output wire, which is called the axon. Don't worry about these biological terms. If you saw them in a biology class, you may remember them, but you don't really need to memorize any of these terms for the purpose of building artificial neural networks. But this biological neuron may then send electrical impulses that becomes the input to another neuron. So the artificial neural network uses a very simplified mathematical model of what a biological neuron does. And I'm going to draw a little circle here to denote a single neuron. And what a neuron does is it takes some inputs, one or more inputs, which are just numbers, and it does some computation and it outputs some other number, which then could be an input to a second neuron shown here on the right. When you're building an artificial neural network or a deep learning algorithm, rather than building one neuron at a time, you often want to simulate many such neurons at the same time. And so when in this diagram, I'm drawing three neurons, and what these neurons do collectively is input a few numbers, carry out some computation, and output some other numbers. Now at this point, I'd like to give one big caveat, which is that even though I made a loose analogy between biological neurons and artificial neurons, I think that today we have almost no idea how the human brain works. In fact, every few years, neuroscientists make some fundamental breakthrough about how the brain works, and I think we'll continue to do so for the foreseeable future. And that to me is a sign that there are many breakthroughs that are yet to be discovered about how the brain actually works, and thus attempts to blindly mimic
| 500
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4.2 Neural Networks Intuition |Neurons and the brain --[Machine Learning | Andrew Ng]: does the brain work? Here's a diagram illustrating what neurons in a brain looks like. All of human thought is from neurons like these in your brain and mind sending electrical impulses and sometimes forming new connections with other neurons. Given a neuron like this one, it has a number of inputs where it receives electrical impulses from other neurons. And then this neuron that I've circled carries out some computations and will then send this output to other neurons via these electrical impulses. And this upper neurons output in turn becomes the input to this neuron down below, which again, advocates inputs from multiple other neurons to then maybe send his own output to yet other neurons. And this is the stuff of which human thought is made. Here's a simplified diagram of a biological neuron. A neuron comprises a cell body shown here on the left. And if you have taken a class in biology, you may recognize this to be the nucleus of the neuron. And as we saw on the previous slide, the neuron has different inputs. And in a biological neuron, the input wires are called the dendrites. And it then occasionally sends electrical impulses to other neurons via the output wire, which is called the axon. Don't worry about these biological terms. If you saw them in a biology class, you may remember them, but you don't really need to memorize any of these terms for the purpose of building artificial neural networks. But this biological neuron may then send electrical impulses that becomes the input to another neuron. So the artificial neural network uses a very simplified mathematical model of what a biological neuron does. And I'm going to draw a little circle here to denote a single neuron. And what a neuron does is it takes some inputs, one or more inputs, which are just numbers, and it does some computation and it outputs some other number, which then could be an input to a second neuron shown here on the right. When you're building an artificial neural network or a deep learning algorithm, rather than building one neuron at a time, you often want to simulate many such neurons at the same time. And so when in this diagram, I'm drawing three neurons, and what these neurons do collectively is input a few numbers, carry out some computation, and output some other numbers. Now at this point, I'd like to give one big caveat, which is that even though I made a loose analogy between biological neurons and artificial neurons, I think that today we have almost no idea how the human brain works. In fact, every few years, neuroscientists make some fundamental breakthrough about how the brain works, and I think we'll continue to do so for the foreseeable future. And that to me is a sign that there are many breakthroughs that are yet to be discovered about how the brain actually works, and thus attempts to blindly mimic
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4.2 Neural Networks Intuition |Neurons and the brain --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=QpJ35mMLIOA
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what we know of the human brain today, which is frankly very little, probably won't get us that far to our building role intelligence, certainly not with our current level of knowledge in neuroscience. Having said that, even with these extremely simplified models of a neuron, which we'll talk about, we'll be able to build really powerful deep learning algorithms. And so as you go deeper into neural networks and into deep learning, even though the origins were biologically motivated, don't take the biological motivation too seriously. In fact, those of us that do research in deep learning have shifted away from looking to biological motivation that much, but instead are just using engineering principles to figure out how to build algorithms that are more effective. But I think it might still be fun to speculate and think about how biological neurons work every now and then. The ideas of neural networks have been around for many decades. So a few people have asked me, hey, Andrew, why now? Why is it that only in the last handful of years that neural networks have really taken off? This is a picture I draw for them when I'm asked that question, and that maybe you could draw for others as well if they ask you that question. Let me plot on the horizontal axis, the amount of data you have for a problem, and on the vertical axis, the performance or the accuracy of a learning algorithm applied to that problem. Over the last couple of decades, with the rise of the internet, the rise of mobile phones, the digitalization of our society, the amount of data we have for a lot of applications has steadily marched to the right. A lot of records that used to be on paper, such as if you order something, rather than it being on a piece of paper, that's much more likely to be a digital record. Or health record, if you see a doctor, it's much more likely to be digital now compared to on pieces of paper. And so in many application areas, the amount of digital data has exploded. And what we saw was with traditional machine learning algorithms, such as logistic regression and linear regression, even as you fed those algorithms more data, it was very difficult to get the performance to keep on going up. So it was as if the traditional learning algorithms like linear regression and logistic regression, they just weren't able to scale with the amount of data we could now feed it. And they weren't able to take effective advantage of all this data we had for different applications. And what AI researchers started to observe was that if you were to train a small neural network on this data set, then the performance maybe looks like this. And if you were to train a medium sized neural network, meaning one with more neurons in it, this performance may look like that. And if you
| 500
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4.2 Neural Networks Intuition |Neurons and the brain --[Machine Learning | Andrew Ng]: what we know of the human brain today, which is frankly very little, probably won't get us that far to our building role intelligence, certainly not with our current level of knowledge in neuroscience. Having said that, even with these extremely simplified models of a neuron, which we'll talk about, we'll be able to build really powerful deep learning algorithms. And so as you go deeper into neural networks and into deep learning, even though the origins were biologically motivated, don't take the biological motivation too seriously. In fact, those of us that do research in deep learning have shifted away from looking to biological motivation that much, but instead are just using engineering principles to figure out how to build algorithms that are more effective. But I think it might still be fun to speculate and think about how biological neurons work every now and then. The ideas of neural networks have been around for many decades. So a few people have asked me, hey, Andrew, why now? Why is it that only in the last handful of years that neural networks have really taken off? This is a picture I draw for them when I'm asked that question, and that maybe you could draw for others as well if they ask you that question. Let me plot on the horizontal axis, the amount of data you have for a problem, and on the vertical axis, the performance or the accuracy of a learning algorithm applied to that problem. Over the last couple of decades, with the rise of the internet, the rise of mobile phones, the digitalization of our society, the amount of data we have for a lot of applications has steadily marched to the right. A lot of records that used to be on paper, such as if you order something, rather than it being on a piece of paper, that's much more likely to be a digital record. Or health record, if you see a doctor, it's much more likely to be digital now compared to on pieces of paper. And so in many application areas, the amount of digital data has exploded. And what we saw was with traditional machine learning algorithms, such as logistic regression and linear regression, even as you fed those algorithms more data, it was very difficult to get the performance to keep on going up. So it was as if the traditional learning algorithms like linear regression and logistic regression, they just weren't able to scale with the amount of data we could now feed it. And they weren't able to take effective advantage of all this data we had for different applications. And what AI researchers started to observe was that if you were to train a small neural network on this data set, then the performance maybe looks like this. And if you were to train a medium sized neural network, meaning one with more neurons in it, this performance may look like that. And if you
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4.2 Neural Networks Intuition |Neurons and the brain --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=QpJ35mMLIOA
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were to train a very large neural network, meaning one with a lot of these artificial neurons, then for some applications, the performance would just keep on going up. And so this meant two things. It meant that for a certain class of applications where you do have a lot of data, sometimes you hear the term big data toss around, if you're able to train a very large neural network to take advantage of that huge amount of data you have, then you could attain performance on anything ranging from speech recognition to image recognition to natural language processing applications and many more that just were not possible with earlier generations of learning algorithms. And this caused deep learning algorithms to take off. And this too is why faster computer processes, including the rise of GPUs or graphics processor units, this is hardware originally designed to generate nice looking computer graphics, but turned out to be really powerful for deep learning as well. That was also a major force in allowing deep learning algorithms to become what it is today. That's how neural networks got started as well as why they took off so quickly in the last several years. Let's now dive more deeply into the details of how a neural network actually works. Please go on to the next video.
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4.2 Neural Networks Intuition |Neurons and the brain --[Machine Learning | Andrew Ng]: were to train a very large neural network, meaning one with a lot of these artificial neurons, then for some applications, the performance would just keep on going up. And so this meant two things. It meant that for a certain class of applications where you do have a lot of data, sometimes you hear the term big data toss around, if you're able to train a very large neural network to take advantage of that huge amount of data you have, then you could attain performance on anything ranging from speech recognition to image recognition to natural language processing applications and many more that just were not possible with earlier generations of learning algorithms. And this caused deep learning algorithms to take off. And this too is why faster computer processes, including the rise of GPUs or graphics processor units, this is hardware originally designed to generate nice looking computer graphics, but turned out to be really powerful for deep learning as well. That was also a major force in allowing deep learning algorithms to become what it is today. That's how neural networks got started as well as why they took off so quickly in the last several years. Let's now dive more deeply into the details of how a neural network actually works. Please go on to the next video.
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4.3 Neural Networks Intuition | Demand Prediction --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=cHFM92fhpew
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To illustrate how neural networks work, let's start with an example. We'll use an example from demand prediction in which you look at a product and try to predict, will this product be a top seller or not? Let's take a look. In this example, you're selling t-shirts and you would like to know if a particular t-shirt will be a top seller, you know, yes or no. And you have collected data of different t-shirts that were sold at different prices, as well as which ones became a top seller. This type of application is used by retailers today in order to plan better inventory levels as well as marketing campaigns. If you know what's likely to be a top seller, you would plan, for example, to just purchase more of that stock in advance. So in this example, the input feature X is the price of the t-shirt. And so that's the input to the learning algorithm. And if you apply logistic regression to fit a sigmoid function to the data that might look like that, then the output of your prediction might look like this, 1 over 1 plus e to the negative wx plus b. Previously we had written this as f of x as the output of the learning algorithm. In order to set this up to build a neural network, I'm going to switch the terminology a little bit and use the alphabet a to denounce the output of this logistic regression algorithm. The term a stands for activation, and it's actually a term from neuroscience, and it refers to how much a neuron is sending a high output to other neurons downstream from it. It turns out that this logistic regression unit, or this little logistic regression algorithm, can be thought of as a very simplified model of a single neuron in the brain, where what the neuron does is it takes as input the price x, and then it computes this formula on top, and it outputs the number a, which is computed via this formula, and it outputs the probability of this t-shirt being a top seller. Another way to think of a neuron is as a tiny little computer whose only job is to input one number, or a few numbers, such as a price, and then to output one number or maybe a few other numbers, which in this case is the probability of the t-shirt being a top seller. As I alluded in the previous video, a logistic regression algorithm is much simpler than what any biological neuron in your brain or mind does, which is why the artificial neural network is such a vastly oversimplified model of the human brain, even though in practice, as you know, deep learning algorithms do work very well. Given this description of a single neuron, building a neural network now just requires taking a bunch of these neurons and wiring them together or putting them together. Let's now look at a
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4.3 Neural Networks Intuition | Demand Prediction --[Machine Learning | Andrew Ng]: To illustrate how neural networks work, let's start with an example. We'll use an example from demand prediction in which you look at a product and try to predict, will this product be a top seller or not? Let's take a look. In this example, you're selling t-shirts and you would like to know if a particular t-shirt will be a top seller, you know, yes or no. And you have collected data of different t-shirts that were sold at different prices, as well as which ones became a top seller. This type of application is used by retailers today in order to plan better inventory levels as well as marketing campaigns. If you know what's likely to be a top seller, you would plan, for example, to just purchase more of that stock in advance. So in this example, the input feature X is the price of the t-shirt. And so that's the input to the learning algorithm. And if you apply logistic regression to fit a sigmoid function to the data that might look like that, then the output of your prediction might look like this, 1 over 1 plus e to the negative wx plus b. Previously we had written this as f of x as the output of the learning algorithm. In order to set this up to build a neural network, I'm going to switch the terminology a little bit and use the alphabet a to denounce the output of this logistic regression algorithm. The term a stands for activation, and it's actually a term from neuroscience, and it refers to how much a neuron is sending a high output to other neurons downstream from it. It turns out that this logistic regression unit, or this little logistic regression algorithm, can be thought of as a very simplified model of a single neuron in the brain, where what the neuron does is it takes as input the price x, and then it computes this formula on top, and it outputs the number a, which is computed via this formula, and it outputs the probability of this t-shirt being a top seller. Another way to think of a neuron is as a tiny little computer whose only job is to input one number, or a few numbers, such as a price, and then to output one number or maybe a few other numbers, which in this case is the probability of the t-shirt being a top seller. As I alluded in the previous video, a logistic regression algorithm is much simpler than what any biological neuron in your brain or mind does, which is why the artificial neural network is such a vastly oversimplified model of the human brain, even though in practice, as you know, deep learning algorithms do work very well. Given this description of a single neuron, building a neural network now just requires taking a bunch of these neurons and wiring them together or putting them together. Let's now look at a
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4.3 Neural Networks Intuition | Demand Prediction --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=cHFM92fhpew
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more complex example of demand prediction. In this example, we're going to have four features to predict whether or not a t-shirt is a top seller. The features are the price of the t-shirt, the shipping costs, the amount of marketing of that particular t-shirt, as well as the material quality. Is this a high quality thick cotton, or is this maybe a lower quality material? Now you might suspect that whether or not a t-shirt becomes a top seller actually depends on a few factors. First, what is the affordability of this t-shirt? Second is what's the degree of awareness of this t-shirt that potential buyers have? And third is perceived quality. Do buyers or potential buyers think this is a high quality t-shirt? And so what I'm going to do is create one artificial neuron to try to estimate the probability that this t-shirt is perceived as highly affordable. And affordability is mainly a function of price and shipping costs because the total amount you have to pay is the sum of the price plus the shipping costs. And so we're going to use a little neuron here, a logistic regression unit, to input price and shipping costs and predict, do people think this is affordable? Second, I'm going to create another artificial neuron here to estimate, is there high awareness of this? And awareness in this case is mainly a function of the marketing of the t-shirt. And finally, going to create another neuron to estimate, do people perceive this to be of high quality? And that may mainly be a function of the price of the t-shirt and of the material quality. Price is a factor here because fortunately or unfortunately, if there's a very high price t-shirt, people will sometimes perceive that to be of high quality because if it's very expensive then maybe people think it's got to be of high quality. Given these estimates of affordability, awareness, and perceived quality, we then wire the outputs of these three neurons to another neuron here on the right that then is another logistic regression unit that finally inputs those three numbers and outputs the probability of this t-shirt being a top seller. So in the terminology of neural networks, we're going to group these three neurons together into what's called a layer, and a layer is a grouping of neurons which take as input the same or similar features and that in turn outputs a few numbers together. So these three neurons on the left form one layer, which is why I drew them on top of each other. And the single neuron on the right is also one layer. The layer on the left has three neurons, so a layer can have multiple neurons, or it can also have a single neuron, as in the case of this layer on the right. This layer on the right is also called the output layer because the output of this final neuron is the output probability
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4.3 Neural Networks Intuition | Demand Prediction --[Machine Learning | Andrew Ng]: more complex example of demand prediction. In this example, we're going to have four features to predict whether or not a t-shirt is a top seller. The features are the price of the t-shirt, the shipping costs, the amount of marketing of that particular t-shirt, as well as the material quality. Is this a high quality thick cotton, or is this maybe a lower quality material? Now you might suspect that whether or not a t-shirt becomes a top seller actually depends on a few factors. First, what is the affordability of this t-shirt? Second is what's the degree of awareness of this t-shirt that potential buyers have? And third is perceived quality. Do buyers or potential buyers think this is a high quality t-shirt? And so what I'm going to do is create one artificial neuron to try to estimate the probability that this t-shirt is perceived as highly affordable. And affordability is mainly a function of price and shipping costs because the total amount you have to pay is the sum of the price plus the shipping costs. And so we're going to use a little neuron here, a logistic regression unit, to input price and shipping costs and predict, do people think this is affordable? Second, I'm going to create another artificial neuron here to estimate, is there high awareness of this? And awareness in this case is mainly a function of the marketing of the t-shirt. And finally, going to create another neuron to estimate, do people perceive this to be of high quality? And that may mainly be a function of the price of the t-shirt and of the material quality. Price is a factor here because fortunately or unfortunately, if there's a very high price t-shirt, people will sometimes perceive that to be of high quality because if it's very expensive then maybe people think it's got to be of high quality. Given these estimates of affordability, awareness, and perceived quality, we then wire the outputs of these three neurons to another neuron here on the right that then is another logistic regression unit that finally inputs those three numbers and outputs the probability of this t-shirt being a top seller. So in the terminology of neural networks, we're going to group these three neurons together into what's called a layer, and a layer is a grouping of neurons which take as input the same or similar features and that in turn outputs a few numbers together. So these three neurons on the left form one layer, which is why I drew them on top of each other. And the single neuron on the right is also one layer. The layer on the left has three neurons, so a layer can have multiple neurons, or it can also have a single neuron, as in the case of this layer on the right. This layer on the right is also called the output layer because the output of this final neuron is the output probability
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4.3 Neural Networks Intuition | Demand Prediction --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=cHFM92fhpew
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predicted by the neural network. In the terminology of neural networks, we're also going to call affordability, awareness, and perceived quality to be activations. The term activations comes from biological neurons, and it refers to the degree that a biological neuron is sending a high output value or sending many electrical impulses to other neurons, to the downstream from it. And so these numbers on affordability, awareness, and perceived quality are the activations of these three neurons in this layer. And also, this output probability is the activation of this neuron shown here on the right. So, this particular neural network, therefore, carries out computations as follows. It inputs four numbers, then this layer of the neural network uses those four numbers to compute three new numbers, also called activation values, and then the final layer, the output layer of the neural network, uses those three numbers to compute one number. And in a neural network, this list of four numbers is also called the input layer, and that's just a list of four numbers. Now, there's one simplification I'd like to make to this neural network, which is the way I've described it so far, we had to go through the neurons one at a time and decide what inputs it would take from the previous layer. So for example, we said affordability is a function of just price and shipping costs and awareness is a function of just marketing and so on. But if you're building a large neural network, it'd be a lot of work to go through and manually decide which neurons should take which features as inputs. The way a neural network is implemented in practice, each neuron in a certain layer, say this layer in the middle, will have access to every feature, to every value from the previous layer, from the input layer, which is why I'm now drawing arrows from every input feature to every one of these neurons shown here in the middle. And you can imagine that if you're trying to predict affordability and it knows what the price shipping costs, marketing and material may be, or learn to ignore marketing and material and just figure out through setting the parameters appropriately to only focus on the subset of features that are most relevant to affordability. To further simplify the notation and the description of this neural network, I'm going to take these four input features and write them as a vector x and we're going to view the neural network as having four features that comprise this feature vector x and this feature vector is fed to this layer in the middle, which then computes three activation values, that is these three numbers, and these three activation values in turn becomes another vector, which is fed to this final output layer that finally outputs the probability of this t-shirt being a top seller. So that's all a neural network is. It has a few layers where each layer inputs
| 500
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4.3 Neural Networks Intuition | Demand Prediction --[Machine Learning | Andrew Ng]: predicted by the neural network. In the terminology of neural networks, we're also going to call affordability, awareness, and perceived quality to be activations. The term activations comes from biological neurons, and it refers to the degree that a biological neuron is sending a high output value or sending many electrical impulses to other neurons, to the downstream from it. And so these numbers on affordability, awareness, and perceived quality are the activations of these three neurons in this layer. And also, this output probability is the activation of this neuron shown here on the right. So, this particular neural network, therefore, carries out computations as follows. It inputs four numbers, then this layer of the neural network uses those four numbers to compute three new numbers, also called activation values, and then the final layer, the output layer of the neural network, uses those three numbers to compute one number. And in a neural network, this list of four numbers is also called the input layer, and that's just a list of four numbers. Now, there's one simplification I'd like to make to this neural network, which is the way I've described it so far, we had to go through the neurons one at a time and decide what inputs it would take from the previous layer. So for example, we said affordability is a function of just price and shipping costs and awareness is a function of just marketing and so on. But if you're building a large neural network, it'd be a lot of work to go through and manually decide which neurons should take which features as inputs. The way a neural network is implemented in practice, each neuron in a certain layer, say this layer in the middle, will have access to every feature, to every value from the previous layer, from the input layer, which is why I'm now drawing arrows from every input feature to every one of these neurons shown here in the middle. And you can imagine that if you're trying to predict affordability and it knows what the price shipping costs, marketing and material may be, or learn to ignore marketing and material and just figure out through setting the parameters appropriately to only focus on the subset of features that are most relevant to affordability. To further simplify the notation and the description of this neural network, I'm going to take these four input features and write them as a vector x and we're going to view the neural network as having four features that comprise this feature vector x and this feature vector is fed to this layer in the middle, which then computes three activation values, that is these three numbers, and these three activation values in turn becomes another vector, which is fed to this final output layer that finally outputs the probability of this t-shirt being a top seller. So that's all a neural network is. It has a few layers where each layer inputs
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4.3 Neural Networks Intuition | Demand Prediction --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=cHFM92fhpew
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a vector and outputs another vector of numbers, where for example, this layer in the middle inputs four numbers x and outputs three numbers corresponding to affordability, awareness and perceived quality. To add a little bit more terminology, you've seen that this layer is called the output layer and this layer is called the input layer. To give the layer in the middle a name as well, this layer in the middle is called a hidden layer. I know that this is maybe not the best or the most intuitive name, but that terminology comes from that when you have a training set, in the training set, you get to observe both x and y, your data set tells you what is x and what is y, and so you get data that tells you what are the correct inputs and the correct outputs, but your data set doesn't tell you what are the correct values for affordability, awareness and perceived quality. And so the correct values for those are hidden, you don't see them in the training set, which is why this layer in the middle is called a hidden layer. I'd like to share with you another way of thinking about neural networks that I found useful for building my intuition about it. Just let me cover up the left half of this diagram and see what we're left with. What you see here is that there is a logistic regression algorithm or logistic regression unit that is taking as input affordability, awareness and perceived quality of a t-shirt and using these three features to estimate the probability of the t-shirt being a top seller. So this is just logistic regression. But the cool thing about this is rather than using the original features, price, shipping, cost, marketing and so on, is using a new, maybe better set of features, affordability, awareness and perceived quality that are hopefully more predictive of whether or not this t-shirt will be a top seller. So one way to think of this neural network is just logistic regression, but it is a version of logistic regression that can learn its own features that makes it easier to make accurate predictions. In fact, you might remember from the previous week, this housing example where we said that if you want to predict the price of the house, you might take the frontage or the width of lots and multiply that by the depth of a lot to construct a more complex feature, X1 times X2, which was the size of the lot. So there we were doing manual feature engineering where we had to look at the features X1 and X2 and decide by hand how to combine them together to come up with better features. What the neural network does is instead of you needing to manually engineer the features, it can learn, as you see later, its own features to make the learning problem easier for itself. So this is what
| 500
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4.3 Neural Networks Intuition | Demand Prediction --[Machine Learning | Andrew Ng]: a vector and outputs another vector of numbers, where for example, this layer in the middle inputs four numbers x and outputs three numbers corresponding to affordability, awareness and perceived quality. To add a little bit more terminology, you've seen that this layer is called the output layer and this layer is called the input layer. To give the layer in the middle a name as well, this layer in the middle is called a hidden layer. I know that this is maybe not the best or the most intuitive name, but that terminology comes from that when you have a training set, in the training set, you get to observe both x and y, your data set tells you what is x and what is y, and so you get data that tells you what are the correct inputs and the correct outputs, but your data set doesn't tell you what are the correct values for affordability, awareness and perceived quality. And so the correct values for those are hidden, you don't see them in the training set, which is why this layer in the middle is called a hidden layer. I'd like to share with you another way of thinking about neural networks that I found useful for building my intuition about it. Just let me cover up the left half of this diagram and see what we're left with. What you see here is that there is a logistic regression algorithm or logistic regression unit that is taking as input affordability, awareness and perceived quality of a t-shirt and using these three features to estimate the probability of the t-shirt being a top seller. So this is just logistic regression. But the cool thing about this is rather than using the original features, price, shipping, cost, marketing and so on, is using a new, maybe better set of features, affordability, awareness and perceived quality that are hopefully more predictive of whether or not this t-shirt will be a top seller. So one way to think of this neural network is just logistic regression, but it is a version of logistic regression that can learn its own features that makes it easier to make accurate predictions. In fact, you might remember from the previous week, this housing example where we said that if you want to predict the price of the house, you might take the frontage or the width of lots and multiply that by the depth of a lot to construct a more complex feature, X1 times X2, which was the size of the lot. So there we were doing manual feature engineering where we had to look at the features X1 and X2 and decide by hand how to combine them together to come up with better features. What the neural network does is instead of you needing to manually engineer the features, it can learn, as you see later, its own features to make the learning problem easier for itself. So this is what
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4.3 Neural Networks Intuition | Demand Prediction --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=cHFM92fhpew
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makes neural networks one of the most powerful learning algorithms in the world today. So to summarize, a neural network does this. The input layer has a vector of features, four numbers in this example. It is input to the hidden layer, which outputs three numbers. And I'm going to use a vector to denote this vector of activations that this hidden layer outputs. And then the output layer takes this input, those three numbers and outputs one number, which would be the final activation or the final prediction of the neural network. One note, even though I previously described this neural network as computing affordability, awareness, and perceived quality, one of the really nice properties of a neural network is when you train it from data, you don't need to go and explicitly decide what are the features such as affordability and so on that a neural network should compute. Then it will figure out all by itself what are the features it wants to use in this hidden layer. And that's what makes it such a powerful learning algorithm. So you've seen here one example of a neural network, and this neural network has a single layer that is a hidden layer. Let's take a look at some other examples of neural networks, specifically examples with more than one hidden layer. Here's an example. This neural network has an input feature vector X that is fed to one hidden layer, and I'm going to call this the first hidden layer. And so if this hidden layer has three neurons, it will then output a vector of three activation values. These three numbers can then be input to a second hidden layer. And if the second hidden layer has two neurons, two logistic units, then this second hidden layer will output another vector of now two activation values that maybe goes to the output layer that then outputs the neural network's final prediction. Or here's another example. Here's a neural network that has this input go to the first hidden layer, that the output to the first hidden layer goes to the second hidden layer, goes to the third hidden layer, and then finally to the output layer. When you're building your own neural network, one of the decisions you need to make is how many hidden layers do you want and how many neurons do you want each hidden layer to have. And this question of how many hidden layers and how many neurons per hidden layer is a question of the architecture of the neural network. You learn later in this course some tips for choosing an appropriate architecture for a neural network. But choosing the right number of hidden layers and number of hidden units per layer can have an impact on the performance of your learning algorithm as well. So later in this course, you learn how to choose a good architecture for your neural network as well. Oh, and by the way, in some
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4.3 Neural Networks Intuition | Demand Prediction --[Machine Learning | Andrew Ng]: makes neural networks one of the most powerful learning algorithms in the world today. So to summarize, a neural network does this. The input layer has a vector of features, four numbers in this example. It is input to the hidden layer, which outputs three numbers. And I'm going to use a vector to denote this vector of activations that this hidden layer outputs. And then the output layer takes this input, those three numbers and outputs one number, which would be the final activation or the final prediction of the neural network. One note, even though I previously described this neural network as computing affordability, awareness, and perceived quality, one of the really nice properties of a neural network is when you train it from data, you don't need to go and explicitly decide what are the features such as affordability and so on that a neural network should compute. Then it will figure out all by itself what are the features it wants to use in this hidden layer. And that's what makes it such a powerful learning algorithm. So you've seen here one example of a neural network, and this neural network has a single layer that is a hidden layer. Let's take a look at some other examples of neural networks, specifically examples with more than one hidden layer. Here's an example. This neural network has an input feature vector X that is fed to one hidden layer, and I'm going to call this the first hidden layer. And so if this hidden layer has three neurons, it will then output a vector of three activation values. These three numbers can then be input to a second hidden layer. And if the second hidden layer has two neurons, two logistic units, then this second hidden layer will output another vector of now two activation values that maybe goes to the output layer that then outputs the neural network's final prediction. Or here's another example. Here's a neural network that has this input go to the first hidden layer, that the output to the first hidden layer goes to the second hidden layer, goes to the third hidden layer, and then finally to the output layer. When you're building your own neural network, one of the decisions you need to make is how many hidden layers do you want and how many neurons do you want each hidden layer to have. And this question of how many hidden layers and how many neurons per hidden layer is a question of the architecture of the neural network. You learn later in this course some tips for choosing an appropriate architecture for a neural network. But choosing the right number of hidden layers and number of hidden units per layer can have an impact on the performance of your learning algorithm as well. So later in this course, you learn how to choose a good architecture for your neural network as well. Oh, and by the way, in some
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4.3 Neural Networks Intuition | Demand Prediction --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=cHFM92fhpew
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of the literature, you see this type of neural network with multiple layers like this called a multi-layer perceptron. So if you see that, that just refers to a neural network that looks like what you're seeing here on the slide. So that's a neural network. I know we went through a lot of this video. So thank you for sticking with me, but you now know how a neural network works. In the next video, let's take a look at how these ideas can be applied to other applications as well. In particular, we'll take a look at the computer vision application of face recognition. Let's go on to the next video.
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4.3 Neural Networks Intuition | Demand Prediction --[Machine Learning | Andrew Ng]: of the literature, you see this type of neural network with multiple layers like this called a multi-layer perceptron. So if you see that, that just refers to a neural network that looks like what you're seeing here on the slide. So that's a neural network. I know we went through a lot of this video. So thank you for sticking with me, but you now know how a neural network works. In the next video, let's take a look at how these ideas can be applied to other applications as well. In particular, we'll take a look at the computer vision application of face recognition. Let's go on to the next video.
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4.4 Neural Networks Intuition | Example Recognizing Images--[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=3RIUt73mj3Q
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In the last video, you saw how a neural network works in a demand prediction example. Let's take a look at how you can apply a similar type of idea to a computer vision application. Let's dive in. If you're building a face recognition application, you might want to train, say, a neural network that takes this input to picture like this and outputs the identity of the person in the picture. This image is a thousand by a thousand pixels and so its representation in the computer is actually as a thousand by a thousand grid or also called a thousand by a thousand matrix of pixel intensity values. In this example, my pixel intensity values or pixel brightness values goes from 0 to 255 and so 197 here would be the brightness of the pixel in the very upper left of the image, 185 is the brightness of the pixel, one pixel over and so on, down to 214 would be the lower right corner of this image. If you were to take these pixel intensity values and unroll them into a vector, you end up with a list or a vector of a million pixel intensity values, one million because a thousand by a thousand squared gives you a million numbers. The face recognition problem is, can you train a neural network that takes this input, a feature vector with a million pixel brightness values and outputs the identity of the person in the picture? This is how you might build a neural network to carry out this task. The input image X is fed to this layer of neurons, this is the first hidden layer, which then extracts some features and the output of this first hidden layer is fed to a second hidden layer and that output is fed to a third hidden layer and then finally to the output layer which then estimates, say, the probability of this being a particular person. One interesting thing would be if you look at a neural network that's been trained on a lot of images of faces and to try to visualize what are these hidden layers trying to compute. It turns out that when you train a system like this on a lot of pictures of faces and you peer at the different neurons in the hidden layers to figure what they may be computing, this is what you might find. In the first hidden layer, you might find one neuron that is looking for a little vertical line or a vertical edge like that and a second neuron looking for a oriented line or oriented edge like that and a third neuron looking for a line at that orientation and so on. In the earliest layers of a neural network, you might find that the neurons are looking for very short lines or very short edges in the image. If you look at the next hidden layer, you find that these neurons might learn to
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4.4 Neural Networks Intuition | Example Recognizing Images--[Machine Learning | Andrew Ng]: In the last video, you saw how a neural network works in a demand prediction example. Let's take a look at how you can apply a similar type of idea to a computer vision application. Let's dive in. If you're building a face recognition application, you might want to train, say, a neural network that takes this input to picture like this and outputs the identity of the person in the picture. This image is a thousand by a thousand pixels and so its representation in the computer is actually as a thousand by a thousand grid or also called a thousand by a thousand matrix of pixel intensity values. In this example, my pixel intensity values or pixel brightness values goes from 0 to 255 and so 197 here would be the brightness of the pixel in the very upper left of the image, 185 is the brightness of the pixel, one pixel over and so on, down to 214 would be the lower right corner of this image. If you were to take these pixel intensity values and unroll them into a vector, you end up with a list or a vector of a million pixel intensity values, one million because a thousand by a thousand squared gives you a million numbers. The face recognition problem is, can you train a neural network that takes this input, a feature vector with a million pixel brightness values and outputs the identity of the person in the picture? This is how you might build a neural network to carry out this task. The input image X is fed to this layer of neurons, this is the first hidden layer, which then extracts some features and the output of this first hidden layer is fed to a second hidden layer and that output is fed to a third hidden layer and then finally to the output layer which then estimates, say, the probability of this being a particular person. One interesting thing would be if you look at a neural network that's been trained on a lot of images of faces and to try to visualize what are these hidden layers trying to compute. It turns out that when you train a system like this on a lot of pictures of faces and you peer at the different neurons in the hidden layers to figure what they may be computing, this is what you might find. In the first hidden layer, you might find one neuron that is looking for a little vertical line or a vertical edge like that and a second neuron looking for a oriented line or oriented edge like that and a third neuron looking for a line at that orientation and so on. In the earliest layers of a neural network, you might find that the neurons are looking for very short lines or very short edges in the image. If you look at the next hidden layer, you find that these neurons might learn to
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4.4 Neural Networks Intuition | Example Recognizing Images--[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=3RIUt73mj3Q
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group together lots of little short lines, little short edge segments in order to look for parts of faces. For example, each of these little square boxes is a visualization of what that neuron is trying to detect. This first neuron looks like it's trying to detect the presence or absence of an eye in a certain position of the image and the second neuron looks like it's trying to detect the horn of a nose and maybe this neuron over here is trying to detect the bottom of a nose. And then as you look at the next hidden layer, in this example, the neural network is aggregating different parts of faces to then try to detect presence or absence of larger, coarser face shapes. And then finally, detecting how much the face corresponds to different face shapes creates a rich set of features that then helps the upper layer try to determine the identity of the person pictured. And a remarkable thing about the neural network is it can learn these feature detectors at the different hidden layers all by itself. In this example, no one ever told it to look for short little edges in the first layer and eyes and noses and face parts in the second layer and then more complete face shapes at the third layer. The neural network is able to figure out these things all by itself from data. Just one note, in this visualization, the neurons in the first hidden layer are shown looking at relatively small windows to look for these edges. And then the second hidden layer is looking at a bigger window and the third hidden layer is looking at an even bigger window. So these little neurons visualizations actually correspond to differently sized regions in the image. Just for fun, let's see what happens if you were to train this neural network on a different data set, say on lots of pictures of cars pictured on the side. The same learning algorithm, if it's asked to detect cars, will then learn edges in the first layer, so pretty similar, but then they'll learn to detect parts of cars in the second hidden layer and then more complete car shapes in the third hidden layer. So just by feeding it different data, the neural network automatically learns to detect very different features so as to try to make the predictions of car detection or person recognition or whatever is a particular given task that it's trained on. So that's how a neural network works for a computer vision application. And in fact, later this week, you see how you can build a neural network yourself and apply it to a handwritten digit recognition application. So far, we've been going over the description of intuitions of neural networks to give you a feel for how they work. In the next video, let's look more deeply into the concrete mathematics and the concrete implementational details of how you actually build
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4.4 Neural Networks Intuition | Example Recognizing Images--[Machine Learning | Andrew Ng]: group together lots of little short lines, little short edge segments in order to look for parts of faces. For example, each of these little square boxes is a visualization of what that neuron is trying to detect. This first neuron looks like it's trying to detect the presence or absence of an eye in a certain position of the image and the second neuron looks like it's trying to detect the horn of a nose and maybe this neuron over here is trying to detect the bottom of a nose. And then as you look at the next hidden layer, in this example, the neural network is aggregating different parts of faces to then try to detect presence or absence of larger, coarser face shapes. And then finally, detecting how much the face corresponds to different face shapes creates a rich set of features that then helps the upper layer try to determine the identity of the person pictured. And a remarkable thing about the neural network is it can learn these feature detectors at the different hidden layers all by itself. In this example, no one ever told it to look for short little edges in the first layer and eyes and noses and face parts in the second layer and then more complete face shapes at the third layer. The neural network is able to figure out these things all by itself from data. Just one note, in this visualization, the neurons in the first hidden layer are shown looking at relatively small windows to look for these edges. And then the second hidden layer is looking at a bigger window and the third hidden layer is looking at an even bigger window. So these little neurons visualizations actually correspond to differently sized regions in the image. Just for fun, let's see what happens if you were to train this neural network on a different data set, say on lots of pictures of cars pictured on the side. The same learning algorithm, if it's asked to detect cars, will then learn edges in the first layer, so pretty similar, but then they'll learn to detect parts of cars in the second hidden layer and then more complete car shapes in the third hidden layer. So just by feeding it different data, the neural network automatically learns to detect very different features so as to try to make the predictions of car detection or person recognition or whatever is a particular given task that it's trained on. So that's how a neural network works for a computer vision application. And in fact, later this week, you see how you can build a neural network yourself and apply it to a handwritten digit recognition application. So far, we've been going over the description of intuitions of neural networks to give you a feel for how they work. In the next video, let's look more deeply into the concrete mathematics and the concrete implementational details of how you actually build
|
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4.5 Neural Networks Model | Neural Network Layer --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=YmDM4Bq-dyA
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The fundamental building block of most modern neural networks is a layer of neurons. In this video, you learn how to construct a layer of neurons, and once you have that down, you'll be able to take those building blocks and put them together to form a large neural network. Let's take a look at how a layer of neurons works. Here's the example we had from the demand prediction example where we had four input features that were fed to this layer of three neurons in this hidden layer that then sends this output to this output layer with just one neuron. Let's zoom in to the hidden layer to look at its computations. This hidden layer inputs four numbers, and these four numbers are inputs to each of three neurons, and each of these three neurons is just implementing a little logistic regression unit or a little logistic regression function. Take this first neuron. It has two parameters, W and B, and in fact, to denote that this is the first hidden unit, I'm going to subscript this as W1B1, and what it does is it'll output some activation value A, which is G of W1 in a product of X plus B1, where this is the familiar Z value that you had learned about in logistic regression in the previous course, and G of Z is the familiar logistic function, one over one plus E to the negative Z, and so maybe this ends up being a number 0.3, and that's the activation value A of the first neuron. To denote that this is the first neuron, I'm also going to add a subscript A1 over here, and so A1 may be a number like 0.3. That's a 0.3 chance of this being highly affordable based on the input features. Now let's look at the second neuron. The second neuron has parameters W2 and B2, and this WB or W2B2 are the parameters of the second logistic unit. So it computes A2 equals the logistic function G applied to W2 dot product X plus B2, and this may be some other number, say 0.7, because in this example there's a 0.7 chance that we think the potential buyers will be aware of this t-shirt. And similarly, the third neuron has a third set of parameters, W3B3, and similarly computes an activation value A3 equals G of W3 dot product X plus B3, and that may be, say, 0.2. So in this example, these three neurons output 0.3, 0.7, and 0.2, and this vector of three numbers becomes the vector of activation values A that is then passed to the final output layer of this neural network. Now when you build neural networks with multiple layers, it will be useful to give the layers different numbers. So by convention, this layer is called layer one of the neural network, and this layer is called layer two of the neural network, and the input layer is also sometimes called
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4.5 Neural Networks Model | Neural Network Layer --[Machine Learning | Andrew Ng]: The fundamental building block of most modern neural networks is a layer of neurons. In this video, you learn how to construct a layer of neurons, and once you have that down, you'll be able to take those building blocks and put them together to form a large neural network. Let's take a look at how a layer of neurons works. Here's the example we had from the demand prediction example where we had four input features that were fed to this layer of three neurons in this hidden layer that then sends this output to this output layer with just one neuron. Let's zoom in to the hidden layer to look at its computations. This hidden layer inputs four numbers, and these four numbers are inputs to each of three neurons, and each of these three neurons is just implementing a little logistic regression unit or a little logistic regression function. Take this first neuron. It has two parameters, W and B, and in fact, to denote that this is the first hidden unit, I'm going to subscript this as W1B1, and what it does is it'll output some activation value A, which is G of W1 in a product of X plus B1, where this is the familiar Z value that you had learned about in logistic regression in the previous course, and G of Z is the familiar logistic function, one over one plus E to the negative Z, and so maybe this ends up being a number 0.3, and that's the activation value A of the first neuron. To denote that this is the first neuron, I'm also going to add a subscript A1 over here, and so A1 may be a number like 0.3. That's a 0.3 chance of this being highly affordable based on the input features. Now let's look at the second neuron. The second neuron has parameters W2 and B2, and this WB or W2B2 are the parameters of the second logistic unit. So it computes A2 equals the logistic function G applied to W2 dot product X plus B2, and this may be some other number, say 0.7, because in this example there's a 0.7 chance that we think the potential buyers will be aware of this t-shirt. And similarly, the third neuron has a third set of parameters, W3B3, and similarly computes an activation value A3 equals G of W3 dot product X plus B3, and that may be, say, 0.2. So in this example, these three neurons output 0.3, 0.7, and 0.2, and this vector of three numbers becomes the vector of activation values A that is then passed to the final output layer of this neural network. Now when you build neural networks with multiple layers, it will be useful to give the layers different numbers. So by convention, this layer is called layer one of the neural network, and this layer is called layer two of the neural network, and the input layer is also sometimes called
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4.5 Neural Networks Model | Neural Network Layer --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=YmDM4Bq-dyA
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layer zero, and today there are neural networks that can have dozens or even hundreds of layers. But in order to introduce notation to help us distinguish between the different layers, I'm going to use superscript square bracket one to index into different layers. So in particular, A superscript in square brackets one, I'm going to use as a notation to denote the output of layer one of this hidden layer of this neural network, and similarly, W1B1 here are the parameters of the first unit in layer one of the neural network, so I'm also going to add a superscript in square brackets one here, and W2B2 are the parameters of the second hidden unit or the second hidden neuron in layer one, and so those parameters are also denoted here, W superscript square bracket one, like so. And similarly, I can add superscript square brackets like so to denote that these are the activation values of the hidden units of layer one of this neural network. I know maybe this notation is getting a little bit cluttered, but the thing to remember is whenever you see this superscript square bracket one, that just refers to a quantity that is associated with layer one of the neural network, and if you see superscript square bracket two, that refers to a quantity associated with layer two of the neural network, and similarly for other layers as well, including layer three, layer four, and so on for neural networks with more layers. So that's the computation of layer one of this neural network. This output is this activation vector, A superscript square bracket one, and I'm going to copy this over here because this output, A1, becomes the input to layer two. So now let's zoom into the computation of layer two of this neural network, which is also the output layer. So the input to layer two is the output of layer one. So A1 is this vector 0.3, 0.7, 0.2 that we just computed on the previous part of the slide, and so because the output layer has just a single neuron, all it does is it computes a subscript one that is the output of its first and only neuron as G, the sigmoid function, applied to W subscript one in a product with A superscript square bracket one, so this is the input into this layer, and then plus B1. Here this is the quantity Z that you're familiar with, and G as before is the sigmoid function that you apply to this, and if this results in a number, say 0.84, then that becomes the output of this output layer of the neural network, and in this example, because the output layer has just a single neuron, this output is just a scalar, it's a single number rather than a vector of numbers. Sticking with our notational convention from before, we're going to use A superscript in square brackets two to denote the quantities associated with
| 500
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4.5 Neural Networks Model | Neural Network Layer --[Machine Learning | Andrew Ng]: layer zero, and today there are neural networks that can have dozens or even hundreds of layers. But in order to introduce notation to help us distinguish between the different layers, I'm going to use superscript square bracket one to index into different layers. So in particular, A superscript in square brackets one, I'm going to use as a notation to denote the output of layer one of this hidden layer of this neural network, and similarly, W1B1 here are the parameters of the first unit in layer one of the neural network, so I'm also going to add a superscript in square brackets one here, and W2B2 are the parameters of the second hidden unit or the second hidden neuron in layer one, and so those parameters are also denoted here, W superscript square bracket one, like so. And similarly, I can add superscript square brackets like so to denote that these are the activation values of the hidden units of layer one of this neural network. I know maybe this notation is getting a little bit cluttered, but the thing to remember is whenever you see this superscript square bracket one, that just refers to a quantity that is associated with layer one of the neural network, and if you see superscript square bracket two, that refers to a quantity associated with layer two of the neural network, and similarly for other layers as well, including layer three, layer four, and so on for neural networks with more layers. So that's the computation of layer one of this neural network. This output is this activation vector, A superscript square bracket one, and I'm going to copy this over here because this output, A1, becomes the input to layer two. So now let's zoom into the computation of layer two of this neural network, which is also the output layer. So the input to layer two is the output of layer one. So A1 is this vector 0.3, 0.7, 0.2 that we just computed on the previous part of the slide, and so because the output layer has just a single neuron, all it does is it computes a subscript one that is the output of its first and only neuron as G, the sigmoid function, applied to W subscript one in a product with A superscript square bracket one, so this is the input into this layer, and then plus B1. Here this is the quantity Z that you're familiar with, and G as before is the sigmoid function that you apply to this, and if this results in a number, say 0.84, then that becomes the output of this output layer of the neural network, and in this example, because the output layer has just a single neuron, this output is just a scalar, it's a single number rather than a vector of numbers. Sticking with our notational convention from before, we're going to use A superscript in square brackets two to denote the quantities associated with
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4.5 Neural Networks Model | Neural Network Layer --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=YmDM4Bq-dyA
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layer two of this neural network, so A superscript square bracket two is the output of this layer, and so I'm going to also copy this here as the final output of the neural network, and to make the notation consistent, you can also add these superscript square bracket twos to denote that these are the parameters and activation values associated with layer two of the neural network. Once the neural network has computed A2, there's one final optional step that you can choose to implement or not, which is if you want a binary prediction, one or zero, is this a top seller, yes or no, is you can take the number A superscript square brackets two, subscript one, and this is the number 0.84 that we computed, and threshold is at 0.5, so if it's greater than 0.5, you can predict y hat equals one, and if it's less than 0.5, then predict y hat equals zero, and we saw this thresholding as well when we learned about logistic regression in the first course of the specialization. So if you wish, this then gives you the final prediction y hat as either one or zero if you don't want just a probability of it being a top seller. So that's how a neural network works. Every layer inputs a vector of numbers and applies a bunch of logistic regression units to it and then computes another vector of numbers that then gets passed from layer to layer until you get the final output layers computation, which is a prediction of the neural network that you can either threshold at 0.5 or not to come up with the final prediction, and with that, let's go on to use this foundation we've built now to look at some even more complex, even larger neural network models, and I hope that by seeing more examples, this concept of layers and how to put them together to build a neural network will become even clearer. So let's go on to the next video.
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4.5 Neural Networks Model | Neural Network Layer --[Machine Learning | Andrew Ng]: layer two of this neural network, so A superscript square bracket two is the output of this layer, and so I'm going to also copy this here as the final output of the neural network, and to make the notation consistent, you can also add these superscript square bracket twos to denote that these are the parameters and activation values associated with layer two of the neural network. Once the neural network has computed A2, there's one final optional step that you can choose to implement or not, which is if you want a binary prediction, one or zero, is this a top seller, yes or no, is you can take the number A superscript square brackets two, subscript one, and this is the number 0.84 that we computed, and threshold is at 0.5, so if it's greater than 0.5, you can predict y hat equals one, and if it's less than 0.5, then predict y hat equals zero, and we saw this thresholding as well when we learned about logistic regression in the first course of the specialization. So if you wish, this then gives you the final prediction y hat as either one or zero if you don't want just a probability of it being a top seller. So that's how a neural network works. Every layer inputs a vector of numbers and applies a bunch of logistic regression units to it and then computes another vector of numbers that then gets passed from layer to layer until you get the final output layers computation, which is a prediction of the neural network that you can either threshold at 0.5 or not to come up with the final prediction, and with that, let's go on to use this foundation we've built now to look at some even more complex, even larger neural network models, and I hope that by seeing more examples, this concept of layers and how to put them together to build a neural network will become even clearer. So let's go on to the next video.
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4.6 Neural Networks Model | More complex neural networks --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=4-2FOgsMOpk
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In the last video, you learned about the neural network layer and how that takes as input a vector of numbers and in turn outputs another vector of numbers. In this video, let's use that layer to build a more complex neural network. And through this, I hope that the notation that we're using for neural networks will become clearer and more concrete as well. Let's take a look. This is the running example that I'm going to use throughout this video as an example of a more complex neural network. This network has four layers, not counting the input layer, which is also called layer zero, where layers one, two, and three are hidden layers and layer four is the output layer and layer zero as usual is the input layer. By convention, when we say that a neural network has four layers, that includes all the hidden layers and the output layer, but we don't count the input layer. So this is a neural network with four layers in the conventional way of counting layers in the network. Let's zoom in to layer three, which is the third and final hidden layer to look at the computations of that layer. Layer three inputs a vector, a superscript square bracket two that was computed by the previous layer and it outputs a three, which is another vector. So what is the computation that layer three does in order to go from a two to a three? If it has three neurons or we call it three hidden units, then it has parameters w1, b1, w2, b2, and w3, b3 and it computes a1 equals sigmoid of w1 dot product with this input to the layer plus b1 and it computes a2 equals sigmoid of w2 dot product with again a2, the input to the layer plus b2 and so on to get a3 and then the output of this layer is a vector comprising a1, a2, and a3. And again by convention if we want to more explicitly denote that all of these are quantities associated with layer three, then we add in all of these superscript square brackets three here to denote that these parameters w and b are the parameters associated with neurons in layer three and that these activations are activations with layer three. Notice that this term here is w1 superscript square bracket three meaning the parameters associated with layer three dot product with a superscript square bracket two, which was the output of layer two, which became the input to layer three. So that's why there's a three here because there's a parameter associated with layer three dot product with and there's a two there because it's the output of layer two. Now let's just do a quick double check of our understanding of this. I'm going to hide the superscripts and subscripts associated with the second neuron and without rewinding this video, go ahead and rewind if you want but you know prefer
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4.6 Neural Networks Model | More complex neural networks --[Machine Learning | Andrew Ng]: In the last video, you learned about the neural network layer and how that takes as input a vector of numbers and in turn outputs another vector of numbers. In this video, let's use that layer to build a more complex neural network. And through this, I hope that the notation that we're using for neural networks will become clearer and more concrete as well. Let's take a look. This is the running example that I'm going to use throughout this video as an example of a more complex neural network. This network has four layers, not counting the input layer, which is also called layer zero, where layers one, two, and three are hidden layers and layer four is the output layer and layer zero as usual is the input layer. By convention, when we say that a neural network has four layers, that includes all the hidden layers and the output layer, but we don't count the input layer. So this is a neural network with four layers in the conventional way of counting layers in the network. Let's zoom in to layer three, which is the third and final hidden layer to look at the computations of that layer. Layer three inputs a vector, a superscript square bracket two that was computed by the previous layer and it outputs a three, which is another vector. So what is the computation that layer three does in order to go from a two to a three? If it has three neurons or we call it three hidden units, then it has parameters w1, b1, w2, b2, and w3, b3 and it computes a1 equals sigmoid of w1 dot product with this input to the layer plus b1 and it computes a2 equals sigmoid of w2 dot product with again a2, the input to the layer plus b2 and so on to get a3 and then the output of this layer is a vector comprising a1, a2, and a3. And again by convention if we want to more explicitly denote that all of these are quantities associated with layer three, then we add in all of these superscript square brackets three here to denote that these parameters w and b are the parameters associated with neurons in layer three and that these activations are activations with layer three. Notice that this term here is w1 superscript square bracket three meaning the parameters associated with layer three dot product with a superscript square bracket two, which was the output of layer two, which became the input to layer three. So that's why there's a three here because there's a parameter associated with layer three dot product with and there's a two there because it's the output of layer two. Now let's just do a quick double check of our understanding of this. I'm going to hide the superscripts and subscripts associated with the second neuron and without rewinding this video, go ahead and rewind if you want but you know prefer
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4.6 Neural Networks Model | More complex neural networks --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=4-2FOgsMOpk
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you not, but without rewinding this video, are you able to think through what are the missing superscripts and subscripts in this equation and fill them in yourself? Why don't you take a look at the end video quiz and see if you can figure out what are the appropriate superscripts and subscripts for this equation over here. So to recap, A3 is activation associated with layer three for the second neuron, hence this is a two. There's a parameter associated with the third layer for the second neuron. This is A2, same as above, and then plus B3, two. So hopefully that makes sense. Here's the more general form of this equation for an arbitrary layer L and for an arbitrary unit J, which is that A deactivation output of layer L unit J like A32, that's going to be the sigmoid function applied to this term, which is the weight vector of layer L such as layer three for the Jth unit. So there's two again in the example above. And so that's dot product with A deactivation value of, and notice this is not L, this is L minus one, like the two above here, because you're dot producting with the output from the previous layer, and then plus B, the parameter for this layer for that unit J. And so this gives you the activation of layer L unit J, where the superscript in square brackets L denotes layer L and the subscript J denotes unit J. And when building neural networks, unit J refers to the Jth neuron. So use those terms a little bit interchangeably, where each unit is a single neuron in a layer. G here is the sigmoid function. In the context of a neural network, G has another name, which is also called the activation function because G outputs this activation value. So when I say activation function, I mean this function G here. And so far, the only activation function you've seen is the sigmoid function. But next week, we'll look at when other functions than the sigmoid function can be plugged in in place of G as well. But so the activation function is just that function that outputs these activation values. And just one last piece of notation in order to make all this notation consistent. I'm also doing to give the input vector X another name, which is a zero. So this way, the same equation also works for the first layer, where when L is equal to one deactivations of the first layer that is a one will be sigmoid times the weights dot product with a zero, which is just this input feature vector X. So with this notation, you now know how to compute the activation values of any layer in a neural network as a function of the parameters as well as the activations of the previous layer. So you now know how to compute the activations of any layer given the activations
| 500
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4.6 Neural Networks Model | More complex neural networks --[Machine Learning | Andrew Ng]: you not, but without rewinding this video, are you able to think through what are the missing superscripts and subscripts in this equation and fill them in yourself? Why don't you take a look at the end video quiz and see if you can figure out what are the appropriate superscripts and subscripts for this equation over here. So to recap, A3 is activation associated with layer three for the second neuron, hence this is a two. There's a parameter associated with the third layer for the second neuron. This is A2, same as above, and then plus B3, two. So hopefully that makes sense. Here's the more general form of this equation for an arbitrary layer L and for an arbitrary unit J, which is that A deactivation output of layer L unit J like A32, that's going to be the sigmoid function applied to this term, which is the weight vector of layer L such as layer three for the Jth unit. So there's two again in the example above. And so that's dot product with A deactivation value of, and notice this is not L, this is L minus one, like the two above here, because you're dot producting with the output from the previous layer, and then plus B, the parameter for this layer for that unit J. And so this gives you the activation of layer L unit J, where the superscript in square brackets L denotes layer L and the subscript J denotes unit J. And when building neural networks, unit J refers to the Jth neuron. So use those terms a little bit interchangeably, where each unit is a single neuron in a layer. G here is the sigmoid function. In the context of a neural network, G has another name, which is also called the activation function because G outputs this activation value. So when I say activation function, I mean this function G here. And so far, the only activation function you've seen is the sigmoid function. But next week, we'll look at when other functions than the sigmoid function can be plugged in in place of G as well. But so the activation function is just that function that outputs these activation values. And just one last piece of notation in order to make all this notation consistent. I'm also doing to give the input vector X another name, which is a zero. So this way, the same equation also works for the first layer, where when L is equal to one deactivations of the first layer that is a one will be sigmoid times the weights dot product with a zero, which is just this input feature vector X. So with this notation, you now know how to compute the activation values of any layer in a neural network as a function of the parameters as well as the activations of the previous layer. So you now know how to compute the activations of any layer given the activations
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4.7 Neural Networks Model |Inference : making predictions (forward propagation)- [ML- Andrew Ng]
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https://www.youtube.com/watch?v=L1fsfAbq-q8
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Let's take what we've learned and put it together into an algorithm to let your neural network make inferences or make predictions. This will be an algorithm called forward propagation. Let's take a look. I'm going to use as a multi-fingered example, handwritten digit recognition. And for simplicity, we're just going to distinguish between the handwritten digits 0 and 1. So it's just a binary classification problem where we're going to input an image and classify is this the digit 0 or the digit 1. And you get to play with this yourself later this week in the practice lab as well. For the example on this slide, I'm going to use an 8 by 8 image. And so this image of a 1 is this grid or matrix of 8 by 8 or 64 pixel intensity values where 255 denotes a bright white pixel and 0 would denote a black pixel and different numbers are different shades of gray in between the shades of black and white. Given these 64 input features, we're going to use a neural network with two hidden layers where the first hidden layer has 25 neurons or 25 units. Second hidden layer has 15 neurons or 15 units, and then finally the output layer outputs. What's the chance of this being 1 versus 0? So let's step through the sequence of computations that the neural network would need to make to go from the input x, this 8 by 8 or 64 numbers, to the predicted probability a3. The first computation is to go from x to a1. And that's what the first layer or the first hidden layer does. It carries out a computation of a superscript square bracket 1 equals this formula on the right. Notice that a1 has 25 numbers because this hidden layer has 25 units, which is why the parameters go from w1 through w25 as well as b1 through b25. And I've written x here, but I could also have written a0 here because by convention deactivation of layer 0, that is a0, is equal to the input feature value x. So that lets us compute a1. The next step is to compute a2. Looking at the second hidden layer, it then carries out this computation where a2 is a function of a1 and is computed as the sigmoid activation function applied to w dot product a1 plus the corresponding value of b. Notice that layer 2 has 15 neurons, so 15 units, which is why the parameters here run from w1 through w15 and b1 through b15. Now we've computed a2. The final step is then to compute a3. And we do so using a very similar computation. Only now this third layer, the output layer, has just one unit, which is why there's just one output here. So a3 is just a scalar. And finally, you can optionally take a3 subscript 1 and threshold it at 0.5 to come up with a binary classification label. Is
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4.7 Neural Networks Model |Inference : making predictions (forward propagation)- [ML- Andrew Ng]: Let's take what we've learned and put it together into an algorithm to let your neural network make inferences or make predictions. This will be an algorithm called forward propagation. Let's take a look. I'm going to use as a multi-fingered example, handwritten digit recognition. And for simplicity, we're just going to distinguish between the handwritten digits 0 and 1. So it's just a binary classification problem where we're going to input an image and classify is this the digit 0 or the digit 1. And you get to play with this yourself later this week in the practice lab as well. For the example on this slide, I'm going to use an 8 by 8 image. And so this image of a 1 is this grid or matrix of 8 by 8 or 64 pixel intensity values where 255 denotes a bright white pixel and 0 would denote a black pixel and different numbers are different shades of gray in between the shades of black and white. Given these 64 input features, we're going to use a neural network with two hidden layers where the first hidden layer has 25 neurons or 25 units. Second hidden layer has 15 neurons or 15 units, and then finally the output layer outputs. What's the chance of this being 1 versus 0? So let's step through the sequence of computations that the neural network would need to make to go from the input x, this 8 by 8 or 64 numbers, to the predicted probability a3. The first computation is to go from x to a1. And that's what the first layer or the first hidden layer does. It carries out a computation of a superscript square bracket 1 equals this formula on the right. Notice that a1 has 25 numbers because this hidden layer has 25 units, which is why the parameters go from w1 through w25 as well as b1 through b25. And I've written x here, but I could also have written a0 here because by convention deactivation of layer 0, that is a0, is equal to the input feature value x. So that lets us compute a1. The next step is to compute a2. Looking at the second hidden layer, it then carries out this computation where a2 is a function of a1 and is computed as the sigmoid activation function applied to w dot product a1 plus the corresponding value of b. Notice that layer 2 has 15 neurons, so 15 units, which is why the parameters here run from w1 through w15 and b1 through b15. Now we've computed a2. The final step is then to compute a3. And we do so using a very similar computation. Only now this third layer, the output layer, has just one unit, which is why there's just one output here. So a3 is just a scalar. And finally, you can optionally take a3 subscript 1 and threshold it at 0.5 to come up with a binary classification label. Is
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4.7 Neural Networks Model |Inference : making predictions (forward propagation)- [ML- Andrew Ng]
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https://www.youtube.com/watch?v=L1fsfAbq-q8
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this the digit 1? Yes or no? So the sequence of computations first takes x and then computes a1 and then computes a2 and then computes a3, which is also the output of the neural network. You can also write that as f of x. So remember when we learned about linear regression and logistic regression, we used f of x to denote the outputs of linear regression or logistic regression. So we can also use f of x to denote the function computed by the neural network as a function of x. Because this computation goes from left to right, you start from x, then compute a1, then a2, then a3, this algorithm is also called forward propagation because you're propagating the activations of the neurons. So you're making these computations in the forward direction from left to right. And this is in contrast to a different algorithm called backward propagation or back propagation, which is used for learning. And that's something you learn about next week. And by the way, this type of neural network architecture where you have more hidden units initially, and then the number of hidden units decreases as you get closer to the output layer, that's also a pretty typical choice when choosing neural network architectures. And you see more examples of this in the practice lab as well. So that's neural network inference using the forward propagation algorithm. And with this, you'd be able to download the parameters of a neural network that someone else had trained and posted on the internet. And you'd be able to carry out inference on your new data using their neural network. Now that you've seen the math and the algorithm, let's take a look at how you can actually implement this in TensorFlow. Specifically, let's take a look at this in the next video.
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4.7 Neural Networks Model |Inference : making predictions (forward propagation)- [ML- Andrew Ng]: this the digit 1? Yes or no? So the sequence of computations first takes x and then computes a1 and then computes a2 and then computes a3, which is also the output of the neural network. You can also write that as f of x. So remember when we learned about linear regression and logistic regression, we used f of x to denote the outputs of linear regression or logistic regression. So we can also use f of x to denote the function computed by the neural network as a function of x. Because this computation goes from left to right, you start from x, then compute a1, then a2, then a3, this algorithm is also called forward propagation because you're propagating the activations of the neurons. So you're making these computations in the forward direction from left to right. And this is in contrast to a different algorithm called backward propagation or back propagation, which is used for learning. And that's something you learn about next week. And by the way, this type of neural network architecture where you have more hidden units initially, and then the number of hidden units decreases as you get closer to the output layer, that's also a pretty typical choice when choosing neural network architectures. And you see more examples of this in the practice lab as well. So that's neural network inference using the forward propagation algorithm. And with this, you'd be able to download the parameters of a neural network that someone else had trained and posted on the internet. And you'd be able to carry out inference on your new data using their neural network. Now that you've seen the math and the algorithm, let's take a look at how you can actually implement this in TensorFlow. Specifically, let's take a look at this in the next video.
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4.8 TensorFlow implementation | Inference in Code --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=VctGI7Xaogw
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TensorFlow is one of the leading frameworks for implementing deep learning algorithms. When I'm building projects, TensorFlow is actually the tool that I use the most often and the other popular tool is PyTorch. But we're going to focus in this specialization on TensorFlow. In this video, let's take a look at how you can implement inference in code using TensorFlow. Let's dive in. One of the remarkable things about neural networks is the same algorithm can be applied to so many different applications. So in order both for this video and in some of the labs for you to see what a neural network is doing, I'm going to use another example to illustrate inference. So sometimes I do like to roast coffee beans myself at home. My favorite is actually Colombian coffee beans. So kind of learning algorithm help optimize the quality of the beans you get from a roasting process like this. When you're roasting coffee, two parameters you get to control are the temperature at which are heating up the raw coffee beans to turn them into nicely roasted coffee beans, as well as the duration or how long you're going to roast the beans. And in this slightly simplified example, we've created the data sets of different temperatures and different durations, as well as labels showing whether the coffee you roasted is good tasting coffee, where cross here, the positive class, y equals one, corresponds to good coffee, and oh, the negative class corresponds to bad coffee. So it looks like a reasonable way to think of this data set is if you cook it at too low a temperature, it doesn't get roasted and ends up undercooked. If you cook it not for long enough, the duration is too short. There's also not a nicely roasted set of beans. And finally, if you were to cook it either for too long or for too high a temperature, then you end up with overcooked beans, they're a little bit burnt beans. And so that's not good coffee either. There's only points within this little triangle here that corresponds to good coffee. This example is simplified a bit from actual coffee roasting. Even though this example is a simplified one for the purpose of illustration, there have actually been serious projects using machine learning to optimize coffee roasting as well. So the task is given a feature vector x with both temperature and duration, say 200 degrees Celsius for 17 minutes, how can we do inference in a neural network to get it to tell us whether or not this temperature and duration setting will result in good coffee or not? It looks like this. We're going to set x to be an array of two numbers, the input features 200 degrees Celsius and 17 minutes. And this here, layer one equals dense units, three activation equals sigmoid, creates a hidden layer of neurons with three hidden units and using as the activation function, the sigmoid
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4.8 TensorFlow implementation | Inference in Code --[Machine Learning | Andrew Ng]: TensorFlow is one of the leading frameworks for implementing deep learning algorithms. When I'm building projects, TensorFlow is actually the tool that I use the most often and the other popular tool is PyTorch. But we're going to focus in this specialization on TensorFlow. In this video, let's take a look at how you can implement inference in code using TensorFlow. Let's dive in. One of the remarkable things about neural networks is the same algorithm can be applied to so many different applications. So in order both for this video and in some of the labs for you to see what a neural network is doing, I'm going to use another example to illustrate inference. So sometimes I do like to roast coffee beans myself at home. My favorite is actually Colombian coffee beans. So kind of learning algorithm help optimize the quality of the beans you get from a roasting process like this. When you're roasting coffee, two parameters you get to control are the temperature at which are heating up the raw coffee beans to turn them into nicely roasted coffee beans, as well as the duration or how long you're going to roast the beans. And in this slightly simplified example, we've created the data sets of different temperatures and different durations, as well as labels showing whether the coffee you roasted is good tasting coffee, where cross here, the positive class, y equals one, corresponds to good coffee, and oh, the negative class corresponds to bad coffee. So it looks like a reasonable way to think of this data set is if you cook it at too low a temperature, it doesn't get roasted and ends up undercooked. If you cook it not for long enough, the duration is too short. There's also not a nicely roasted set of beans. And finally, if you were to cook it either for too long or for too high a temperature, then you end up with overcooked beans, they're a little bit burnt beans. And so that's not good coffee either. There's only points within this little triangle here that corresponds to good coffee. This example is simplified a bit from actual coffee roasting. Even though this example is a simplified one for the purpose of illustration, there have actually been serious projects using machine learning to optimize coffee roasting as well. So the task is given a feature vector x with both temperature and duration, say 200 degrees Celsius for 17 minutes, how can we do inference in a neural network to get it to tell us whether or not this temperature and duration setting will result in good coffee or not? It looks like this. We're going to set x to be an array of two numbers, the input features 200 degrees Celsius and 17 minutes. And this here, layer one equals dense units, three activation equals sigmoid, creates a hidden layer of neurons with three hidden units and using as the activation function, the sigmoid
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4.8 TensorFlow implementation | Inference in Code --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=VctGI7Xaogw
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function. And dense here is just the name of this layer. And then finally, to compute the activation values a one, you would write a one equals layer one applied to the input features x. Then you create layer one as this first hidden layer of the neural network as dense, open prints units three, that means three units or three hidden units in this layer using as the activation function, the sigmoid function. And dense is another name for the layers of a neural network that we've learned about so far. And as you learn more about neural networks, you learn about other types of layers as well. But for now, we just use the dense layer, which is the layer type you've learned about in the last few videos for all of our examples. So next, you compute a one by taking layer one, which is actually a function and applying this function layer one to the values of x. So that's how you get a one, which is going to be a list of three numbers because they one had three units. And so a one here may just for the sake of illustration be point 2.7.3. Next, for the second hidden layer, layer two would be dense of. Now this time as one unit and again to signal and activation function. And you can then compute a two by applying this layer two function to the activation values from layer one to a one. And that will give you the value of a two, which for the sake of illustration is maybe zero point eight. Finally, if you wish to threshold is at 0.5, then you can just test if a two is greater than or equal to 0.5 and set y hat equals to one or zero positive or negative class accordingly. So that's how you do inference in the neural network using TensorFlow. There are some additional details that I didn't go over here, such as how to load the TensorFlow library and how to also load the parameters W and B of the neural network. But we'll go over that in the lab. So please be sure to take a look at the lab. But these are the key steps for for propagation and how you compute a one and a two and optionally threshold a two. Let's look at one more example. And we're going to go back to the handwritten digit classification problem. So in this example, X is a list of the pixel intensity values. So X is equal to a non-pi array of this list of pixel intensity values. And then to initialize and carry out one step before propagation, layer one is a dense layer with 25 units and a sigmoid activation function. And you then compute a one equals the layer one function applied to X to build and carry out inference through the second layer. Similarly, you set up layer two as follows and then compute
| 500
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4.8 TensorFlow implementation | Inference in Code --[Machine Learning | Andrew Ng]: function. And dense here is just the name of this layer. And then finally, to compute the activation values a one, you would write a one equals layer one applied to the input features x. Then you create layer one as this first hidden layer of the neural network as dense, open prints units three, that means three units or three hidden units in this layer using as the activation function, the sigmoid function. And dense is another name for the layers of a neural network that we've learned about so far. And as you learn more about neural networks, you learn about other types of layers as well. But for now, we just use the dense layer, which is the layer type you've learned about in the last few videos for all of our examples. So next, you compute a one by taking layer one, which is actually a function and applying this function layer one to the values of x. So that's how you get a one, which is going to be a list of three numbers because they one had three units. And so a one here may just for the sake of illustration be point 2.7.3. Next, for the second hidden layer, layer two would be dense of. Now this time as one unit and again to signal and activation function. And you can then compute a two by applying this layer two function to the activation values from layer one to a one. And that will give you the value of a two, which for the sake of illustration is maybe zero point eight. Finally, if you wish to threshold is at 0.5, then you can just test if a two is greater than or equal to 0.5 and set y hat equals to one or zero positive or negative class accordingly. So that's how you do inference in the neural network using TensorFlow. There are some additional details that I didn't go over here, such as how to load the TensorFlow library and how to also load the parameters W and B of the neural network. But we'll go over that in the lab. So please be sure to take a look at the lab. But these are the key steps for for propagation and how you compute a one and a two and optionally threshold a two. Let's look at one more example. And we're going to go back to the handwritten digit classification problem. So in this example, X is a list of the pixel intensity values. So X is equal to a non-pi array of this list of pixel intensity values. And then to initialize and carry out one step before propagation, layer one is a dense layer with 25 units and a sigmoid activation function. And you then compute a one equals the layer one function applied to X to build and carry out inference through the second layer. Similarly, you set up layer two as follows and then compute
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4.8 TensorFlow implementation | Inference in Code --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=VctGI7Xaogw
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a two as layer two applied to a one. And then finally, layer three is the third and final dense layer. And then finally, you can optionally threshold a three to come up with a binary prediction for Y hat. So that's the syntax for carrying out inference in TensorFlow. One thing I briefly alluded to is the structure of the non-pi arrays. TensorFlow treats data in a certain way that is important to get right. So in the next video, let's take a look at how TensorFlow handles data.
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4.8 TensorFlow implementation | Inference in Code --[Machine Learning | Andrew Ng]: a two as layer two applied to a one. And then finally, layer three is the third and final dense layer. And then finally, you can optionally threshold a three to come up with a binary prediction for Y hat. So that's the syntax for carrying out inference in TensorFlow. One thing I briefly alluded to is the structure of the non-pi arrays. TensorFlow treats data in a certain way that is important to get right. So in the next video, let's take a look at how TensorFlow handles data.
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4.9 TensorFlow implementation | Data in TensorFlow --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=Q3cLU1trK_E
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In this video, I want to step through with you how data is represented in NumPy and IntensiveLow so that as you're implementing new neural networks, you can have a consistent framework to think about how to represent your data. One of the unfortunate things about the way things are done in code today is that many, many years ago, NumPy was first created and became a standard library for linear algebra in Python. And then much later, the Google Brain team, the team that I had started and once led, created TensorFlow. And so unfortunately, there are some inconsistencies between how data is represented in NumPy and IntensiveLow. So it's good to be aware of these conventions so that you can implement correct code and hopefully get things running in your neural networks. Let's start by taking a look at how TensorFlow represents data. Let's say you have a data set like this from the coffee example. I mentioned that you would write x as follows. So why do you have this double square bracket here? Let's take a look at how NumPy stores vectors and matrices. In case you think matrices and vectors are complicated mathematical concepts, don't worry about it. We'll go through a few concrete examples and you'll be able to do everything you need to do with matrices and vectors in order to implement neural networks. Let's start with an example of a matrix. Here is a matrix with two rows and three columns. Notice that there are one, two rows and one, two, three columns. So we call this a two by three matrix. And so the convention is the dimension of the matrix is written as the number of rows by the number of columns. So in code to store this matrix, this two by three matrix, you just write x equals NP dot array of these numbers like these, where you notice that the square brackets tells you that one, two, three is the first row of this matrix and four, five, six is the second row of this matrix. And then this alphas square bracket groups the first and the second row together. So this says x to be this 2D array of numbers. So a matrix is just a 2D array of numbers. Let's look at one more example. Here I've written out another matrix. How many rows and how many columns does this have? Well we count this as one, two, three, four rows and it has one, two columns. So this is a number of rows by number of columns matrix. So it's a four by two matrix. And so to store this in code, you would write x equals NP dot array and then this syntax over here to store these four rows of a matrix in the variable x. So this creates a 2D array of these eight numbers. Matrices can have different dimensions. We saw an example of a two by three matrix and a
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4.9 TensorFlow implementation | Data in TensorFlow --[Machine Learning | Andrew Ng]: In this video, I want to step through with you how data is represented in NumPy and IntensiveLow so that as you're implementing new neural networks, you can have a consistent framework to think about how to represent your data. One of the unfortunate things about the way things are done in code today is that many, many years ago, NumPy was first created and became a standard library for linear algebra in Python. And then much later, the Google Brain team, the team that I had started and once led, created TensorFlow. And so unfortunately, there are some inconsistencies between how data is represented in NumPy and IntensiveLow. So it's good to be aware of these conventions so that you can implement correct code and hopefully get things running in your neural networks. Let's start by taking a look at how TensorFlow represents data. Let's say you have a data set like this from the coffee example. I mentioned that you would write x as follows. So why do you have this double square bracket here? Let's take a look at how NumPy stores vectors and matrices. In case you think matrices and vectors are complicated mathematical concepts, don't worry about it. We'll go through a few concrete examples and you'll be able to do everything you need to do with matrices and vectors in order to implement neural networks. Let's start with an example of a matrix. Here is a matrix with two rows and three columns. Notice that there are one, two rows and one, two, three columns. So we call this a two by three matrix. And so the convention is the dimension of the matrix is written as the number of rows by the number of columns. So in code to store this matrix, this two by three matrix, you just write x equals NP dot array of these numbers like these, where you notice that the square brackets tells you that one, two, three is the first row of this matrix and four, five, six is the second row of this matrix. And then this alphas square bracket groups the first and the second row together. So this says x to be this 2D array of numbers. So a matrix is just a 2D array of numbers. Let's look at one more example. Here I've written out another matrix. How many rows and how many columns does this have? Well we count this as one, two, three, four rows and it has one, two columns. So this is a number of rows by number of columns matrix. So it's a four by two matrix. And so to store this in code, you would write x equals NP dot array and then this syntax over here to store these four rows of a matrix in the variable x. So this creates a 2D array of these eight numbers. Matrices can have different dimensions. We saw an example of a two by three matrix and a
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4.9 TensorFlow implementation | Data in TensorFlow --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=Q3cLU1trK_E
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four by two matrix. A matrix can also be other dimensions like one by two or two by one. And we'll see examples of these on the next slide. So what we did previously when setting x to be input feature vectors was set x to be equal to NP array with two square brackets 200 comma 17. And what that does is this creates a one by two matrix that is just one row and two columns. Let's look at a different example. If you were to define x to be NP array, but now written like this, this creates a two by one matrix that has two rows and one column. Because the first row is just a number 200 and the second row is just the number 17. And so this has the same numbers, but in a two by one instead of a one by two matrix. In nav, this example on top is also called a row vector. It's a vector that is just a single row. And this example is also called a column vector because it's a vector that just has a single column. And the difference between using double square brackets like this versus a single square bracket like this is that whereas the two examples on top are two D arrays where one of the dimensions happens to be one. This example results in a one D vector. So this is just a one D array that has no rows or columns. Although by convention, we may write x as a column like this. So on a contrast this with what we had previously done in the first course, which was to write x like this with a single square bracket. And that resulted in what's called in Python a one D vector instead of a two D matrix. And this technically is not one by two or two by one. It's just a linear array with no rows or no columns. And it's just a list of numbers. So whereas in course one, when we're working with linear regression, logistic regression, we use these one D vectors to represent the input features x. With TensorFlow, the convention is to use matrices to represent the data. And why is there this switch in conventions? Well, it turns out that TensorFlow was designed to handle very large datasets. And by representing the data in matrices instead of one D arrays, it lets TensorFlow be a bit more computationally efficient internally. So going back to our original example, for the first training example in this dataset with features 200 degrees Celsius in 17 minutes, we would represent it like this. And so this is actually a one by two matrix that happens to have one row and two columns to store the numbers 217. And in case this seems like a lot of details and really complicated conventions, don't worry about it. All this will become clearer. And you get to see the
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4.9 TensorFlow implementation | Data in TensorFlow --[Machine Learning | Andrew Ng]: four by two matrix. A matrix can also be other dimensions like one by two or two by one. And we'll see examples of these on the next slide. So what we did previously when setting x to be input feature vectors was set x to be equal to NP array with two square brackets 200 comma 17. And what that does is this creates a one by two matrix that is just one row and two columns. Let's look at a different example. If you were to define x to be NP array, but now written like this, this creates a two by one matrix that has two rows and one column. Because the first row is just a number 200 and the second row is just the number 17. And so this has the same numbers, but in a two by one instead of a one by two matrix. In nav, this example on top is also called a row vector. It's a vector that is just a single row. And this example is also called a column vector because it's a vector that just has a single column. And the difference between using double square brackets like this versus a single square bracket like this is that whereas the two examples on top are two D arrays where one of the dimensions happens to be one. This example results in a one D vector. So this is just a one D array that has no rows or columns. Although by convention, we may write x as a column like this. So on a contrast this with what we had previously done in the first course, which was to write x like this with a single square bracket. And that resulted in what's called in Python a one D vector instead of a two D matrix. And this technically is not one by two or two by one. It's just a linear array with no rows or no columns. And it's just a list of numbers. So whereas in course one, when we're working with linear regression, logistic regression, we use these one D vectors to represent the input features x. With TensorFlow, the convention is to use matrices to represent the data. And why is there this switch in conventions? Well, it turns out that TensorFlow was designed to handle very large datasets. And by representing the data in matrices instead of one D arrays, it lets TensorFlow be a bit more computationally efficient internally. So going back to our original example, for the first training example in this dataset with features 200 degrees Celsius in 17 minutes, we would represent it like this. And so this is actually a one by two matrix that happens to have one row and two columns to store the numbers 217. And in case this seems like a lot of details and really complicated conventions, don't worry about it. All this will become clearer. And you get to see the
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4.9 TensorFlow implementation | Data in TensorFlow --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=Q3cLU1trK_E
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concrete implementations of the code yourself in the optional labs and in the practice labs. Going back to the code for carrying out for propagation or inference in the neural network, when you compute a one equals layer one applied to X, what is a one? Well, a one is actually going to be because there's three numbers is actually going to be a one by three matrix. And if you print out a one, you will get something like this is TF dot tensor point 2.7.3 is a shape of one by three, one three refers to that this is a one by three matrix. And this is TensorFlow's way of saying that this is a floating point number, meaning that it's a number that can have a decimal point represented using 32 bits of memory in your computer. That's what a float 32 is. And what is a tensor? A tensor here is a data type that the TensorFlow team had created in order to store and carry out computations on matrices efficiently. So whenever you see tensor, just think of it as matrix on these few slides. Technically a tensor is a little bit more general than the matrix, but for the purposes of this course, think of tensor as just a way of representing matrices. So remember I said at the start of this video that there's the TensorFlow way of representing a matrix and the NumPy way of representing matrix. This is an artifact of the history of how NumPy and TensorFlow were created. And unfortunately, there are two ways of representing a matrix that have been baked into these systems. And in fact, if you want to take a one, which is a tensor and want to convert it back to NumPy array, you can do so with this function a one dot NumPy and it will take the same data and return it in the form of a NumPy array rather than in the form of a TensorFlow array or TensorFlow matrix. Now let's take a look at what the activations output by the second layer would look like. Here's the code that we had from before. Layer two is a dense layer with one unit and a six point activation. And a two is computed by taking layer two and applying it to a one. So what is a two? A two may be a number like 0.8 and technically this is a one by one matrix. It's a 2D array with one row and one column. And so it's equal to this number 0.8. And if you print out a two, you see that it is a TensorFlow tensor with just one element, one number 0.8. And it is a one by one matrix. Again it is a flow 32 decimal point number taking up 32 bits in computer memory. Once again you can convert from a TensorFlow tensor to a NumPy matrix using a two dot NumPy and that will
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4.9 TensorFlow implementation | Data in TensorFlow --[Machine Learning | Andrew Ng]: concrete implementations of the code yourself in the optional labs and in the practice labs. Going back to the code for carrying out for propagation or inference in the neural network, when you compute a one equals layer one applied to X, what is a one? Well, a one is actually going to be because there's three numbers is actually going to be a one by three matrix. And if you print out a one, you will get something like this is TF dot tensor point 2.7.3 is a shape of one by three, one three refers to that this is a one by three matrix. And this is TensorFlow's way of saying that this is a floating point number, meaning that it's a number that can have a decimal point represented using 32 bits of memory in your computer. That's what a float 32 is. And what is a tensor? A tensor here is a data type that the TensorFlow team had created in order to store and carry out computations on matrices efficiently. So whenever you see tensor, just think of it as matrix on these few slides. Technically a tensor is a little bit more general than the matrix, but for the purposes of this course, think of tensor as just a way of representing matrices. So remember I said at the start of this video that there's the TensorFlow way of representing a matrix and the NumPy way of representing matrix. This is an artifact of the history of how NumPy and TensorFlow were created. And unfortunately, there are two ways of representing a matrix that have been baked into these systems. And in fact, if you want to take a one, which is a tensor and want to convert it back to NumPy array, you can do so with this function a one dot NumPy and it will take the same data and return it in the form of a NumPy array rather than in the form of a TensorFlow array or TensorFlow matrix. Now let's take a look at what the activations output by the second layer would look like. Here's the code that we had from before. Layer two is a dense layer with one unit and a six point activation. And a two is computed by taking layer two and applying it to a one. So what is a two? A two may be a number like 0.8 and technically this is a one by one matrix. It's a 2D array with one row and one column. And so it's equal to this number 0.8. And if you print out a two, you see that it is a TensorFlow tensor with just one element, one number 0.8. And it is a one by one matrix. Again it is a flow 32 decimal point number taking up 32 bits in computer memory. Once again you can convert from a TensorFlow tensor to a NumPy matrix using a two dot NumPy and that will
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4.9 TensorFlow implementation | Data in TensorFlow --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=Q3cLU1trK_E
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turn this back into a NumPy array that looks like this. So that hopefully gives you a sense of how data is represented in TensorFlow and in NumPy. I'm used to loading data and manipulating data in NumPy. But when you pass a NumPy array into TensorFlow, TensorFlow likes to convert it to its own internal format, the tensor, and then operate efficiently using tensors. And when you read the data back out, you can keep it as a tensor or convert it back to a NumPy array. I think it's a bit unfortunate that the history of how these libraries evolve has led us to do this extra conversion work when actually the two libraries can work quite well together. But when you convert back and forth, whether you're using a NumPy array or a tensor, it's just something to be aware of when you're writing code. Next, let's take what we've learned and put it together to actually build a neural network. Let's go see that in the next video.
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4.9 TensorFlow implementation | Data in TensorFlow --[Machine Learning | Andrew Ng]: turn this back into a NumPy array that looks like this. So that hopefully gives you a sense of how data is represented in TensorFlow and in NumPy. I'm used to loading data and manipulating data in NumPy. But when you pass a NumPy array into TensorFlow, TensorFlow likes to convert it to its own internal format, the tensor, and then operate efficiently using tensors. And when you read the data back out, you can keep it as a tensor or convert it back to a NumPy array. I think it's a bit unfortunate that the history of how these libraries evolve has led us to do this extra conversion work when actually the two libraries can work quite well together. But when you convert back and forth, whether you're using a NumPy array or a tensor, it's just something to be aware of when you're writing code. Next, let's take what we've learned and put it together to actually build a neural network. Let's go see that in the next video.
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4.10 TensorFlow implementation | Building a neural network --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=8GAQXo6SRws
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So you've seen a bunch of TensorFlow code by now, learned about how to build a layer in TensorFlow, how to do forward prop through a single layer in TensorFlow, and also learned about data in TensorFlow. Let's put it all together and talk about how to build a neural network in TensorFlow. This is also the last video on TensorFlow for this week, and in this video, you also learned about a different way of building a neural network that will be even a little bit simpler than what you've seen so far. So let's dive in. What you saw previously was if you want to do forward prop, you initialize the data x, create layer one like so, then compute A1, then create layer two, and compute A2. So this was an explicit way of carrying out forward prop one layer of computation at the time. It turns out that TensorFlow has a different way of implementing forward prop as well as learning. Let me show you a different way of building a neural network in TensorFlow, which is that same as before, you're going to create layer one and create layer two. But now, instead of you manually taking the data and passing it to layer one, and then taking the activations from layer one and passing it to layer two, we can instead tell TensorFlow that we would like it to take layer one and layer two and string them together to form a neural network. That's what the sequential function in TensorFlow does, which is it says, dear TensorFlow, please create a neural network for me by sequentially stringing together these two layers that I just created. It turns out that with the sequential framework, TensorFlow can do a lot of work for you. Let's say you have a training set like this on the left. This is for the coffee example. You can then take the training data's inputs X and put them into a NumPy array. This here is a four by two matrix, and the target labels Y can then be written as follows. This is just a four dimensional array. Y, the set of targets, can then be stored as a 1D array like this, 1 0 0 1, corresponding to the four training examples. It turns out that given the data X and Y stored in this matrix X and this array Y, if you want to train this neural network, all you need to do is call two functions. You need to call model.compile with some parameters. We'll talk more about this next week, so don't worry about it for now. And then you need to call model.fit X Y, which tells TensorFlow to take this neural network that it created by sequentially stringing together layers one and two, and to train it on the data X and Y. But we'll learn the details of how to do this next week. And then finally, how do you do inference
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4.10 TensorFlow implementation | Building a neural network --[Machine Learning | Andrew Ng]: So you've seen a bunch of TensorFlow code by now, learned about how to build a layer in TensorFlow, how to do forward prop through a single layer in TensorFlow, and also learned about data in TensorFlow. Let's put it all together and talk about how to build a neural network in TensorFlow. This is also the last video on TensorFlow for this week, and in this video, you also learned about a different way of building a neural network that will be even a little bit simpler than what you've seen so far. So let's dive in. What you saw previously was if you want to do forward prop, you initialize the data x, create layer one like so, then compute A1, then create layer two, and compute A2. So this was an explicit way of carrying out forward prop one layer of computation at the time. It turns out that TensorFlow has a different way of implementing forward prop as well as learning. Let me show you a different way of building a neural network in TensorFlow, which is that same as before, you're going to create layer one and create layer two. But now, instead of you manually taking the data and passing it to layer one, and then taking the activations from layer one and passing it to layer two, we can instead tell TensorFlow that we would like it to take layer one and layer two and string them together to form a neural network. That's what the sequential function in TensorFlow does, which is it says, dear TensorFlow, please create a neural network for me by sequentially stringing together these two layers that I just created. It turns out that with the sequential framework, TensorFlow can do a lot of work for you. Let's say you have a training set like this on the left. This is for the coffee example. You can then take the training data's inputs X and put them into a NumPy array. This here is a four by two matrix, and the target labels Y can then be written as follows. This is just a four dimensional array. Y, the set of targets, can then be stored as a 1D array like this, 1 0 0 1, corresponding to the four training examples. It turns out that given the data X and Y stored in this matrix X and this array Y, if you want to train this neural network, all you need to do is call two functions. You need to call model.compile with some parameters. We'll talk more about this next week, so don't worry about it for now. And then you need to call model.fit X Y, which tells TensorFlow to take this neural network that it created by sequentially stringing together layers one and two, and to train it on the data X and Y. But we'll learn the details of how to do this next week. And then finally, how do you do inference
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4.10 TensorFlow implementation | Building a neural network --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=8GAQXo6SRws
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on this neural network? How do you do forward prop? If you have a new example, say X new, which is NP array with these two features, then to carry out forward prop, instead of having to do it one layer at a time yourself, you just have to call model predict on X new, and this will output the corresponding value of A2 for you, given this input value of X. So model predict carries out forward propagation or carries out inference for you using this neural network that you compiled using the sequential function. Now, I want to take these three lines of code on top and just simplify it a little bit further, which is when coding in TensorFlow, by convention, we don't explicitly assign the two layers to two variables, layer one and layer two as follows, but by convention, I would usually just write the code like this. What we say the model is a sequential model of a few layers strung together sequentially, where the first layer, layer one, is a dense layer with three units and activation of sigmoid, and the second layer is a dense layer with one unit and again a sigmoid activation function. So if you look at others TensorFlow code, you often see it look more like this, rather than having an explicit assignment to these layer one and layer two variables. And so that's it. This is pretty much the code you need in order to train as well as to inference on a neural network in TensorFlow. Where again, we'll talk more about the training bits of this, the compound, the compile and the fit function next week. Let's redo this for the digit classification example as well. So previously we had X is this input, layer one is a layer, A1 equals layer one applied to X and so on through layer two and layer three in order to try to classify a digit. With this new coding convention, with using TensorFlow's sequential function, you can instead specify what are layer one, layer two, layer three, and tell TensorFlow to string the layers together for you into a neural network. And same as before, you can then store the data in a matrix and run the compile function and fit the model as follows. Again, more on this next week. And finally, to do inference or to make predictions, you can use model predict on X new. And similar to what you saw before with the coffee classification network, by convention, instead of assigning layer one, layer two, layer three explicitly like this, we would more calmly just take these layers and put them directly into the sequential function to end up with this more compact code where you just tell TensorFlow, create a model for me that sequentially strings together these three layers and then the rest of the code works same as before. So that's how you would build a neural network in TensorFlow.
| 500
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4.10 TensorFlow implementation | Building a neural network --[Machine Learning | Andrew Ng]: on this neural network? How do you do forward prop? If you have a new example, say X new, which is NP array with these two features, then to carry out forward prop, instead of having to do it one layer at a time yourself, you just have to call model predict on X new, and this will output the corresponding value of A2 for you, given this input value of X. So model predict carries out forward propagation or carries out inference for you using this neural network that you compiled using the sequential function. Now, I want to take these three lines of code on top and just simplify it a little bit further, which is when coding in TensorFlow, by convention, we don't explicitly assign the two layers to two variables, layer one and layer two as follows, but by convention, I would usually just write the code like this. What we say the model is a sequential model of a few layers strung together sequentially, where the first layer, layer one, is a dense layer with three units and activation of sigmoid, and the second layer is a dense layer with one unit and again a sigmoid activation function. So if you look at others TensorFlow code, you often see it look more like this, rather than having an explicit assignment to these layer one and layer two variables. And so that's it. This is pretty much the code you need in order to train as well as to inference on a neural network in TensorFlow. Where again, we'll talk more about the training bits of this, the compound, the compile and the fit function next week. Let's redo this for the digit classification example as well. So previously we had X is this input, layer one is a layer, A1 equals layer one applied to X and so on through layer two and layer three in order to try to classify a digit. With this new coding convention, with using TensorFlow's sequential function, you can instead specify what are layer one, layer two, layer three, and tell TensorFlow to string the layers together for you into a neural network. And same as before, you can then store the data in a matrix and run the compile function and fit the model as follows. Again, more on this next week. And finally, to do inference or to make predictions, you can use model predict on X new. And similar to what you saw before with the coffee classification network, by convention, instead of assigning layer one, layer two, layer three explicitly like this, we would more calmly just take these layers and put them directly into the sequential function to end up with this more compact code where you just tell TensorFlow, create a model for me that sequentially strings together these three layers and then the rest of the code works same as before. So that's how you would build a neural network in TensorFlow.
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4.10 TensorFlow implementation | Building a neural network --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=8GAQXo6SRws
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Now I know that when you're learning about these techniques, sometimes someone may ask you to, hey, implement these five lines of code, and then you type five lines of code and then someone says, congratulations with just five lines of code, you've built this crazy complicated state of the art neural network. And sometimes that makes you wonder what exactly did I do with just these five lines of code. One thing I want you to take away from the machine learning specialization is the ability to use cutting edge libraries, like TensorFlow to do your work efficiently. But I don't really want you to just call five lines of code and not really also know what the code is actually doing underneath the hood. So in the next video, I'd like to go back and share with you how you can implement from scratch by yourself for propagation in Python so that you can understand the whole thing for yourself. In practice, most machine learning engineers don't actually implement for a problem in Python that often. We just use libraries like TensorFlow and PyTorch. But because I want you to understand how these algorithms work yourself, so that if something goes wrong, you can think through for yourself what you might need to change, what's likely to work, what's less likely to work. Let's also go through what it would take for you to implement for propagation from scratch. Because that way, even when you're calling a library and having it run efficiently and do great things in your application, I want you in the back of your mind to also have that deeper understanding of what your code is actually doing. So with that, let's go on to the next video.
| 294
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4.10 TensorFlow implementation | Building a neural network --[Machine Learning | Andrew Ng]: Now I know that when you're learning about these techniques, sometimes someone may ask you to, hey, implement these five lines of code, and then you type five lines of code and then someone says, congratulations with just five lines of code, you've built this crazy complicated state of the art neural network. And sometimes that makes you wonder what exactly did I do with just these five lines of code. One thing I want you to take away from the machine learning specialization is the ability to use cutting edge libraries, like TensorFlow to do your work efficiently. But I don't really want you to just call five lines of code and not really also know what the code is actually doing underneath the hood. So in the next video, I'd like to go back and share with you how you can implement from scratch by yourself for propagation in Python so that you can understand the whole thing for yourself. In practice, most machine learning engineers don't actually implement for a problem in Python that often. We just use libraries like TensorFlow and PyTorch. But because I want you to understand how these algorithms work yourself, so that if something goes wrong, you can think through for yourself what you might need to change, what's likely to work, what's less likely to work. Let's also go through what it would take for you to implement for propagation from scratch. Because that way, even when you're calling a library and having it run efficiently and do great things in your application, I want you in the back of your mind to also have that deeper understanding of what your code is actually doing. So with that, let's go on to the next video.
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4.11 Neural network implementation in Python | Forward prop in a single layer-[ML| Andrew Ng]
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https://www.youtube.com/watch?v=ydKVT95-Ufo
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If you had to implement forward propagation yourself from scratch in Python, how would you go about doing so? In addition to gaining intuition about what's really going on in libraries like TensorFlow and PyTorch, if ever someday you decide you want to build something even better than TensorFlow and PyTorch, maybe now you'd have a better idea how. I don't really recommend doing this for most people, but maybe someday someone will come up with an even better framework than TensorFlow and PyTorch, and whoever does that may end up having to implement these things from scratch themselves. So let's take a look. On this slide, I'm going to go through quite a bit of code, and you'll see all this code again later in the optional lab as well as in the practice lab. So don't worry about having to take notes on every line of code or memorize every line of code. You see this code written down in the Jupyter Notebook in the lab, and the goal of this video is to just show you the code to make sure you can understand what it's doing so that when you go to the optional lab in the practice lab and see the code there, you know what to do. So don't worry about taking detailed notes on every line. If you can read through the code on this slide and understand what it's doing, that's all you need. So let's take a look at how you implement forward prop in a single layer. We're going to continue using the coffee roasting model shown here. And let's look at how you would take an input feature vector x and implement forward prop to get this output a2. In this Python implementation, I'm going to use one D arrays to represent all of these vectors and parameters, which is why there's only a single square bracket here. This is a one D array in Python rather than a 2D matrix, which is what we had when we had double square brackets. So the first value you need to compute is a superscript square bracket one, subscript one, which is the first activation value of a one. And that's G of this expression over here. So I'm going to use the convention on this slide that a term like w two one, I'm going to represent as a variable w two and then subscript one, this underscore one denotes subscript one. So w two means w superscript two in square brackets and then subscript one. So to compute a one one, we have parameters w one one and b one one, which are say one two and negative one. You would then compute z one one as the dot product between that parameter w one one and the input X and add it to be one one and then finally a one one is equal to G the safe point function applied to z one one. Next let's go
| 500
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4.11 Neural network implementation in Python | Forward prop in a single layer-[ML| Andrew Ng]: If you had to implement forward propagation yourself from scratch in Python, how would you go about doing so? In addition to gaining intuition about what's really going on in libraries like TensorFlow and PyTorch, if ever someday you decide you want to build something even better than TensorFlow and PyTorch, maybe now you'd have a better idea how. I don't really recommend doing this for most people, but maybe someday someone will come up with an even better framework than TensorFlow and PyTorch, and whoever does that may end up having to implement these things from scratch themselves. So let's take a look. On this slide, I'm going to go through quite a bit of code, and you'll see all this code again later in the optional lab as well as in the practice lab. So don't worry about having to take notes on every line of code or memorize every line of code. You see this code written down in the Jupyter Notebook in the lab, and the goal of this video is to just show you the code to make sure you can understand what it's doing so that when you go to the optional lab in the practice lab and see the code there, you know what to do. So don't worry about taking detailed notes on every line. If you can read through the code on this slide and understand what it's doing, that's all you need. So let's take a look at how you implement forward prop in a single layer. We're going to continue using the coffee roasting model shown here. And let's look at how you would take an input feature vector x and implement forward prop to get this output a2. In this Python implementation, I'm going to use one D arrays to represent all of these vectors and parameters, which is why there's only a single square bracket here. This is a one D array in Python rather than a 2D matrix, which is what we had when we had double square brackets. So the first value you need to compute is a superscript square bracket one, subscript one, which is the first activation value of a one. And that's G of this expression over here. So I'm going to use the convention on this slide that a term like w two one, I'm going to represent as a variable w two and then subscript one, this underscore one denotes subscript one. So w two means w superscript two in square brackets and then subscript one. So to compute a one one, we have parameters w one one and b one one, which are say one two and negative one. You would then compute z one one as the dot product between that parameter w one one and the input X and add it to be one one and then finally a one one is equal to G the safe point function applied to z one one. Next let's go
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4.11 Neural network implementation in Python | Forward prop in a single layer-[ML| Andrew Ng]
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https://www.youtube.com/watch?v=ydKVT95-Ufo
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on to compute a one two, which again by the convention I described here is going to be a one two written like that. So similar as what we did on the left, w one two is your two parameters minus three four, b one two is the term b one two over there. So you compute z as this term in the middle and then apply the safe point function and then you end up with a one two and finally you do the same thing to compute a one three. Now you've computed these three values a one one, a one two and a one three and we like to take these three numbers and group them together into an array to give you a one up here, which is the output of the first layer. And so you do that by grouping them together using a NumPy array as follows. So now you've computed a one. Let's implement the second layer as well to compute the output a two. So a two is computed using this expression and so we would have parameters w two one and b two one corresponding to these parameters and then you would compute z as the dot product between w two one and a one and add b two one and then apply the safe point function to get a two one and that's it. That's how you implement Ford prop using just Python and NumPy. Now there are a lot of expressions in this page of code that you just saw. Let's in the next video look at how you can simplify this to implement Ford prop for a more general neural network rather than hard coding it for every single neuron like we just did. So let's go see that in the next video.
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4.11 Neural network implementation in Python | Forward prop in a single layer-[ML| Andrew Ng]: on to compute a one two, which again by the convention I described here is going to be a one two written like that. So similar as what we did on the left, w one two is your two parameters minus three four, b one two is the term b one two over there. So you compute z as this term in the middle and then apply the safe point function and then you end up with a one two and finally you do the same thing to compute a one three. Now you've computed these three values a one one, a one two and a one three and we like to take these three numbers and group them together into an array to give you a one up here, which is the output of the first layer. And so you do that by grouping them together using a NumPy array as follows. So now you've computed a one. Let's implement the second layer as well to compute the output a two. So a two is computed using this expression and so we would have parameters w two one and b two one corresponding to these parameters and then you would compute z as the dot product between w two one and a one and add b two one and then apply the safe point function to get a two one and that's it. That's how you implement Ford prop using just Python and NumPy. Now there are a lot of expressions in this page of code that you just saw. Let's in the next video look at how you can simplify this to implement Ford prop for a more general neural network rather than hard coding it for every single neuron like we just did. So let's go see that in the next video.
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4.12 Neural network implementation in Python | General implementation of forward propagation-ML Ng
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https://www.youtube.com/watch?v=snDJExVtMMQ
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In the last video, you saw how to implement forward prop in Python, but by hard coding lines of code for every single neuron. Let's now take a look at the more general implementation of forward prop in Python. Similar to the previous video, my goal in this video is to show you the code so that when you see it again in the practice lab and the optional labs, you know how to interpret it. So as we walk through this example, don't worry about taking notes on every single line of code. If you can read through the code and understand it, that's definitely enough. So what you can do is write a function to implement a dense layer that is a single layer of a neural network. So I'm going to define the dense function, which takes as input the activation from the previous layer, as well as the parameters w and b for the neurons in a given layer. Using the example from the previous video, if layer one has three neurons, and if w1 and w2 and w3 are these, then what we'll do is stack all of these weight vectors into a matrix. This is going to be a two by three matrix, where the first column is the parameter w11, the second column is the parameter w12, and the third column is the parameter w13. And then in a similar way, if you have parameters b, b11 equals negative 1, b12 equals 1, and so on, then we're going to stack these three numbers into one D array b as follows, negative 1, 1, 2. So what the dense function will do is take as input the activation from the previous layer, and a here could be a0, which is equal to x, or the activation from a later layer, as well as the w parameters stacked in columns, like shown on the right, as well as the b parameters also stacked into one D array, like shown to the left over there. And what this function will do is input a, the activation from the previous layer, and will output the activations from the columns layer. So let's step through the code for doing this. Here's the code. First, units equals w dot shape one. So w here is a two by three matrix, and so the number of columns is three. That's equal to the number of units in this layer. So here, units would be equal to three. And looking at the shape of w, it's just a way of pulling out the number of hidden units, or the number of units in this layer. Next, we set a to be an array of zeros with as many elements as there are units. So in this example, we need to output three activation values. So this just initializes a to be zero, zero, zero, an array of three zeros. Next, we go through a for loop to compute the first,
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4.12 Neural network implementation in Python | General implementation of forward propagation-ML Ng: In the last video, you saw how to implement forward prop in Python, but by hard coding lines of code for every single neuron. Let's now take a look at the more general implementation of forward prop in Python. Similar to the previous video, my goal in this video is to show you the code so that when you see it again in the practice lab and the optional labs, you know how to interpret it. So as we walk through this example, don't worry about taking notes on every single line of code. If you can read through the code and understand it, that's definitely enough. So what you can do is write a function to implement a dense layer that is a single layer of a neural network. So I'm going to define the dense function, which takes as input the activation from the previous layer, as well as the parameters w and b for the neurons in a given layer. Using the example from the previous video, if layer one has three neurons, and if w1 and w2 and w3 are these, then what we'll do is stack all of these weight vectors into a matrix. This is going to be a two by three matrix, where the first column is the parameter w11, the second column is the parameter w12, and the third column is the parameter w13. And then in a similar way, if you have parameters b, b11 equals negative 1, b12 equals 1, and so on, then we're going to stack these three numbers into one D array b as follows, negative 1, 1, 2. So what the dense function will do is take as input the activation from the previous layer, and a here could be a0, which is equal to x, or the activation from a later layer, as well as the w parameters stacked in columns, like shown on the right, as well as the b parameters also stacked into one D array, like shown to the left over there. And what this function will do is input a, the activation from the previous layer, and will output the activations from the columns layer. So let's step through the code for doing this. Here's the code. First, units equals w dot shape one. So w here is a two by three matrix, and so the number of columns is three. That's equal to the number of units in this layer. So here, units would be equal to three. And looking at the shape of w, it's just a way of pulling out the number of hidden units, or the number of units in this layer. Next, we set a to be an array of zeros with as many elements as there are units. So in this example, we need to output three activation values. So this just initializes a to be zero, zero, zero, an array of three zeros. Next, we go through a for loop to compute the first,
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4.12 Neural network implementation in Python | General implementation of forward propagation-ML Ng
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https://www.youtube.com/watch?v=snDJExVtMMQ
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second, and third elements of a. So for j and range units, so j goes from zero to units minus one, so it goes from zero, one, two, indexing from zero and Python as usual. This command, w equals capital W colon comma j, this is how you pull out the j column of a matrix in Python. So the first time through this loop, this will pull out the first column of w, and so will pull out w one one. The second time through this loop, when you're computing the activation of the second unit, the pull out the second column corresponding to w one two, and so on for the third time through this loop. And then you compute z using the usual formula as a thought product between that parameter w and the activation that you had received plus b j. And then you compute the activation a j equals g sigmoid function applied to z. So three times through this loop and you computed the values for all three values of this vector of activations a, and then finally you return a. So what the dense function does is it inputs the activations from the previous layer and given the parameters for the current layer, it returns the activations for the next layer. So given the dense function, here's how you can string together a few dense layers sequentially in order to implement forward prop in the neural network. Given the input features x, you can then compute the activations a one to be a one equals dens of x w one b one, where here w one b one are the parameters, sometimes also called the weights of the first hidden layer. Then you can compute a two as dens of now a one, which you just computed above and w two b two, which are the parameters or weights of this second hidden layer and then compute a three and a four. And if this is a neural network with four layers, then the final output f of x is just equal to a four. And so you return f of x. Notice that here I'm using a capital W because one of the notational conventions from linear algebra is to use uppercase or a capital alphabets when it's referring to a matrix and lower case to refer to vectors and scalars. So because it's a matrix, this is capital W. So that's it. You now know how to implement forward prop yourself from scratch and you get to see all this code and run it and practice it yourself in the practice lab coming after this as well. I think that even when you're using powerful libraries like TensorFlow, it's helpful to know how it works under the hood. Because in case something goes wrong, in case something runs really slowly or you have a strange result, or it looks like there's a bug, your ability to understand what's actually going on
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4.12 Neural network implementation in Python | General implementation of forward propagation-ML Ng: second, and third elements of a. So for j and range units, so j goes from zero to units minus one, so it goes from zero, one, two, indexing from zero and Python as usual. This command, w equals capital W colon comma j, this is how you pull out the j column of a matrix in Python. So the first time through this loop, this will pull out the first column of w, and so will pull out w one one. The second time through this loop, when you're computing the activation of the second unit, the pull out the second column corresponding to w one two, and so on for the third time through this loop. And then you compute z using the usual formula as a thought product between that parameter w and the activation that you had received plus b j. And then you compute the activation a j equals g sigmoid function applied to z. So three times through this loop and you computed the values for all three values of this vector of activations a, and then finally you return a. So what the dense function does is it inputs the activations from the previous layer and given the parameters for the current layer, it returns the activations for the next layer. So given the dense function, here's how you can string together a few dense layers sequentially in order to implement forward prop in the neural network. Given the input features x, you can then compute the activations a one to be a one equals dens of x w one b one, where here w one b one are the parameters, sometimes also called the weights of the first hidden layer. Then you can compute a two as dens of now a one, which you just computed above and w two b two, which are the parameters or weights of this second hidden layer and then compute a three and a four. And if this is a neural network with four layers, then the final output f of x is just equal to a four. And so you return f of x. Notice that here I'm using a capital W because one of the notational conventions from linear algebra is to use uppercase or a capital alphabets when it's referring to a matrix and lower case to refer to vectors and scalars. So because it's a matrix, this is capital W. So that's it. You now know how to implement forward prop yourself from scratch and you get to see all this code and run it and practice it yourself in the practice lab coming after this as well. I think that even when you're using powerful libraries like TensorFlow, it's helpful to know how it works under the hood. Because in case something goes wrong, in case something runs really slowly or you have a strange result, or it looks like there's a bug, your ability to understand what's actually going on
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4.12 Neural network implementation in Python | General implementation of forward propagation-ML Ng
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https://www.youtube.com/watch?v=snDJExVtMMQ
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will make you much more effective when debugging your code. When I run machine learning algorithms a lot of the time, frankly, it doesn't work. So if you're not the first time. And so I find that my ability to debug my code, be it TensorFlow code or something else, is really important to being an effective machine learning engineer. So even when you're using TensorFlow or some other framework, I hope that you find this deeper understanding useful for your own applications and for debugging your own machine learning algorithms as well. So that's it. That's the last required video of this week with code in it. In the next video, I'd like to dive into what I think is a fun and fascinating topic, which is what is the relationship between neural networks and AI or AGI, artificial general intelligence. This is a controversial topic, but because it's been so widely discussed, I want to share with you some thoughts on this. So when you are asked, are neural networks at all on the path to human level intelligence, you have a framework for thinking about that question. Let's go take a look at that fun topic, I think, in the next video.
| 205
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4.12 Neural network implementation in Python | General implementation of forward propagation-ML Ng: will make you much more effective when debugging your code. When I run machine learning algorithms a lot of the time, frankly, it doesn't work. So if you're not the first time. And so I find that my ability to debug my code, be it TensorFlow code or something else, is really important to being an effective machine learning engineer. So even when you're using TensorFlow or some other framework, I hope that you find this deeper understanding useful for your own applications and for debugging your own machine learning algorithms as well. So that's it. That's the last required video of this week with code in it. In the next video, I'd like to dive into what I think is a fun and fascinating topic, which is what is the relationship between neural networks and AI or AGI, artificial general intelligence. This is a controversial topic, but because it's been so widely discussed, I want to share with you some thoughts on this. So when you are asked, are neural networks at all on the path to human level intelligence, you have a framework for thinking about that question. Let's go take a look at that fun topic, I think, in the next video.
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4.13 Speculations on artificial general intellignece (AGI) | Is there a path to AGI ? - ML Andrew Ng
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https://www.youtube.com/watch?v=nxf7N2ZMdlI
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Ever since I was a teenager, starting to play around with neural networks, I always felt that the dream of maybe someday building an AI system that's as intelligent as myself or as intelligent as a typical human, that that was one of the most inspiring dreams of AI. I still hold that dream alive today, but I think that the path to get there is not clear and could be very difficult. And I don't know whether it'll take us mere decades and whether we'll see breakthroughs in a lifetime, or if it may take centuries or even longer to get there. But let's take a look at what this AGI, artificial general intelligence dream is like and speculate a bit on what might be possible paths, unclear paths, difficult paths to get there someday. I think there's been a lot of unnecessary hype about AGI or artificial general intelligence. And maybe one reason for that is AI actually includes two very different things. One is ANI, which stands for artificial narrow intelligence. This is an AI system that does one thing, a narrow task, sometimes really, really well and can be incredibly valuable, such as a smart speaker or self-driving car or web search, or AI applied to specific applications such as farming or factories. Over the last several years, ANI has made tremendous progress and is creating, as you know, tremendous value in the world today. Because ANI is a subset of AI, the rapid progress in ANI makes it logically true that AI has also made tremendous progress in the last decade. There's a different idea in AI, which is AGI, artificial general intelligence, this hope of building AI systems that could do anything a typical human can do. And despite all the progress in ANI and therefore tremendous progress in AI, I'm not sure how much progress, if any, we're really making to what AGI. And I think all the progress in ANI has made people conclude correctly that there's tremendous progress in AI, but that has caused some people to conclude, I think, incorrectly, that a lot of progress in AI necessarily means that there's a lot of progress toward AGI. So if you ever are asked about AI and AGI, sometimes you might find drawing this picture useful for explaining some of the things going on in AI as well, and some of the sources of unnecessary hype about AGI. And with the rise of modern deep learning, we started to simulate neurons. And with faster and faster computers and even GPUs, we could simulate even more neurons. So I think there was this vague hope many years ago that, boy, if only we could simulate a lot of neurons, then we could simulate the human brain or something like a human brain and with really intelligent systems, right? Sadly, it's turned out not to be quite as simple as that. And I think two reasons for this is, first, if you look
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4.13 Speculations on artificial general intellignece (AGI) | Is there a path to AGI ? - ML Andrew Ng: Ever since I was a teenager, starting to play around with neural networks, I always felt that the dream of maybe someday building an AI system that's as intelligent as myself or as intelligent as a typical human, that that was one of the most inspiring dreams of AI. I still hold that dream alive today, but I think that the path to get there is not clear and could be very difficult. And I don't know whether it'll take us mere decades and whether we'll see breakthroughs in a lifetime, or if it may take centuries or even longer to get there. But let's take a look at what this AGI, artificial general intelligence dream is like and speculate a bit on what might be possible paths, unclear paths, difficult paths to get there someday. I think there's been a lot of unnecessary hype about AGI or artificial general intelligence. And maybe one reason for that is AI actually includes two very different things. One is ANI, which stands for artificial narrow intelligence. This is an AI system that does one thing, a narrow task, sometimes really, really well and can be incredibly valuable, such as a smart speaker or self-driving car or web search, or AI applied to specific applications such as farming or factories. Over the last several years, ANI has made tremendous progress and is creating, as you know, tremendous value in the world today. Because ANI is a subset of AI, the rapid progress in ANI makes it logically true that AI has also made tremendous progress in the last decade. There's a different idea in AI, which is AGI, artificial general intelligence, this hope of building AI systems that could do anything a typical human can do. And despite all the progress in ANI and therefore tremendous progress in AI, I'm not sure how much progress, if any, we're really making to what AGI. And I think all the progress in ANI has made people conclude correctly that there's tremendous progress in AI, but that has caused some people to conclude, I think, incorrectly, that a lot of progress in AI necessarily means that there's a lot of progress toward AGI. So if you ever are asked about AI and AGI, sometimes you might find drawing this picture useful for explaining some of the things going on in AI as well, and some of the sources of unnecessary hype about AGI. And with the rise of modern deep learning, we started to simulate neurons. And with faster and faster computers and even GPUs, we could simulate even more neurons. So I think there was this vague hope many years ago that, boy, if only we could simulate a lot of neurons, then we could simulate the human brain or something like a human brain and with really intelligent systems, right? Sadly, it's turned out not to be quite as simple as that. And I think two reasons for this is, first, if you look
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4.13 Speculations on artificial general intellignece (AGI) | Is there a path to AGI ? - ML Andrew Ng
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https://www.youtube.com/watch?v=nxf7N2ZMdlI
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at the artificial neural networks we're building, they are so simple that a logistic regression unit is really nothing like what any biological neuron is doing. It's so much simpler than what any neuron in your brain or mine is doing. And second, even to this day, I think we have almost no idea how the brain works. There's still fundamental questions about how exactly does a neuron map from inputs to outputs that we just don't know today. So trying to simulate that in a computer, much less a single logistic function, is just so far from an accurate model of what the human brain actually does. Given our very limited understanding, both now and probably for the near future, of how the human brain works, I think just trying to simulate the human brain as a path to AGI will be an incredibly difficult path. Having said that, is there any hope of within our lifetimes seeing breakthroughs in AGI? Let me share with you some evidence that helps me keep that hope alive, at least for myself. There have been some fascinating experiments done on animals that shows or strongly suggests that the same piece of biological brain tissue can do a surprisingly wide range of tasks. And this has led to the one learning algorithm hypothesis that maybe a lot of intelligence could be due to one or a small handful of learning algorithms. And if only we could figure out what that one or small handful of algorithms are, we may be able to implement that in a computer someday. Let me share with you some details of those experiments. This is a result due to Ro et al from many decades ago. The part of your brain shown here is your auditory cortex, and your brain is wired to feed signals from your ears in the form of electrical impulses depending on what sound your ear is detecting to that auditory cortex. It turns out that if you were to rewire an animal brain to cut the wire between the ear and the auditory cortex, and instead feed in images to the auditory cortex, then the auditory cortex learns to see. Auditory refers to sound, and so this piece of the brain that in most people learns to hear when it is fed different data, it instead learns to see. Here's another example. This part of your brain is your somatosensory cortex. Somatosensory refers to touch processing. If you were to similarly rewire the brain to cut the connection from the touch sensors to that part of the brain, and instead rewire the brain to feed in images, then the somatosensory cortex learns to see. So there's been a sequence of experiments like this showing that many different parts of the brain, just depending on what data it is given, can learn to see or learn to feel or learn to hear as if there was one, maybe one algorithm that just depending on
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4.13 Speculations on artificial general intellignece (AGI) | Is there a path to AGI ? - ML Andrew Ng: at the artificial neural networks we're building, they are so simple that a logistic regression unit is really nothing like what any biological neuron is doing. It's so much simpler than what any neuron in your brain or mine is doing. And second, even to this day, I think we have almost no idea how the brain works. There's still fundamental questions about how exactly does a neuron map from inputs to outputs that we just don't know today. So trying to simulate that in a computer, much less a single logistic function, is just so far from an accurate model of what the human brain actually does. Given our very limited understanding, both now and probably for the near future, of how the human brain works, I think just trying to simulate the human brain as a path to AGI will be an incredibly difficult path. Having said that, is there any hope of within our lifetimes seeing breakthroughs in AGI? Let me share with you some evidence that helps me keep that hope alive, at least for myself. There have been some fascinating experiments done on animals that shows or strongly suggests that the same piece of biological brain tissue can do a surprisingly wide range of tasks. And this has led to the one learning algorithm hypothesis that maybe a lot of intelligence could be due to one or a small handful of learning algorithms. And if only we could figure out what that one or small handful of algorithms are, we may be able to implement that in a computer someday. Let me share with you some details of those experiments. This is a result due to Ro et al from many decades ago. The part of your brain shown here is your auditory cortex, and your brain is wired to feed signals from your ears in the form of electrical impulses depending on what sound your ear is detecting to that auditory cortex. It turns out that if you were to rewire an animal brain to cut the wire between the ear and the auditory cortex, and instead feed in images to the auditory cortex, then the auditory cortex learns to see. Auditory refers to sound, and so this piece of the brain that in most people learns to hear when it is fed different data, it instead learns to see. Here's another example. This part of your brain is your somatosensory cortex. Somatosensory refers to touch processing. If you were to similarly rewire the brain to cut the connection from the touch sensors to that part of the brain, and instead rewire the brain to feed in images, then the somatosensory cortex learns to see. So there's been a sequence of experiments like this showing that many different parts of the brain, just depending on what data it is given, can learn to see or learn to feel or learn to hear as if there was one, maybe one algorithm that just depending on
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4.13 Speculations on artificial general intellignece (AGI) | Is there a path to AGI ? - ML Andrew Ng
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https://www.youtube.com/watch?v=nxf7N2ZMdlI
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what data it is given, learns to process that input accordingly. There have been systems built which take a camera, maybe mounted to someone's forehead, and maps it to a pattern of voltages in a grid on someone's tongue, and by mapping a grayscale image to a pattern of voltages on your tongue, this can help people that are not sighted, blind individuals, learn to see with your tongue. Or there have been fascinating experiments with human echolocation or human sonar. So animals like dolphins and bats use sonar to see, and researchers have found that if you train humans to make clicking sounds and listen to how that bounces off surroundings, humans can sometimes learn some degree of human echolocation. Or this is a haptic belt, and my research lab at Stanford once built something like this before as well, but if you mount a ring of buzzes around your waist and program it using a magnetic compass so that say the buzzes to the northmost direction are always vibrating slightly, then you somehow gain a direction sense which some animals have but humans don't, and it just feels like you're walking around and you just know where north is. It doesn't feel like, oh that part of my waist is buzzing. It feels like, oh I know where that north is. Or surgeries implant a third eye onto a frog and the brain just learns to deal with this input. There have been a variety of experiments like these showing that the human brain is amazingly adaptable. Neuroscientists say it's amazingly plastic. That just means adaptable to deal with a bewildering range of sensor inputs. And so the question is, if the same piece of brain tissue can learn to see or touch or feel or even other things, what is the algorithm it uses and can we replicate this algorithm and implement it in a computer? I do feel bad for the frog and other animals on which these experiments were done, although I think the conclusions are also quite fascinating. So even to this day, I think working on AGI is one of the most fascinating signs in engineering problems of our time and maybe you will choose someday to do research on it. However, I think it's important to avoid overhyping. I don't know if the brain is really one or a small handful of algorithms and even if it were, I have no idea and I don't think anyone knows what the algorithm is. But I still hold this hope alive and maybe it is and maybe we could, through a lot of hard work, someday discover an approximation to it. I still find this one of the most fascinating topics and I still often idly think about it in my spare time and maybe someday you will be the one to make a contribution to this problem. So in the short term, I think even without pursuing AGI, machine learning and neural networks
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4.13 Speculations on artificial general intellignece (AGI) | Is there a path to AGI ? - ML Andrew Ng: what data it is given, learns to process that input accordingly. There have been systems built which take a camera, maybe mounted to someone's forehead, and maps it to a pattern of voltages in a grid on someone's tongue, and by mapping a grayscale image to a pattern of voltages on your tongue, this can help people that are not sighted, blind individuals, learn to see with your tongue. Or there have been fascinating experiments with human echolocation or human sonar. So animals like dolphins and bats use sonar to see, and researchers have found that if you train humans to make clicking sounds and listen to how that bounces off surroundings, humans can sometimes learn some degree of human echolocation. Or this is a haptic belt, and my research lab at Stanford once built something like this before as well, but if you mount a ring of buzzes around your waist and program it using a magnetic compass so that say the buzzes to the northmost direction are always vibrating slightly, then you somehow gain a direction sense which some animals have but humans don't, and it just feels like you're walking around and you just know where north is. It doesn't feel like, oh that part of my waist is buzzing. It feels like, oh I know where that north is. Or surgeries implant a third eye onto a frog and the brain just learns to deal with this input. There have been a variety of experiments like these showing that the human brain is amazingly adaptable. Neuroscientists say it's amazingly plastic. That just means adaptable to deal with a bewildering range of sensor inputs. And so the question is, if the same piece of brain tissue can learn to see or touch or feel or even other things, what is the algorithm it uses and can we replicate this algorithm and implement it in a computer? I do feel bad for the frog and other animals on which these experiments were done, although I think the conclusions are also quite fascinating. So even to this day, I think working on AGI is one of the most fascinating signs in engineering problems of our time and maybe you will choose someday to do research on it. However, I think it's important to avoid overhyping. I don't know if the brain is really one or a small handful of algorithms and even if it were, I have no idea and I don't think anyone knows what the algorithm is. But I still hold this hope alive and maybe it is and maybe we could, through a lot of hard work, someday discover an approximation to it. I still find this one of the most fascinating topics and I still often idly think about it in my spare time and maybe someday you will be the one to make a contribution to this problem. So in the short term, I think even without pursuing AGI, machine learning and neural networks
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4.13 Speculations on artificial general intellignece (AGI) | Is there a path to AGI ? - ML Andrew Ng
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https://www.youtube.com/watch?v=nxf7N2ZMdlI
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are a very powerful tool and even without trying to go all the way to build human level intelligence, I think you find neural networks to be an incredibly powerful and useful set of tools for applications that you might build. And so that's it for the required videos of this week. Congratulations on getting to this point of the lessons. After this, we'll also have a few optional videos to dive a little bit more deeply into efficient implementations of neural networks. And in particular, in the optional videos to come, I'd like to share with you some details of how to implement vectorized implementations of neural networks. I hope you also take a look at those videos.
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4.13 Speculations on artificial general intellignece (AGI) | Is there a path to AGI ? - ML Andrew Ng: are a very powerful tool and even without trying to go all the way to build human level intelligence, I think you find neural networks to be an incredibly powerful and useful set of tools for applications that you might build. And so that's it for the required videos of this week. Congratulations on getting to this point of the lessons. After this, we'll also have a few optional videos to dive a little bit more deeply into efficient implementations of neural networks. And in particular, in the optional videos to come, I'd like to share with you some details of how to implement vectorized implementations of neural networks. I hope you also take a look at those videos.
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4.14 Vectorization (optional) | How neural networks are implemented efficiently-- [ML- Andrew Ng]
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https://www.youtube.com/watch?v=3iDMb-EUQPA
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One of the reasons that deep learning researchers have been able to scale up neural networks and build really large neural networks over the last decade is because neural networks can be vectorized. They can be implemented very efficiently using matrix multiplications. And it turns out that parallel computing hardware, including GPUs, but also some CPU functions are very good at doing very large matrix multiplications. In this video, we'll take a look at how these vectorized implementations of neural networks work. Without these ideas, I don't think deep learning would be anywhere near success in scale today. Here on the left is the code that you had seen previously of how you would implement for a prop or for a propagation in a single layer. This here is the input w, the weights of the first, second, and third neurons, say, parameters b. And then this is the same code as what you saw before. And this will output three numbers, say, like that. And if you actually implement this computation, you get 101. It turns out you can develop a vectorized implementation of this function as follows. Set x to be equal to this. Notice the double square brackets. So this is now a 2D array, like in TensorFlow. W is the same as before. And b, I'm now using capital B, is also a 1 by 3 2D array. And then it turns out that all of these steps, this for loop inside, can be replaced with just a couple lines of code. Z equals np.matmul. Matmul is how NumPy carries out matrix multiplication, where now x and w are both matrices. And so you just multiply them together. And it turns out that this for loop, all of these lines of code, can be replaced with just a couple lines of code, which gives a vectorized implementation of this function. So you compute z, which is now a matrix again, as NumPy.matmul between a in and w, where here a in and w are both matrices. And matmul is how NumPy carries out a matrix multiplication. It multiplies two matrices together and then adds the matrix b to it. And then a here, or a out, is equal to the activation function g, that is a sigmoid function, multiplied element-wise to this matrix z. And then you finally return a out. So this is what the code looks like. Notice that in the vectorized implementation, all of these quantities, x, which is fed into the value of a in, as well as w, b, as well as z, and a out, all of these are now 2D arrays. All of these are matrices. And this turns out to be a very efficient implementation of one step of forward propagation through a dense layer in the neural network. So this is code for a vectorized implementation of forward prop in a neural network. But what is this code doing and how does it actually work? And what
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4.14 Vectorization (optional) | How neural networks are implemented efficiently-- [ML- Andrew Ng]: One of the reasons that deep learning researchers have been able to scale up neural networks and build really large neural networks over the last decade is because neural networks can be vectorized. They can be implemented very efficiently using matrix multiplications. And it turns out that parallel computing hardware, including GPUs, but also some CPU functions are very good at doing very large matrix multiplications. In this video, we'll take a look at how these vectorized implementations of neural networks work. Without these ideas, I don't think deep learning would be anywhere near success in scale today. Here on the left is the code that you had seen previously of how you would implement for a prop or for a propagation in a single layer. This here is the input w, the weights of the first, second, and third neurons, say, parameters b. And then this is the same code as what you saw before. And this will output three numbers, say, like that. And if you actually implement this computation, you get 101. It turns out you can develop a vectorized implementation of this function as follows. Set x to be equal to this. Notice the double square brackets. So this is now a 2D array, like in TensorFlow. W is the same as before. And b, I'm now using capital B, is also a 1 by 3 2D array. And then it turns out that all of these steps, this for loop inside, can be replaced with just a couple lines of code. Z equals np.matmul. Matmul is how NumPy carries out matrix multiplication, where now x and w are both matrices. And so you just multiply them together. And it turns out that this for loop, all of these lines of code, can be replaced with just a couple lines of code, which gives a vectorized implementation of this function. So you compute z, which is now a matrix again, as NumPy.matmul between a in and w, where here a in and w are both matrices. And matmul is how NumPy carries out a matrix multiplication. It multiplies two matrices together and then adds the matrix b to it. And then a here, or a out, is equal to the activation function g, that is a sigmoid function, multiplied element-wise to this matrix z. And then you finally return a out. So this is what the code looks like. Notice that in the vectorized implementation, all of these quantities, x, which is fed into the value of a in, as well as w, b, as well as z, and a out, all of these are now 2D arrays. All of these are matrices. And this turns out to be a very efficient implementation of one step of forward propagation through a dense layer in the neural network. So this is code for a vectorized implementation of forward prop in a neural network. But what is this code doing and how does it actually work? And what
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4.14 Vectorization (optional) | How neural networks are implemented efficiently-- [ML- Andrew Ng]
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https://www.youtube.com/watch?v=3iDMb-EUQPA
|
is this matmul actually doing? In the next two videos, both also optional, we'll go over matrix multiplication and how that works. If you're familiar with linear algebra, if you're familiar with vectors, matrices, transposes, and matrix matrix multiplications, you can safely just quickly skim over these two videos and jump to the last video of this week. And then in the last video of this week, also optional, we'll dive into more detail to explain how matmul gives you this vectorized implementation. And so with that, let's go on to the next video where we'll take a look at what matrix multiplication is.
| 104
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4.14 Vectorization (optional) | How neural networks are implemented efficiently-- [ML- Andrew Ng]: is this matmul actually doing? In the next two videos, both also optional, we'll go over matrix multiplication and how that works. If you're familiar with linear algebra, if you're familiar with vectors, matrices, transposes, and matrix matrix multiplications, you can safely just quickly skim over these two videos and jump to the last video of this week. And then in the last video of this week, also optional, we'll dive into more detail to explain how matmul gives you this vectorized implementation. And so with that, let's go on to the next video where we'll take a look at what matrix multiplication is.
|
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4.15 Vectorization (optional) | Matrix multiplication-- [Machine Learning - Andrew Ng]
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https://www.youtube.com/watch?v=CYnwhHKnuwY
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So you know that a matrix is just a block or 2D array of numbers. What does it mean to multiply two matrices? Let's take a look. In order to build up to multiplying matrices, let's start by looking at how we take dot products between vectors. Let's use the example of taking the dot product between this vector 1, 2, and this vector 3, 4. If z is the dot product between these two vectors, then you compute z by multiplying the first element by this first element here, so it's 1 times 3, plus the second element times the second element, plus 2 times 4, and so that's just 3 plus 8, which is equal to 11. In the more general case, if z is the dot product between a vector a and a vector w, then you compute z by multiplying the first element together and then the second element together and the third and so on, and then adding up all of these products. So that's the vector vector dot product. It turns out there's another equivalent way of writing a dot product, which is given a vector a, that is 1, 2 written as a column, you can turn this into a row, that is, you can turn it from what's called a column vector to a row vector by taking the transpose of a. So the transpose of a vector a means you take this vector and lay its elements on the side like this. And it turns out that if you multiply a transpose, this is a row vector, or you can think of this as a 1 by 2 matrix, with w, which you can now think of as a 2 by 1 matrix, then z equals a transpose times w, and this is the same as taking the dot product between a and w. So to recap, z equals the dot product between a and w is the same as z equals a transpose, that is a laid on the side, multiplied by w. And this will be useful for understanding matrix multiplication, that these are just two ways of writing the exact same computation to arrive at z. Now let's look at vector matrix multiplication, which is when you take a vector and you multiply a vector by a matrix. Here again is the vector a, 1, 2, and a transpose is a laid on the side, so rather than just kind of think of this as a 2 by 1 matrix, it becomes a 1 by 2 matrix. And let me now create a 2 by 2 matrix w with these four elements, v4, v5, v6. If you want to compute capital Z as a transpose times w, so let's see how you would go about doing so. It turns out that z is going to be a 2 by 1 matrix, and to compute the first value of z, we're going to take a transpose, 1, 2 here,
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4.15 Vectorization (optional) | Matrix multiplication-- [Machine Learning - Andrew Ng]: So you know that a matrix is just a block or 2D array of numbers. What does it mean to multiply two matrices? Let's take a look. In order to build up to multiplying matrices, let's start by looking at how we take dot products between vectors. Let's use the example of taking the dot product between this vector 1, 2, and this vector 3, 4. If z is the dot product between these two vectors, then you compute z by multiplying the first element by this first element here, so it's 1 times 3, plus the second element times the second element, plus 2 times 4, and so that's just 3 plus 8, which is equal to 11. In the more general case, if z is the dot product between a vector a and a vector w, then you compute z by multiplying the first element together and then the second element together and the third and so on, and then adding up all of these products. So that's the vector vector dot product. It turns out there's another equivalent way of writing a dot product, which is given a vector a, that is 1, 2 written as a column, you can turn this into a row, that is, you can turn it from what's called a column vector to a row vector by taking the transpose of a. So the transpose of a vector a means you take this vector and lay its elements on the side like this. And it turns out that if you multiply a transpose, this is a row vector, or you can think of this as a 1 by 2 matrix, with w, which you can now think of as a 2 by 1 matrix, then z equals a transpose times w, and this is the same as taking the dot product between a and w. So to recap, z equals the dot product between a and w is the same as z equals a transpose, that is a laid on the side, multiplied by w. And this will be useful for understanding matrix multiplication, that these are just two ways of writing the exact same computation to arrive at z. Now let's look at vector matrix multiplication, which is when you take a vector and you multiply a vector by a matrix. Here again is the vector a, 1, 2, and a transpose is a laid on the side, so rather than just kind of think of this as a 2 by 1 matrix, it becomes a 1 by 2 matrix. And let me now create a 2 by 2 matrix w with these four elements, v4, v5, v6. If you want to compute capital Z as a transpose times w, so let's see how you would go about doing so. It turns out that z is going to be a 2 by 1 matrix, and to compute the first value of z, we're going to take a transpose, 1, 2 here,
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4.15 Vectorization (optional) | Matrix multiplication-- [Machine Learning - Andrew Ng]
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https://www.youtube.com/watch?v=CYnwhHKnuwY
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and multiply that by the first column of w, so that's 3, 4. And so to compute the first element of z, you end up with 1 times 3 plus 2 times 4, which we saw earlier is equal to 11, and so the first element of z is 11. Let's figure out what's the second element of z. Turns out you just repeat this process, but now multiply a transpose by the second column of w. And so to do that computation, you have 1 times 5 plus 2 times 6, which is equal to 5 plus 12, which is 17, so that's equal to 17. So z is equal to this 1 by 2 matrix, 11 and 17. Now just one last thing, and then that'll take us to the end of this video, which is how to take vector matrix multiplication and generalize it to matrix matrix multiplication. I have a matrix A with these four elements, the first column is 1, 2, and the second column is negative 1, negative 2. And I want to know how to compute a transpose times w. Unlike the previous slide, A now is a matrix rather than just a vector, but the matrix is just a set of different vectors stacked together in columns. So first, let's figure out what is A transpose. In order to compute A transpose, we're going to take the columns of A, and similar to what happened when you transpose a vector, we're going to take the columns and lay them on the side one column at a time. So the first column, 1, 2, becomes the first row, 1, 2, because it's just laid on the side, and this second column, negative 1, negative 2, becomes laid on the side, negative 1, negative 2, like this. So the way you transpose a matrix is you take the columns and you just lay the columns on the side one column at a time. So you end up with this being A transpose. Next, we have this matrix, w, which we're going to write as 3, 4, 5, 6. So there's a column 3, 4, and a column 5, 6. One way I encourage you to think of matrices, at least that's useful for neural network implementations is if you see a matrix, think of the columns of the matrix, and if you see the transpose of a matrix, think of the rows of that matrix as being grouped together, as illustrated here with A and A transpose, as well as w. And now, let me show you how to multiply A transpose and w. In order to carry out this computation, let me call the columns of A, A1 and A2, and that means that A1 transpose is the first row of A transpose, and A2 transpose is the second row of A transpose. And then, same as before, let me call the columns of w to be w1 and w2. So it turns
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4.15 Vectorization (optional) | Matrix multiplication-- [Machine Learning - Andrew Ng]: and multiply that by the first column of w, so that's 3, 4. And so to compute the first element of z, you end up with 1 times 3 plus 2 times 4, which we saw earlier is equal to 11, and so the first element of z is 11. Let's figure out what's the second element of z. Turns out you just repeat this process, but now multiply a transpose by the second column of w. And so to do that computation, you have 1 times 5 plus 2 times 6, which is equal to 5 plus 12, which is 17, so that's equal to 17. So z is equal to this 1 by 2 matrix, 11 and 17. Now just one last thing, and then that'll take us to the end of this video, which is how to take vector matrix multiplication and generalize it to matrix matrix multiplication. I have a matrix A with these four elements, the first column is 1, 2, and the second column is negative 1, negative 2. And I want to know how to compute a transpose times w. Unlike the previous slide, A now is a matrix rather than just a vector, but the matrix is just a set of different vectors stacked together in columns. So first, let's figure out what is A transpose. In order to compute A transpose, we're going to take the columns of A, and similar to what happened when you transpose a vector, we're going to take the columns and lay them on the side one column at a time. So the first column, 1, 2, becomes the first row, 1, 2, because it's just laid on the side, and this second column, negative 1, negative 2, becomes laid on the side, negative 1, negative 2, like this. So the way you transpose a matrix is you take the columns and you just lay the columns on the side one column at a time. So you end up with this being A transpose. Next, we have this matrix, w, which we're going to write as 3, 4, 5, 6. So there's a column 3, 4, and a column 5, 6. One way I encourage you to think of matrices, at least that's useful for neural network implementations is if you see a matrix, think of the columns of the matrix, and if you see the transpose of a matrix, think of the rows of that matrix as being grouped together, as illustrated here with A and A transpose, as well as w. And now, let me show you how to multiply A transpose and w. In order to carry out this computation, let me call the columns of A, A1 and A2, and that means that A1 transpose is the first row of A transpose, and A2 transpose is the second row of A transpose. And then, same as before, let me call the columns of w to be w1 and w2. So it turns
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4.15 Vectorization (optional) | Matrix multiplication-- [Machine Learning - Andrew Ng]
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https://www.youtube.com/watch?v=CYnwhHKnuwY
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out that to compute A transpose w, the first thing we need to do is let's just ignore the second row of A, and let's just pay attention to the first row of A, and let's take this row 1, 2, that is A1 transpose, and multiply that with w. So you already know how to do that from the previous slide. The first element is 1, 2, inner product or dot product with 3, 4, so that ends up with 3 times 1 plus 2 times 4, which is 11. And then the second element is 1, 2, A transpose, inner product with 5, 6, so that's 5 times 1 plus 6 times 2, which is 5 plus 12, which is 17. So that gives you the first row of z equals A transpose w. So all we've done is take A1 transpose and multiply that by w. That's exactly what we did on the previous slide. Next, let's forget A1 for now, and let's just look at A2 and take A2 transpose and multiply that by w. So now we have A2 transpose times w, and to compute that, first we take negative 1 and negative 2 and dot product that with 3, 4, so that's negative 1 times 3 plus negative 2 times 4, and that turns out to be negative 11. And then we have to compute A2 transpose times the second column, and that's negative 1 times 5 plus negative 2 times 6, and that turns out to be negative 17. So you end up with A transpose times w is equal to this 2 by 2 matrix over here. Let's talk about the general form of matrix-matrix multiplication. So this was an example of how you multiply a vector with a matrix or a matrix with a matrix. There's a lot of dot products between vectors, but ordered in a certain way to construct the elements of the output z one element at a time. I know this was a lot, but in the next video, let's look at the general form of how a matrix-matrix multiplication is defined, and I hope that that will make all this clear as well. Let's go on to the next video.
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4.15 Vectorization (optional) | Matrix multiplication-- [Machine Learning - Andrew Ng]: out that to compute A transpose w, the first thing we need to do is let's just ignore the second row of A, and let's just pay attention to the first row of A, and let's take this row 1, 2, that is A1 transpose, and multiply that with w. So you already know how to do that from the previous slide. The first element is 1, 2, inner product or dot product with 3, 4, so that ends up with 3 times 1 plus 2 times 4, which is 11. And then the second element is 1, 2, A transpose, inner product with 5, 6, so that's 5 times 1 plus 6 times 2, which is 5 plus 12, which is 17. So that gives you the first row of z equals A transpose w. So all we've done is take A1 transpose and multiply that by w. That's exactly what we did on the previous slide. Next, let's forget A1 for now, and let's just look at A2 and take A2 transpose and multiply that by w. So now we have A2 transpose times w, and to compute that, first we take negative 1 and negative 2 and dot product that with 3, 4, so that's negative 1 times 3 plus negative 2 times 4, and that turns out to be negative 11. And then we have to compute A2 transpose times the second column, and that's negative 1 times 5 plus negative 2 times 6, and that turns out to be negative 17. So you end up with A transpose times w is equal to this 2 by 2 matrix over here. Let's talk about the general form of matrix-matrix multiplication. So this was an example of how you multiply a vector with a matrix or a matrix with a matrix. There's a lot of dot products between vectors, but ordered in a certain way to construct the elements of the output z one element at a time. I know this was a lot, but in the next video, let's look at the general form of how a matrix-matrix multiplication is defined, and I hope that that will make all this clear as well. Let's go on to the next video.
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4.16 Vectorization (optional) | Matrix multiplication rules-- [Machine Learning - Andrew Ng]
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https://www.youtube.com/watch?v=7GFKFng9gyM
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So, let's take a look at the general form of how you multiply two matrices together. And then in the last video after this one, we'll take this and apply it to the vectorized implementation of a neural network. Let's dive in. Here's a matrix A, which is a two by three matrix, because it has two rows and three columns. As before, I'd encourage you to think of the columns of this matrix as three vectors, vectors A1, A2, and A3. And what we're going to do is take A transpose and multiply that with a matrix W. The first, what is A transpose? Well, A transpose is obtained by taking the first column of A and laying it on the side like this, and then taking the second column of A and laying it on the side like this, and then the third column of A and laying it on the side like that. And so these rows are now A1 transpose, A2 transpose, and A3 transpose. Next, here's a matrix W. I encourage you to think of W as vectors W1, W2, W3, and W4 stacked together as so. Let's look at how you then compute A transpose times W. Now, notice that I've also used slightly different shades of orange to denote the different columns of A, where the same shade corresponds to numbers that we think of as grouped together into a vector. And that same shade is used to indicate different rows of A transpose, because the different rows of A transpose are A1 transpose, A2 transpose, and A3 transpose. And in a similar way, I've used different shades to denote the different columns of W, because the numbers of the same shade of blue are the ones that are grouped together to form the vectors W1, or W2, or W3, or W4. Now, let's look at how you can compute A transpose times W. I'm going to draw vertical bars with the different shades of blue, and horizontal bars with the different shades of orange to indicate which elements of Z, that is A transpose W, are influenced or affected by the different rows of A transpose, and which are influenced or affected by the different columns of W. So, for example, let's look at the first column of W. So, that's W1, as indicated by the lightest shade of blue here. So, W1 will influence or will correspond to this first column of Z, shown here, by this lightest shade of blue. And the values of this second column of W, that is W2, as indicated by this second lightest shade of blue, will affect the values computed in the second column of Z, and so on. For the third and fourth columns. Correspondingly, let's look at A transpose. A1 transpose is the first row of A transpose, as indicated by the lightest shade of orange, and A1 transpose will affect or influence or correspond to the values in the first row
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4.16 Vectorization (optional) | Matrix multiplication rules-- [Machine Learning - Andrew Ng]: So, let's take a look at the general form of how you multiply two matrices together. And then in the last video after this one, we'll take this and apply it to the vectorized implementation of a neural network. Let's dive in. Here's a matrix A, which is a two by three matrix, because it has two rows and three columns. As before, I'd encourage you to think of the columns of this matrix as three vectors, vectors A1, A2, and A3. And what we're going to do is take A transpose and multiply that with a matrix W. The first, what is A transpose? Well, A transpose is obtained by taking the first column of A and laying it on the side like this, and then taking the second column of A and laying it on the side like this, and then the third column of A and laying it on the side like that. And so these rows are now A1 transpose, A2 transpose, and A3 transpose. Next, here's a matrix W. I encourage you to think of W as vectors W1, W2, W3, and W4 stacked together as so. Let's look at how you then compute A transpose times W. Now, notice that I've also used slightly different shades of orange to denote the different columns of A, where the same shade corresponds to numbers that we think of as grouped together into a vector. And that same shade is used to indicate different rows of A transpose, because the different rows of A transpose are A1 transpose, A2 transpose, and A3 transpose. And in a similar way, I've used different shades to denote the different columns of W, because the numbers of the same shade of blue are the ones that are grouped together to form the vectors W1, or W2, or W3, or W4. Now, let's look at how you can compute A transpose times W. I'm going to draw vertical bars with the different shades of blue, and horizontal bars with the different shades of orange to indicate which elements of Z, that is A transpose W, are influenced or affected by the different rows of A transpose, and which are influenced or affected by the different columns of W. So, for example, let's look at the first column of W. So, that's W1, as indicated by the lightest shade of blue here. So, W1 will influence or will correspond to this first column of Z, shown here, by this lightest shade of blue. And the values of this second column of W, that is W2, as indicated by this second lightest shade of blue, will affect the values computed in the second column of Z, and so on. For the third and fourth columns. Correspondingly, let's look at A transpose. A1 transpose is the first row of A transpose, as indicated by the lightest shade of orange, and A1 transpose will affect or influence or correspond to the values in the first row
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4.16 Vectorization (optional) | Matrix multiplication rules-- [Machine Learning - Andrew Ng]
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https://www.youtube.com/watch?v=7GFKFng9gyM
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of Z. And A2 transpose will influence the second row of Z, and A3 transpose will influence or correspond to this third row of Z. So, let's figure out how to compute the matrix Z, which is going to be a 3 by 4 matrix. So, we have 12 numbers altogether. Let's start off and figure out how to compute the number in the first row in the first column of Z. So, this upper left most element here. Because this is the first row in the first column corresponding to the lightest shade of orange and the lightest shade of blue, the way you compute that is to grab the first row of A transpose and the first column of W and take their inner product or the dot product. And so, this number is going to be 1, 2 dot product with 3, 4, which is 1 times 3 plus 2 times 4, which is equal to 11. Let's look at a second example. How would you compute this number, this element of Z? So, this is in the third row, row 1, row 2, row 3. So, this is row 3 and the second column, column 1, column 2. So, to compute the number in row 3, column 2 of Z, you would now grab row 3 of A transpose and column 2 of W and dot product those together. Notice that this corresponds to the darkest shade of orange and the second lightest shade of blue. And to compute this, this is 0.1 times 5 plus 0.2 times 6, which is 0.5 plus 1.2, which is equal to 1.7. So, to compute the number in row 3, column 2 of Z, you grab the third row, row 3 of A transpose and column 2 of W. Let's look at one more example and let's see if you can figure this one out. This is row 2, column 3 of the matrix Z. Why don't you take a look and see if you can figure out which row and which column to grab to dot product together and therefore what is the number that will go in this element of this matrix. Maybe you got that you should be grabbing row 2 of A transpose and column 3 of W and when you dot product that together, you have A2 transpose W3 is negative 1 times 7 plus negative 2 times 8, which is negative 7 plus negative 16, which is equal to negative 23. And so that's how you compute this element of the matrix Z. And it turns out if you do this for every element of the matrix Z, then you can compute all of the numbers in this matrix, which turns out to look like that. Feel free to pause the video if you want and pick any element and double check that the formula we've been going through gives you the right value for Z. I just want to point out one
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4.16 Vectorization (optional) | Matrix multiplication rules-- [Machine Learning - Andrew Ng]: of Z. And A2 transpose will influence the second row of Z, and A3 transpose will influence or correspond to this third row of Z. So, let's figure out how to compute the matrix Z, which is going to be a 3 by 4 matrix. So, we have 12 numbers altogether. Let's start off and figure out how to compute the number in the first row in the first column of Z. So, this upper left most element here. Because this is the first row in the first column corresponding to the lightest shade of orange and the lightest shade of blue, the way you compute that is to grab the first row of A transpose and the first column of W and take their inner product or the dot product. And so, this number is going to be 1, 2 dot product with 3, 4, which is 1 times 3 plus 2 times 4, which is equal to 11. Let's look at a second example. How would you compute this number, this element of Z? So, this is in the third row, row 1, row 2, row 3. So, this is row 3 and the second column, column 1, column 2. So, to compute the number in row 3, column 2 of Z, you would now grab row 3 of A transpose and column 2 of W and dot product those together. Notice that this corresponds to the darkest shade of orange and the second lightest shade of blue. And to compute this, this is 0.1 times 5 plus 0.2 times 6, which is 0.5 plus 1.2, which is equal to 1.7. So, to compute the number in row 3, column 2 of Z, you grab the third row, row 3 of A transpose and column 2 of W. Let's look at one more example and let's see if you can figure this one out. This is row 2, column 3 of the matrix Z. Why don't you take a look and see if you can figure out which row and which column to grab to dot product together and therefore what is the number that will go in this element of this matrix. Maybe you got that you should be grabbing row 2 of A transpose and column 3 of W and when you dot product that together, you have A2 transpose W3 is negative 1 times 7 plus negative 2 times 8, which is negative 7 plus negative 16, which is equal to negative 23. And so that's how you compute this element of the matrix Z. And it turns out if you do this for every element of the matrix Z, then you can compute all of the numbers in this matrix, which turns out to look like that. Feel free to pause the video if you want and pick any element and double check that the formula we've been going through gives you the right value for Z. I just want to point out one
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4.16 Vectorization (optional) | Matrix multiplication rules-- [Machine Learning - Andrew Ng]
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https://www.youtube.com/watch?v=7GFKFng9gyM
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last interesting requirement for multiplying matrices together, which is that X transpose here is a 3 by 2 matrix because it has 3 rows and 2 columns and W here is a 2 by 4 matrix because it has 2 rows and 4 columns. One requirement in order to multiply 2 matrices together is that this number must match that number and that's because you can only take dot products between vectors that are the same length. So you can take the dot product between a vector with 2 numbers and that's because you can take the inner product between a vector of length 2 only with another vector of length 2. You can't take the inner product between a vector of length 2 with a vector of length 3, for example. And that's why matrix multiplication is valid only if the number of columns of the first matrix, that is A transpose here, is equal to the number of rows of the second matrix, that is the number of rows of W here. So that when you take dot products during this process, you're taking dot products of vectors of the same size. And then the other observation is that the output Z equals A transpose W, the dimensions of Z is 3 by 4. And so the output of this multiplication will have the same number of rows as X transpose and the same number of columns as W. And so that too is another property of matrix multiplication. So that's matrix multiplication. All these videos are optional, so thank you for sticking with me through these. And if you're interested, later in this week, there are also some purely optional quizzes to let you practice some more of these calculations yourself as well. So with that, let's take what we've learned about matrix multiplication and apply it back to the vectorized implementation of a neural network. I have to say, the first time I understood the vectorized implementation, I thought it was actually really cool. I've been implementing neural networks for a while myself without the vectorized implementation. And when I finally understood the vectorized implementation and implemented it that way for the first time, it ran blazingly much faster than anything I've ever done before. And I thought, wow, I wish I had figured this out earlier. The vectorized implementation, it is a little bit complicated, but it makes neural networks run much faster. So let's take a look at that in the next video.
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4.16 Vectorization (optional) | Matrix multiplication rules-- [Machine Learning - Andrew Ng]: last interesting requirement for multiplying matrices together, which is that X transpose here is a 3 by 2 matrix because it has 3 rows and 2 columns and W here is a 2 by 4 matrix because it has 2 rows and 4 columns. One requirement in order to multiply 2 matrices together is that this number must match that number and that's because you can only take dot products between vectors that are the same length. So you can take the dot product between a vector with 2 numbers and that's because you can take the inner product between a vector of length 2 only with another vector of length 2. You can't take the inner product between a vector of length 2 with a vector of length 3, for example. And that's why matrix multiplication is valid only if the number of columns of the first matrix, that is A transpose here, is equal to the number of rows of the second matrix, that is the number of rows of W here. So that when you take dot products during this process, you're taking dot products of vectors of the same size. And then the other observation is that the output Z equals A transpose W, the dimensions of Z is 3 by 4. And so the output of this multiplication will have the same number of rows as X transpose and the same number of columns as W. And so that too is another property of matrix multiplication. So that's matrix multiplication. All these videos are optional, so thank you for sticking with me through these. And if you're interested, later in this week, there are also some purely optional quizzes to let you practice some more of these calculations yourself as well. So with that, let's take what we've learned about matrix multiplication and apply it back to the vectorized implementation of a neural network. I have to say, the first time I understood the vectorized implementation, I thought it was actually really cool. I've been implementing neural networks for a while myself without the vectorized implementation. And when I finally understood the vectorized implementation and implemented it that way for the first time, it ran blazingly much faster than anything I've ever done before. And I thought, wow, I wish I had figured this out earlier. The vectorized implementation, it is a little bit complicated, but it makes neural networks run much faster. So let's take a look at that in the next video.
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|
4.17 Vectorization (optional) | Matrix multiplication code-- [Machine Learning - Andrew Ng]
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https://www.youtube.com/watch?v=m7xcF9jXLpc
|
So, without further ado, let's jump into the vectorized implementation of a neural network. We'll look at the code that you have seen in an earlier video, and hopefully, MatMul, that is that matrix multiplication calculation, will make more sense. Let's jump in. So you saw previously how you can take the matrix A and compute A transpose times W, resulting in this matrix here, Z. In code, if this is the matrix A, this is a numpy array with the elements corresponding to what I wrote on top, then A transpose, which I'm going to write as A T, is going to be this matrix here, with, again, the columns of A now laid out in rows instead. And by the way, instead of setting up A T this way, another way to compute A T in numpy array would be to write A T equals A dot T. That's the transpose function that takes the columns of the matrix and lays them on the side. In code, here's how you initialize the matrix W, this is another 2D numpy array. And then to compute Z equals A transpose times W, you would write Z equals NP dot MatMul A T comma W. And that will compute this matrix Z over here, giving you this result down here. And by the way, if you read others' code, sometimes you see Z equals A T and then the at symbol W. This is an alternative way of calling the MatMul function, although I find using NP dot MatMul to be clearer. And so in the code you see in this class, we just use the MatMul function like this, rather than this at symbol. So let's look at what a vectorized implementation of what prop looks like. I'm going to set A transpose to be equal to the input feature values 217. So these are just the usual input feature values, 200 degrees, roasting coffee for 17 minutes. So this is a 1 by 2 matrix, and I'm going to take the parameters W1, W2, and W3 and stack them in columns like this to form this matrix capital W. And the values b1, b2, b3, I'm going to put into a 1 by 3 matrix that is this matrix B as follows. And it turns out that if you were to compute Z equals A transpose W plus B, that will result in these three numbers. And that's computed by taking the input feature values and multiplying that by the first column and then adding B to get 165. Taking these feature values, dot producting with the second column that is a weights W2 and adding B2 to get negative 531. And these feature values dot product with the weights W3 plus B3 to get 900. Feel free to pause the video if you wish to double check these calculations, but this gives you the values of Z11, Z12, and Z13. And then finally, if the function G applies
| 500
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4.17 Vectorization (optional) | Matrix multiplication code-- [Machine Learning - Andrew Ng]: So, without further ado, let's jump into the vectorized implementation of a neural network. We'll look at the code that you have seen in an earlier video, and hopefully, MatMul, that is that matrix multiplication calculation, will make more sense. Let's jump in. So you saw previously how you can take the matrix A and compute A transpose times W, resulting in this matrix here, Z. In code, if this is the matrix A, this is a numpy array with the elements corresponding to what I wrote on top, then A transpose, which I'm going to write as A T, is going to be this matrix here, with, again, the columns of A now laid out in rows instead. And by the way, instead of setting up A T this way, another way to compute A T in numpy array would be to write A T equals A dot T. That's the transpose function that takes the columns of the matrix and lays them on the side. In code, here's how you initialize the matrix W, this is another 2D numpy array. And then to compute Z equals A transpose times W, you would write Z equals NP dot MatMul A T comma W. And that will compute this matrix Z over here, giving you this result down here. And by the way, if you read others' code, sometimes you see Z equals A T and then the at symbol W. This is an alternative way of calling the MatMul function, although I find using NP dot MatMul to be clearer. And so in the code you see in this class, we just use the MatMul function like this, rather than this at symbol. So let's look at what a vectorized implementation of what prop looks like. I'm going to set A transpose to be equal to the input feature values 217. So these are just the usual input feature values, 200 degrees, roasting coffee for 17 minutes. So this is a 1 by 2 matrix, and I'm going to take the parameters W1, W2, and W3 and stack them in columns like this to form this matrix capital W. And the values b1, b2, b3, I'm going to put into a 1 by 3 matrix that is this matrix B as follows. And it turns out that if you were to compute Z equals A transpose W plus B, that will result in these three numbers. And that's computed by taking the input feature values and multiplying that by the first column and then adding B to get 165. Taking these feature values, dot producting with the second column that is a weights W2 and adding B2 to get negative 531. And these feature values dot product with the weights W3 plus B3 to get 900. Feel free to pause the video if you wish to double check these calculations, but this gives you the values of Z11, Z12, and Z13. And then finally, if the function G applies
|
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4.17 Vectorization (optional) | Matrix multiplication code-- [Machine Learning - Andrew Ng]
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https://www.youtube.com/watch?v=m7xcF9jXLpc
|
the sigmoid function to these three numbers element-wise, that this applies the sigmoid function to 165, to negative 531, and to 900, then you end up with A equals G of this matrix Z ends up being 101. And it's 101 because sigmoid of 165 is so close to 1 that, you know, up to numerical round off is basically 1 and these are basically 0 and 1. Now let's look at how you implement this in code. A transpose is equal to this, is this 1 by 2 array of 217. The matrix W is this 2 by 3 matrix and B is this 1 by 3 matrix. And so the way you can implement for prop in the layer is dense input A transpose WB is equal to Z equals matmul A transpose times W plus B. So that just implements this line of code. And then A out, that is the output of this layer, is equal to G, the activation function applied element-wise to this matrix Z. And you return A out and that gives you this value. In case you're comparing the slide with the slides a few videos back, there was just one little difference, which was by convention, the way this is implemented in TensorFlow, rather than calling this variable Xt, we call it just A and rather than calling this variable At, we were calling it An, which is why this too is a correct implementation of the code. And there is a convention in TensorFlow that individual examples are actually laid out in rows in the matrix X rather than in the matrix X transpose, which is why the code implementation actually looks like this in TensorFlow. But this explains why with just a few lines of code, you can implement for prop in the neural network and moreover get a huge speed bonus because matmul matrix multiplication can be done very efficiently using fast hardware and get a huge bonus because modern computers are very good at implementing matrix multiplication such as matmul efficiently. That's the last video of this week. Thanks for sticking with me all the way through the end of these optional videos. For the rest of this week, I hope you also take a look at the quizzes and the practice labs and also the optional labs to exercise this material even more deeply. You now know how to do inference and for prop in a neural network, which I think is really cool. So congratulations. After you have gone through the quizzes in the labs, please also come back and in the next week, we'll look at how to actually train a neural network. So I look forward to seeing you next week.
| 458
|
4.17 Vectorization (optional) | Matrix multiplication code-- [Machine Learning - Andrew Ng]: the sigmoid function to these three numbers element-wise, that this applies the sigmoid function to 165, to negative 531, and to 900, then you end up with A equals G of this matrix Z ends up being 101. And it's 101 because sigmoid of 165 is so close to 1 that, you know, up to numerical round off is basically 1 and these are basically 0 and 1. Now let's look at how you implement this in code. A transpose is equal to this, is this 1 by 2 array of 217. The matrix W is this 2 by 3 matrix and B is this 1 by 3 matrix. And so the way you can implement for prop in the layer is dense input A transpose WB is equal to Z equals matmul A transpose times W plus B. So that just implements this line of code. And then A out, that is the output of this layer, is equal to G, the activation function applied element-wise to this matrix Z. And you return A out and that gives you this value. In case you're comparing the slide with the slides a few videos back, there was just one little difference, which was by convention, the way this is implemented in TensorFlow, rather than calling this variable Xt, we call it just A and rather than calling this variable At, we were calling it An, which is why this too is a correct implementation of the code. And there is a convention in TensorFlow that individual examples are actually laid out in rows in the matrix X rather than in the matrix X transpose, which is why the code implementation actually looks like this in TensorFlow. But this explains why with just a few lines of code, you can implement for prop in the neural network and moreover get a huge speed bonus because matmul matrix multiplication can be done very efficiently using fast hardware and get a huge bonus because modern computers are very good at implementing matrix multiplication such as matmul efficiently. That's the last video of this week. Thanks for sticking with me all the way through the end of these optional videos. For the rest of this week, I hope you also take a look at the quizzes and the practice labs and also the optional labs to exercise this material even more deeply. You now know how to do inference and for prop in a neural network, which I think is really cool. So congratulations. After you have gone through the quizzes in the labs, please also come back and in the next week, we'll look at how to actually train a neural network. So I look forward to seeing you next week.
|
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5.1 Neural Network Training | TensorFlow implementation --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=jdjGdT_jR50
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Welcome back to the second week of this course on advanced learning algorithms. Last week you learned how to carry out inference in a neural network. This week we're going to go over training of a neural network. I think being able to take your own data and train your own neural network on it is really fun. This week we'll look at how you could do that. Let's dive in. Let's continue with our running example of handwritten digit recognition, recognizing this image as 0 or a 1. And here we're using the neural network architecture that you saw last week, where you have an input x that is the image, and then the first hidden layer with 25 units, second hidden layer with 15 units, and then one output unit. If you are given a training set of examples comprising images x as well as the ground truth label y, how would you train the parameters of this neural network? Let me go ahead and show you the code that you can use in TensorFlow to train this network. And then in the next few videos after this, we'll dive into details to explain what the code is actually doing. So this is the code you would write. This first part may look familiar from the previous week, where you are asking TensorFlow to sequentially string together these three layers of a neural network, the first hidden layer with 25 units and sig 1 activation, the second hidden layer, and then finally the output layer. So nothing new here relative to what you saw last week. Second step is you have to ask TensorFlow to compile the model. And the key step in asking TensorFlow to compile the model is to specify what is the loss function you want to use. In this case, we'll use something that goes by the arcane name of sparse categorical cross entropy. We'll say more in the next video what this really is. And then having specified the loss function, the third step is to call the fit function, which tells TensorFlow to fit the model that you specified in step one. Using the loss of the cost function that you specified in step two to the data set x, y. And back in the first course, when we talked about gradient descent, we had to decide how many steps to run gradient descent or how long to run gradient descent. So epochs is a technical term for how many steps of learning algorithm like gradient descent you may want to run. And that's it. Step one is to specify the model, which tells TensorFlow how to compute for the inference. Step two compiles the model using a specific loss function. And step three is to train the model. So that's how you can train a neural network in TensorFlow. As usual, I hope that you'll be able to not just call these lines of code to train the model, but that
| 500
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5.1 Neural Network Training | TensorFlow implementation --[Machine Learning | Andrew Ng]: Welcome back to the second week of this course on advanced learning algorithms. Last week you learned how to carry out inference in a neural network. This week we're going to go over training of a neural network. I think being able to take your own data and train your own neural network on it is really fun. This week we'll look at how you could do that. Let's dive in. Let's continue with our running example of handwritten digit recognition, recognizing this image as 0 or a 1. And here we're using the neural network architecture that you saw last week, where you have an input x that is the image, and then the first hidden layer with 25 units, second hidden layer with 15 units, and then one output unit. If you are given a training set of examples comprising images x as well as the ground truth label y, how would you train the parameters of this neural network? Let me go ahead and show you the code that you can use in TensorFlow to train this network. And then in the next few videos after this, we'll dive into details to explain what the code is actually doing. So this is the code you would write. This first part may look familiar from the previous week, where you are asking TensorFlow to sequentially string together these three layers of a neural network, the first hidden layer with 25 units and sig 1 activation, the second hidden layer, and then finally the output layer. So nothing new here relative to what you saw last week. Second step is you have to ask TensorFlow to compile the model. And the key step in asking TensorFlow to compile the model is to specify what is the loss function you want to use. In this case, we'll use something that goes by the arcane name of sparse categorical cross entropy. We'll say more in the next video what this really is. And then having specified the loss function, the third step is to call the fit function, which tells TensorFlow to fit the model that you specified in step one. Using the loss of the cost function that you specified in step two to the data set x, y. And back in the first course, when we talked about gradient descent, we had to decide how many steps to run gradient descent or how long to run gradient descent. So epochs is a technical term for how many steps of learning algorithm like gradient descent you may want to run. And that's it. Step one is to specify the model, which tells TensorFlow how to compute for the inference. Step two compiles the model using a specific loss function. And step three is to train the model. So that's how you can train a neural network in TensorFlow. As usual, I hope that you'll be able to not just call these lines of code to train the model, but that
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5.1 Neural Network Training | TensorFlow implementation --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=jdjGdT_jR50
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you also understand what's actually going on behind these lines of code. So you don't just call it without really understanding what's going on. And I think this is important because when you're running a learning algorithm, if it doesn't work initially, having that conceptual mental framework of what's really going on will help you debug whenever things don't work the way you expect. So with that, let's go on to the next video, where we'll dive more deeply into what these steps in the TensorFlow implementation are actually doing. I'll see you in the next video.
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5.1 Neural Network Training | TensorFlow implementation --[Machine Learning | Andrew Ng]: you also understand what's actually going on behind these lines of code. So you don't just call it without really understanding what's going on. And I think this is important because when you're running a learning algorithm, if it doesn't work initially, having that conceptual mental framework of what's really going on will help you debug whenever things don't work the way you expect. So with that, let's go on to the next video, where we'll dive more deeply into what these steps in the TensorFlow implementation are actually doing. I'll see you in the next video.
|
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5.2 Neural Network Training | Training Details --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=jzaF4An03Oc
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Let's take a look at the details of what the TensorFlow code for training a neural network is actually doing. Let's dive in. Before looking at the details of training a neural network, let's recall how you had trained a logistic regression model in the previous course. Step 1 of building a logistic regression model was you would specify how to compute the output given the input v to the x and the parameters w and b. So in the first course we said the logistic regression function predicts f of x is equal to g, the sigmoid function applied to w dot product x plus b, which was the sigmoid function applied to w dot x plus b. So if z is the dot product of w of x plus b, then f of x is 1 over 1 plus e to the negative z. So that was the first step, where to specify what is the input to output function of logistic regression, and that depends on both the input x and the parameters of the model. The second step we had to do to train the logistic regression model was to specify the loss function and also the cost function. So you may recall that the loss function said if logistic regression outputs f of x and the ground truth label, the actual label in the training set was y, then the loss on that single training example was negative y log f of x minus 1 minus y times log of 1 minus f of x. So this was a measure of how well is logistic regression doing on a single training example x, y. Given this definition of a loss function, we then define the cost function, and the cost function was a function of the parameters w and b, and that was just the average that is taking an average over all m training examples of the loss function computed on the m training examples x1, y1, through xm, ym. And remember that in the condition we're using, the loss function is a function of the output of the learning algorithm and the ground truth label as computed over a single training example, whereas the cost function j is an average of the loss function computed over your entire training set. So that was step two of what we did when building up logistic regression. And then the third and final step to train the logistic regression model was to use an algorithm, specifically gradient descent, to minimize that cost function j of w, b to minimize it as a function of the parameters w and b. And we minimize the cost j as a function of the parameters using gradient descent, where w is updated as w minus the learning rate alpha times the derivative of j with respect to w, and b similarly is updated as b minus the learning rate alpha times the derivative of j with respect to b. So
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5.2 Neural Network Training | Training Details --[Machine Learning | Andrew Ng]: Let's take a look at the details of what the TensorFlow code for training a neural network is actually doing. Let's dive in. Before looking at the details of training a neural network, let's recall how you had trained a logistic regression model in the previous course. Step 1 of building a logistic regression model was you would specify how to compute the output given the input v to the x and the parameters w and b. So in the first course we said the logistic regression function predicts f of x is equal to g, the sigmoid function applied to w dot product x plus b, which was the sigmoid function applied to w dot x plus b. So if z is the dot product of w of x plus b, then f of x is 1 over 1 plus e to the negative z. So that was the first step, where to specify what is the input to output function of logistic regression, and that depends on both the input x and the parameters of the model. The second step we had to do to train the logistic regression model was to specify the loss function and also the cost function. So you may recall that the loss function said if logistic regression outputs f of x and the ground truth label, the actual label in the training set was y, then the loss on that single training example was negative y log f of x minus 1 minus y times log of 1 minus f of x. So this was a measure of how well is logistic regression doing on a single training example x, y. Given this definition of a loss function, we then define the cost function, and the cost function was a function of the parameters w and b, and that was just the average that is taking an average over all m training examples of the loss function computed on the m training examples x1, y1, through xm, ym. And remember that in the condition we're using, the loss function is a function of the output of the learning algorithm and the ground truth label as computed over a single training example, whereas the cost function j is an average of the loss function computed over your entire training set. So that was step two of what we did when building up logistic regression. And then the third and final step to train the logistic regression model was to use an algorithm, specifically gradient descent, to minimize that cost function j of w, b to minimize it as a function of the parameters w and b. And we minimize the cost j as a function of the parameters using gradient descent, where w is updated as w minus the learning rate alpha times the derivative of j with respect to w, and b similarly is updated as b minus the learning rate alpha times the derivative of j with respect to b. So
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5.2 Neural Network Training | Training Details --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=jzaF4An03Oc
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with these three steps, step one, specifying how to compute the outputs given the input x in parameters, step two, specifying loss in cost, and step three, minimize the cost function, we trained logistic regression. The same three steps is how we can train a neural network in TensorFlow. Now let's look at how these three steps map to training a neural network. We'll go over this in greater detail on the next three slides, but really briefly, step one of specifying how to compute the output given the input x in parameters w and b, that's done with this code snippet, which should be familiar from last week of specifying the neural network, and this was actually enough to specify the computations needed in forward propagation or for the inference algorithm, for example. The second step is to compile the model and to tell it what loss you want to use. And here's a code that you use to specify this loss function, which is the binary cross NGP loss function. And once you specify this loss, taking an average over the entire training set also gives you the cost function for the neural network. And then step three is to call function to try to minimize the cost as a function of the parameters of the neural network. Let's look in greater detail in these three steps in the context of training a neural network. The first step, specify how to compute the output given the input x in parameters w and b. This code snippet specifies the entire architecture of the neural network. It tells you that there are 25 hidden units in the first hidden layer, then 15 in the next one, and then one output unit, and that we're using the sig one activation value. And so based on this code snippet, we know also what are the parameters, w1, b1 of the first layer, parameters of the second layer, and parameters of the third layer. So this code snippet specifies the entire architecture of the neural network and therefore tells TensorFlow everything it needs in order to compute the output x as a function. In order to compute the output a3 or f of x as a function of the input x and the parameters here we have written wl and bl. Let's go on to step two. In the second step, you have to specify what is the loss function, and that will also define the cost function we use to train the neural network. So for the MNIST 01 digit classification problem is a binary classification problem, and the most common by far loss function to use is this one. It's actually the same loss function as what we had for logistic regression is negative y log f of x minus one minus y times log one minus f of x, where y is the ground truth label, sometimes also called the target label y, and f of x is now the
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5.2 Neural Network Training | Training Details --[Machine Learning | Andrew Ng]: with these three steps, step one, specifying how to compute the outputs given the input x in parameters, step two, specifying loss in cost, and step three, minimize the cost function, we trained logistic regression. The same three steps is how we can train a neural network in TensorFlow. Now let's look at how these three steps map to training a neural network. We'll go over this in greater detail on the next three slides, but really briefly, step one of specifying how to compute the output given the input x in parameters w and b, that's done with this code snippet, which should be familiar from last week of specifying the neural network, and this was actually enough to specify the computations needed in forward propagation or for the inference algorithm, for example. The second step is to compile the model and to tell it what loss you want to use. And here's a code that you use to specify this loss function, which is the binary cross NGP loss function. And once you specify this loss, taking an average over the entire training set also gives you the cost function for the neural network. And then step three is to call function to try to minimize the cost as a function of the parameters of the neural network. Let's look in greater detail in these three steps in the context of training a neural network. The first step, specify how to compute the output given the input x in parameters w and b. This code snippet specifies the entire architecture of the neural network. It tells you that there are 25 hidden units in the first hidden layer, then 15 in the next one, and then one output unit, and that we're using the sig one activation value. And so based on this code snippet, we know also what are the parameters, w1, b1 of the first layer, parameters of the second layer, and parameters of the third layer. So this code snippet specifies the entire architecture of the neural network and therefore tells TensorFlow everything it needs in order to compute the output x as a function. In order to compute the output a3 or f of x as a function of the input x and the parameters here we have written wl and bl. Let's go on to step two. In the second step, you have to specify what is the loss function, and that will also define the cost function we use to train the neural network. So for the MNIST 01 digit classification problem is a binary classification problem, and the most common by far loss function to use is this one. It's actually the same loss function as what we had for logistic regression is negative y log f of x minus one minus y times log one minus f of x, where y is the ground truth label, sometimes also called the target label y, and f of x is now the
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5.2 Neural Network Training | Training Details --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=jzaF4An03Oc
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output of the neural network. So in the terminology of TensorFlow, this loss function is called binary cross entropy, and the syntax is to ask TensorFlow to compile the neural network using this loss function. And another historical note, Keras was originally a library that had developed independently of TensorFlow, it was actually a totally separate project from TensorFlow, but eventually it got merged into TensorFlow, which is why we have tf.keraslibrary.losses.the name of this loss function. And by the way, I don't always remember the names of all the loss functions in TensorFlow, but I just do a quick web search myself to find the right name, and then I plug that into my code. Having specified the loss with respect to a single training example, TensorFlow knows that the costs you want to minimize is then the average, taking the average over all m training examples of the loss on all of the training examples. And optimizing this cost function will result in fitting the neural network to your binary classification data. In case you want to solve a regression problem rather than a classification problem, you can also tell TensorFlow to compile your model using a different loss function. For example, if you have a regression problem, and if you want to minimize the squared error loss, so here is the squared error loss, the loss with respect to if you're learning algorithm outputs f of x with a target or ground truth label of y, that's one half of the squared error, then you can use this loss function in TensorFlow, which is to use the maybe more intuitively named mean squared error loss function, and then TensorFlow will try to minimize the mean squared error. In this expression, I'm using J of capital W comma capital B to denote the cost function. The cost function is a function of all of the parameters in the neural network. So you can think of capital W as including W1, W2, W3, so all the W parameters in the entire neural network and B as including B1, B2, and B3. So if you are optimizing the cost function with respect to W and B, you'd be trying to optimize it with respect to all of the parameters in the neural network. And up on top as well, I had written f of x as the output of the neural network, but you want, you can also write f of WB if we want to emphasize that the output of the neural network as a function of x depends on all the parameters in all the layers of the neural network. So that's the loss function and the cost function. And in TensorFlow, this is called the binary cross entropy loss function. Where does that name come from? Well, it turns out in statistics, this function on top is called the cross entropy loss function. So that's what cross entropy means. And the word binary just re-emphasizes or points
| 500
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5.2 Neural Network Training | Training Details --[Machine Learning | Andrew Ng]: output of the neural network. So in the terminology of TensorFlow, this loss function is called binary cross entropy, and the syntax is to ask TensorFlow to compile the neural network using this loss function. And another historical note, Keras was originally a library that had developed independently of TensorFlow, it was actually a totally separate project from TensorFlow, but eventually it got merged into TensorFlow, which is why we have tf.keraslibrary.losses.the name of this loss function. And by the way, I don't always remember the names of all the loss functions in TensorFlow, but I just do a quick web search myself to find the right name, and then I plug that into my code. Having specified the loss with respect to a single training example, TensorFlow knows that the costs you want to minimize is then the average, taking the average over all m training examples of the loss on all of the training examples. And optimizing this cost function will result in fitting the neural network to your binary classification data. In case you want to solve a regression problem rather than a classification problem, you can also tell TensorFlow to compile your model using a different loss function. For example, if you have a regression problem, and if you want to minimize the squared error loss, so here is the squared error loss, the loss with respect to if you're learning algorithm outputs f of x with a target or ground truth label of y, that's one half of the squared error, then you can use this loss function in TensorFlow, which is to use the maybe more intuitively named mean squared error loss function, and then TensorFlow will try to minimize the mean squared error. In this expression, I'm using J of capital W comma capital B to denote the cost function. The cost function is a function of all of the parameters in the neural network. So you can think of capital W as including W1, W2, W3, so all the W parameters in the entire neural network and B as including B1, B2, and B3. So if you are optimizing the cost function with respect to W and B, you'd be trying to optimize it with respect to all of the parameters in the neural network. And up on top as well, I had written f of x as the output of the neural network, but you want, you can also write f of WB if we want to emphasize that the output of the neural network as a function of x depends on all the parameters in all the layers of the neural network. So that's the loss function and the cost function. And in TensorFlow, this is called the binary cross entropy loss function. Where does that name come from? Well, it turns out in statistics, this function on top is called the cross entropy loss function. So that's what cross entropy means. And the word binary just re-emphasizes or points
|
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5.2 Neural Network Training | Training Details --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=jzaF4An03Oc
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out that this is a binary classification problem because each image is either a zero or a one. Finally, you will ask TensorFlow to minimize the cost function. You might remember the gradient descent algorithm from the first course. If you are using gradient descent to train the parameters of a neural network, then you will repeatedly for every layer L and for every unit J, update WLJ according to WLJ minus the learning rate alpha times the partial derivative with respect to that parameter of the cost function J of WB. And similarly for the parameters B as well. And after doing, say, 100 iterations of gradient descent, hopefully you get to a good value of the parameters. So in order to use gradient descent, the key thing you need to compute is these partial derivative terms. And what TensorFlow does, and in fact what is standard in neural network training, is to use an algorithm called backpropagation in order to compute these partial derivative terms. TensorFlow can do all of these things for you. It implements backpropagation all within this function called fit. So all you have to do is call model.fit x, y as your training set and tell it to do so for 100 iterations or 100 epochs. In fact, what you see later is that TensorFlow can use an algorithm that is even a little bit faster than gradient descent. And you see more about that later this week as well. Now I know that we're relying heavily on the TensorFlow library in order to implement a neural network. One pattern I've seen across multiple ideas is as the technology evolves, libraries become more mature and most engineers will use libraries rather than implement code from scratch. And there have been many other examples of this in the history of computing. Once many, many decades ago programmers had to implement their own sorting function from scratch. But now sorting libraries are quite mature that you probably call someone else's sorting function rather than implement it yourself unless you're taking a computing classes, ask you to do it as an exercise. And today if you want to compute the square root of a number, like what is the square root of 7? Well once programmers had to write their own code to compute this, but now pretty much everyone just calls a library to take square roots or matrix operations such as multiplying two matrices together. So when deep learning was younger and less mature, many developers, including me, were implementing things from scratch using Python or C++ or some other library. But today deep learning libraries have matured enough that most developers will use these libraries and in fact most commercial implementations of neural networks today use a library like TensorFlow or PyTorch. But as I've mentioned, it's still useful to understand how they work under the hood so that if something unexpected happens, which still does with today's libraries, you have a better chance
| 500
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5.2 Neural Network Training | Training Details --[Machine Learning | Andrew Ng]: out that this is a binary classification problem because each image is either a zero or a one. Finally, you will ask TensorFlow to minimize the cost function. You might remember the gradient descent algorithm from the first course. If you are using gradient descent to train the parameters of a neural network, then you will repeatedly for every layer L and for every unit J, update WLJ according to WLJ minus the learning rate alpha times the partial derivative with respect to that parameter of the cost function J of WB. And similarly for the parameters B as well. And after doing, say, 100 iterations of gradient descent, hopefully you get to a good value of the parameters. So in order to use gradient descent, the key thing you need to compute is these partial derivative terms. And what TensorFlow does, and in fact what is standard in neural network training, is to use an algorithm called backpropagation in order to compute these partial derivative terms. TensorFlow can do all of these things for you. It implements backpropagation all within this function called fit. So all you have to do is call model.fit x, y as your training set and tell it to do so for 100 iterations or 100 epochs. In fact, what you see later is that TensorFlow can use an algorithm that is even a little bit faster than gradient descent. And you see more about that later this week as well. Now I know that we're relying heavily on the TensorFlow library in order to implement a neural network. One pattern I've seen across multiple ideas is as the technology evolves, libraries become more mature and most engineers will use libraries rather than implement code from scratch. And there have been many other examples of this in the history of computing. Once many, many decades ago programmers had to implement their own sorting function from scratch. But now sorting libraries are quite mature that you probably call someone else's sorting function rather than implement it yourself unless you're taking a computing classes, ask you to do it as an exercise. And today if you want to compute the square root of a number, like what is the square root of 7? Well once programmers had to write their own code to compute this, but now pretty much everyone just calls a library to take square roots or matrix operations such as multiplying two matrices together. So when deep learning was younger and less mature, many developers, including me, were implementing things from scratch using Python or C++ or some other library. But today deep learning libraries have matured enough that most developers will use these libraries and in fact most commercial implementations of neural networks today use a library like TensorFlow or PyTorch. But as I've mentioned, it's still useful to understand how they work under the hood so that if something unexpected happens, which still does with today's libraries, you have a better chance
|
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5.2 Neural Network Training | Training Details --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=jzaF4An03Oc
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of knowing how to fix it. Now that you know how to train a basic neural network, also called a multilayer perceptron, there are some things you can change about the neural network that will make it even more powerful. In the next video, let's take a look at how you can swap in different activation functions as an alternative to the sigmoid activation function we've been using. This will make your neural networks work even much better. So let's go take a look at that in the next video.
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5.2 Neural Network Training | Training Details --[Machine Learning | Andrew Ng]: of knowing how to fix it. Now that you know how to train a basic neural network, also called a multilayer perceptron, there are some things you can change about the neural network that will make it even more powerful. In the next video, let's take a look at how you can swap in different activation functions as an alternative to the sigmoid activation function we've been using. This will make your neural networks work even much better. So let's go take a look at that in the next video.
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5.3 Activation Functions | Alternatives to the sigmoid activation --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=TQ388ISSoI4
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So far, we've been using the sigmoid activation function in all the nodes, in the hidden layers, and in the output layer. And we have started that way because we were building up neural networks by taking logistic regression and creating a lot of logistic regression units and stringing them together. But if you use other activation functions, your neural network can become much more powerful. Let's take a look at how to do that. Recall the demand prediction example from last week, where given price, shipping cost, marketing material, you would try to predict if something is highly affordable, if there's good awareness and high perceived quality, and based on that, try to predict if it's a top seller. But this assumes that awareness is maybe binary, it's either people are aware or they are not. But it seems like the degree to which possible buyers are aware of the t-shirt you're selling may not be binary. They can be a little bit aware, somewhat aware, extremely aware, or it could have gone completely viral. So rather than modeling awareness as a binary number, 0 or 1, that you try to estimate the probability of awareness, or rather than modeling awareness as just a number between 0 and 1, maybe awareness should be any non-negative number, because there can be any non-negative value of awareness going from 0 up to very, very large numbers. So whereas previously we had used this equation to calculate the activation of that second hidden unit estimating awareness, where g was the sigmoid function, and thus goes between 0 and 1. If you want to allow a 1, 2 to potentially take on much larger positive values, we can instead swap in a different activation function. It turns out that a very common choice of activation function in neural networks is this function. It looks like this. It goes, if z is this, then g of z is 0 to the left, and then this is straight line, 35 degrees to the right of 0. And so when z is greater than or equal to 0, g of z is just equal to z, that is to the right half of this diagram. And the mathematical equation for this is g of z equals max of 0, z. Be free to verify for yourself that max of 0, z results in this curve that I've drawn over here. And if a 1, 2 is g of z for this value of z, then a, the activation value, can now take on 0 or any non-negative value. This activation function has a name. It goes by the name ReLU with this funny capitalization. And ReLU stands for, again, a somewhat arcane term, but it stands for rectified linear unit. Don't worry too much about what rectified means and what linear unit means. This was just a name that the authors had given to this particular activation function when they came up with it. But most
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5.3 Activation Functions | Alternatives to the sigmoid activation --[Machine Learning | Andrew Ng]: So far, we've been using the sigmoid activation function in all the nodes, in the hidden layers, and in the output layer. And we have started that way because we were building up neural networks by taking logistic regression and creating a lot of logistic regression units and stringing them together. But if you use other activation functions, your neural network can become much more powerful. Let's take a look at how to do that. Recall the demand prediction example from last week, where given price, shipping cost, marketing material, you would try to predict if something is highly affordable, if there's good awareness and high perceived quality, and based on that, try to predict if it's a top seller. But this assumes that awareness is maybe binary, it's either people are aware or they are not. But it seems like the degree to which possible buyers are aware of the t-shirt you're selling may not be binary. They can be a little bit aware, somewhat aware, extremely aware, or it could have gone completely viral. So rather than modeling awareness as a binary number, 0 or 1, that you try to estimate the probability of awareness, or rather than modeling awareness as just a number between 0 and 1, maybe awareness should be any non-negative number, because there can be any non-negative value of awareness going from 0 up to very, very large numbers. So whereas previously we had used this equation to calculate the activation of that second hidden unit estimating awareness, where g was the sigmoid function, and thus goes between 0 and 1. If you want to allow a 1, 2 to potentially take on much larger positive values, we can instead swap in a different activation function. It turns out that a very common choice of activation function in neural networks is this function. It looks like this. It goes, if z is this, then g of z is 0 to the left, and then this is straight line, 35 degrees to the right of 0. And so when z is greater than or equal to 0, g of z is just equal to z, that is to the right half of this diagram. And the mathematical equation for this is g of z equals max of 0, z. Be free to verify for yourself that max of 0, z results in this curve that I've drawn over here. And if a 1, 2 is g of z for this value of z, then a, the activation value, can now take on 0 or any non-negative value. This activation function has a name. It goes by the name ReLU with this funny capitalization. And ReLU stands for, again, a somewhat arcane term, but it stands for rectified linear unit. Don't worry too much about what rectified means and what linear unit means. This was just a name that the authors had given to this particular activation function when they came up with it. But most
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5.3 Activation Functions | Alternatives to the sigmoid activation --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=TQ388ISSoI4
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people in deep learning just say ReLU to refer to this g of z. More generally, you have a choice of what to use for g of z. And sometimes we'll use a different choice than the sigmoid activation function. Here are the most commonly used activation functions. You saw the sigmoid activation function, g of z equals the sigmoid function. On the last slide, we just looked at the ReLU, or rectified linear unit, g of z equals max of 0, z. There's one other activation function which is worth mentioning, which is called the linear activation function, which is just g of z equals to z. Sometimes if you use the linear activation function, people will say we're not using any activation function because if a is g of z, where g of z equals z, then a is just equal to this, w dot x plus b, say. And so it's as if there was no g in there at all. So when you are using this linear activation function, g of z, sometimes people will say, well, we're not using any activation function. Although in this clause, I will refer to using the linear activation function rather than no activation function. But if you hear someone else use that terminology, that's what they mean. It just refers to the linear activation function. And these three are probably by far the most commonly used activation functions in neural networks. Later this week, we'll touch on the fourth one called the softmax activation function. But with these activation functions, you'll be able to build a rich variety of powerful neural networks. So when building a neural network, for each neuron, do you want to use the sigmoid activation function or the ReLU activation function or a linear activation function? How do you choose between these different activation functions? Let's take a look at that in the next video.
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5.3 Activation Functions | Alternatives to the sigmoid activation --[Machine Learning | Andrew Ng]: people in deep learning just say ReLU to refer to this g of z. More generally, you have a choice of what to use for g of z. And sometimes we'll use a different choice than the sigmoid activation function. Here are the most commonly used activation functions. You saw the sigmoid activation function, g of z equals the sigmoid function. On the last slide, we just looked at the ReLU, or rectified linear unit, g of z equals max of 0, z. There's one other activation function which is worth mentioning, which is called the linear activation function, which is just g of z equals to z. Sometimes if you use the linear activation function, people will say we're not using any activation function because if a is g of z, where g of z equals z, then a is just equal to this, w dot x plus b, say. And so it's as if there was no g in there at all. So when you are using this linear activation function, g of z, sometimes people will say, well, we're not using any activation function. Although in this clause, I will refer to using the linear activation function rather than no activation function. But if you hear someone else use that terminology, that's what they mean. It just refers to the linear activation function. And these three are probably by far the most commonly used activation functions in neural networks. Later this week, we'll touch on the fourth one called the softmax activation function. But with these activation functions, you'll be able to build a rich variety of powerful neural networks. So when building a neural network, for each neuron, do you want to use the sigmoid activation function or the ReLU activation function or a linear activation function? How do you choose between these different activation functions? Let's take a look at that in the next video.
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5.4 Activation Functions | Choosing activation functions--[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=orElYjWScBw
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Let's take a look at how you can choose the activation function for different neurons in your neural network. We'll start with some guidance for how to choose it for the output layer. It turns out that depending on what the target label or the ground truth label Y is, there will be one fairly natural choice for the activation function for the output layer. And we'll then go and look at the choice of the activation function also for the hidden layers of your neural network. Let's take a look. You can choose different activation functions for different neurons in your neural network. And when considering the activation function for the output layer, it turns out that there will often be one fairly natural choice depending on what is the target or the ground truth label Y. Specifically, if you are working on a classification problem where Y is either 0 or 1, so a binary classification problem, then the sigmoid activation function will almost always be the most natural choice because then the neural network learns to predict the probability that Y is equal to 1, just like we had for logistic regression. So my recommendation is if you're working on a binary classification problem, use sigmoid at the output layer. Alternatively, if you're solving a regression problem, then you might choose a different activation function. For example, if you're trying to predict how tomorrow's stock price will change compared to today's stock price, well, it can go up or down. And so in this case, Y would be a number that can be either positive or negative. And in that case, I would recommend you use the linear activation function. Why is that? Well, that's because then the output of your neural network, f of x, which is equal to a3 in the example above, would be g applied to z3. And with the linear activation function, g of z can take on either positive or negative values. So Y can be positive or negative, use the linear activation function. And finally, if Y can only take on non-negative values, such as if you're predicting the price of a house, that can never be negative. Then the most natural choice would be the ReLU activation function. Because as you see here, this activation function only takes on non-negative values, either zero or positive values. So when choosing the activation function to use for your output layer, usually depending on what is the label, why you're trying to predict, there'll be one fairly natural choice. And in fact, the guidance on this slide is how I pretty much always choose my activation function as well for the output layer of a neural network. How about the hidden layers of a neural network? It turns out that the ReLU activation function is by far the most common choice in how neural networks are trained by many, many practitioners today. Even though we had initially described neural networks using the
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5.4 Activation Functions | Choosing activation functions--[Machine Learning | Andrew Ng]: Let's take a look at how you can choose the activation function for different neurons in your neural network. We'll start with some guidance for how to choose it for the output layer. It turns out that depending on what the target label or the ground truth label Y is, there will be one fairly natural choice for the activation function for the output layer. And we'll then go and look at the choice of the activation function also for the hidden layers of your neural network. Let's take a look. You can choose different activation functions for different neurons in your neural network. And when considering the activation function for the output layer, it turns out that there will often be one fairly natural choice depending on what is the target or the ground truth label Y. Specifically, if you are working on a classification problem where Y is either 0 or 1, so a binary classification problem, then the sigmoid activation function will almost always be the most natural choice because then the neural network learns to predict the probability that Y is equal to 1, just like we had for logistic regression. So my recommendation is if you're working on a binary classification problem, use sigmoid at the output layer. Alternatively, if you're solving a regression problem, then you might choose a different activation function. For example, if you're trying to predict how tomorrow's stock price will change compared to today's stock price, well, it can go up or down. And so in this case, Y would be a number that can be either positive or negative. And in that case, I would recommend you use the linear activation function. Why is that? Well, that's because then the output of your neural network, f of x, which is equal to a3 in the example above, would be g applied to z3. And with the linear activation function, g of z can take on either positive or negative values. So Y can be positive or negative, use the linear activation function. And finally, if Y can only take on non-negative values, such as if you're predicting the price of a house, that can never be negative. Then the most natural choice would be the ReLU activation function. Because as you see here, this activation function only takes on non-negative values, either zero or positive values. So when choosing the activation function to use for your output layer, usually depending on what is the label, why you're trying to predict, there'll be one fairly natural choice. And in fact, the guidance on this slide is how I pretty much always choose my activation function as well for the output layer of a neural network. How about the hidden layers of a neural network? It turns out that the ReLU activation function is by far the most common choice in how neural networks are trained by many, many practitioners today. Even though we had initially described neural networks using the
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5.4 Activation Functions | Choosing activation functions--[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=orElYjWScBw
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sigmoid activation function, and in fact, in the early history of the development of neural networks, people use sigmoid activation functions in many places, the field has evolved to use ReLU much more often and sigmoids hardly ever. With the one exception that you do use a sigmoid activation function in the output layer if you have a binary classification problem. So why is that? Well, there are a few reasons. First, if you compare the ReLU and the sigmoid activation functions, the ReLU is a bit faster to compute because it just requires computing max of 0, z, whereas the sigmoid requires taking an exponentiation and an inverse and so on, and so it's a little bit less efficient. But the second reason, which turns out to be even more important, is that the ReLU function kind of goes flat only in one part of the graph, here on the left, it's completely flat. Whereas the sigmoid activation function, it kind of goes flat in two places. It goes flat to the left of the graph and it goes flat to the right of the graph. And if you're using gradient descent to train a neural network, then when you have a function that is flat in a lot of places, gradient descent will be really slow. I know that gradient descent optimizes the cost function J of WB rather than optimizes the activation function, but the activation function is a piece of what goes into computing, and that results in more places in the cost function J of WB that are flat as well and with a smaller gradient and it slows down learning. I know that that was just an intuitive explanation, but researchers have found that using the ReLU activation function can cause your neural network to learn a bit faster as well, which is why for most practitioners, if you're trying to decide what activation function to use for the hidden layer, the ReLU activation function has become now by far the most common choice. In fact, if I'm building a neural network, this is how I choose activation functions for the hidden layers as well. So to summarize, here's what I recommend in terms of how you choose the activation functions for your neural network. For the output layer, use a sigmoid if you have a binary classification problem, linear if Y is a number that can take on positive or negative values, or use ReLU if Y can take on only positive values or zero positive values or non-negative values. Then for the hidden layers, I would recommend just using ReLU as a default activation function. And in TensorFlow, this is how you would implement it. Rather than saying activation equals sigmoid as we had previously, you can then for the hidden layers, that's the first hidden layer, the second hidden layer, ask TensorFlow to use the ReLU activation function. And then for the output layer, in this example, I've asked it to
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5.4 Activation Functions | Choosing activation functions--[Machine Learning | Andrew Ng]: sigmoid activation function, and in fact, in the early history of the development of neural networks, people use sigmoid activation functions in many places, the field has evolved to use ReLU much more often and sigmoids hardly ever. With the one exception that you do use a sigmoid activation function in the output layer if you have a binary classification problem. So why is that? Well, there are a few reasons. First, if you compare the ReLU and the sigmoid activation functions, the ReLU is a bit faster to compute because it just requires computing max of 0, z, whereas the sigmoid requires taking an exponentiation and an inverse and so on, and so it's a little bit less efficient. But the second reason, which turns out to be even more important, is that the ReLU function kind of goes flat only in one part of the graph, here on the left, it's completely flat. Whereas the sigmoid activation function, it kind of goes flat in two places. It goes flat to the left of the graph and it goes flat to the right of the graph. And if you're using gradient descent to train a neural network, then when you have a function that is flat in a lot of places, gradient descent will be really slow. I know that gradient descent optimizes the cost function J of WB rather than optimizes the activation function, but the activation function is a piece of what goes into computing, and that results in more places in the cost function J of WB that are flat as well and with a smaller gradient and it slows down learning. I know that that was just an intuitive explanation, but researchers have found that using the ReLU activation function can cause your neural network to learn a bit faster as well, which is why for most practitioners, if you're trying to decide what activation function to use for the hidden layer, the ReLU activation function has become now by far the most common choice. In fact, if I'm building a neural network, this is how I choose activation functions for the hidden layers as well. So to summarize, here's what I recommend in terms of how you choose the activation functions for your neural network. For the output layer, use a sigmoid if you have a binary classification problem, linear if Y is a number that can take on positive or negative values, or use ReLU if Y can take on only positive values or zero positive values or non-negative values. Then for the hidden layers, I would recommend just using ReLU as a default activation function. And in TensorFlow, this is how you would implement it. Rather than saying activation equals sigmoid as we had previously, you can then for the hidden layers, that's the first hidden layer, the second hidden layer, ask TensorFlow to use the ReLU activation function. And then for the output layer, in this example, I've asked it to
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5.4 Activation Functions | Choosing activation functions--[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=orElYjWScBw
|
use the sigmoid activation function. But if you want it to use the linear activation function instead, that's the syntax for it. Or if you wanted to use the ReLU activation function, that shows the syntax for it. With this richer set of activation functions, you'd be well positioned to build much more powerful neural networks than just once using only the sigmoid activation function. By the way, if you look at the research literature, you sometimes hear of authors using even other activation functions, such as the tanh activation function or the leaky ReLU activation function or the swish activation function. Every few years, researchers sometimes come up with another interesting activation function, and sometimes they do work a little bit better. For example, I've used the leaky ReLU activation function a few times in my work, and sometimes it works a little bit better than the ReLU activation function you learned about in this video. But I think for the most part, and for the vast majority of applications, what you learned about in this video would be good enough. Of course, if you want to learn more about other activation functions, feel free to look on the internet, and there are just a small handful of cases where these other activation functions could be even more powerful as well. With that, I hope you also enjoy practicing these ideas, these activation functions, in the optional labs and in the practice labs. But this raises yet another question. Why do we even need activation functions at all? Why don't we just use the linear activation function, or use no activation function anywhere? It turns out this does not work at all. And in the next video, let's take a look at why that's the case, and why activation functions are so important for getting your neural networks to work.
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5.4 Activation Functions | Choosing activation functions--[Machine Learning | Andrew Ng]: use the sigmoid activation function. But if you want it to use the linear activation function instead, that's the syntax for it. Or if you wanted to use the ReLU activation function, that shows the syntax for it. With this richer set of activation functions, you'd be well positioned to build much more powerful neural networks than just once using only the sigmoid activation function. By the way, if you look at the research literature, you sometimes hear of authors using even other activation functions, such as the tanh activation function or the leaky ReLU activation function or the swish activation function. Every few years, researchers sometimes come up with another interesting activation function, and sometimes they do work a little bit better. For example, I've used the leaky ReLU activation function a few times in my work, and sometimes it works a little bit better than the ReLU activation function you learned about in this video. But I think for the most part, and for the vast majority of applications, what you learned about in this video would be good enough. Of course, if you want to learn more about other activation functions, feel free to look on the internet, and there are just a small handful of cases where these other activation functions could be even more powerful as well. With that, I hope you also enjoy practicing these ideas, these activation functions, in the optional labs and in the practice labs. But this raises yet another question. Why do we even need activation functions at all? Why don't we just use the linear activation function, or use no activation function anywhere? It turns out this does not work at all. And in the next video, let's take a look at why that's the case, and why activation functions are so important for getting your neural networks to work.
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5.5 Activation Functions | Why do we need activation functions? --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=fK5YzGIc2u8
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Let's take a look at why neural networks need activation functions and why they just don't work if we were to use the linear activation function in every neuron in the neural network. Recall this demand prediction example. What would happen if we were to use a linear activation function for all of the nodes in this neural network? It turns out that this big neural network will become no different than just linear regression. And so this would defeat the entire purpose of using a neural network because it would then just not be able to fit anything more complex than the linear regression model that we learned about in the first course. Let's illustrate this with a simpler example. Let's look at the example of a neural network where the input x is just a number and we have one hidden unit with parameters w1 and b1 that outputs a1, which is here just a number. And then the second layer is the output layer and it has also just one output unit with parameters w2 and b2 and that outputs a2, which is also just a number, just a scalar, which is the output of the neural network f of x. Let's see what this neural network would do if we were to use the linear activation function g of z equals z everywhere. So to compute a1 as a function of x, the neural network would use a1 equals g of w1 times x plus b1, but g of z is equal to z, so this is just w1 times x plus b1. Then a2 is equal to w2 times a1 plus b2 because g of z equals z. And let me take this expression for a1 and substitute it in there. So that becomes w2 times w1 x plus b1 plus b2. And if we simplify, this becomes w2 w1 times x plus w2 b1 plus b2. And it turns out that if I were to set w equals w2 times w1 and set b equals this quantity over here, then what we've just shown is that a2 is equal to wx plus b. So w is just a linear function of the input x. And rather than using a neural network with one hidden layer and one output layer, we might as well have just used a linear regression model. If you're familiar with linear algebra, this result comes from the fact that a linear function of a linear function is itself a linear function. And this is why having multiple layers in a neural network doesn't let the neural network compute any more complex features or learn anything more complex than just a linear function. So in the general case, if you had a neural network with multiple layers like this, and say you were to use a linear activation function for all the hidden layers, and also use a linear activation function for the output layer, then it turns out this model
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5.5 Activation Functions | Why do we need activation functions? --[Machine Learning | Andrew Ng]: Let's take a look at why neural networks need activation functions and why they just don't work if we were to use the linear activation function in every neuron in the neural network. Recall this demand prediction example. What would happen if we were to use a linear activation function for all of the nodes in this neural network? It turns out that this big neural network will become no different than just linear regression. And so this would defeat the entire purpose of using a neural network because it would then just not be able to fit anything more complex than the linear regression model that we learned about in the first course. Let's illustrate this with a simpler example. Let's look at the example of a neural network where the input x is just a number and we have one hidden unit with parameters w1 and b1 that outputs a1, which is here just a number. And then the second layer is the output layer and it has also just one output unit with parameters w2 and b2 and that outputs a2, which is also just a number, just a scalar, which is the output of the neural network f of x. Let's see what this neural network would do if we were to use the linear activation function g of z equals z everywhere. So to compute a1 as a function of x, the neural network would use a1 equals g of w1 times x plus b1, but g of z is equal to z, so this is just w1 times x plus b1. Then a2 is equal to w2 times a1 plus b2 because g of z equals z. And let me take this expression for a1 and substitute it in there. So that becomes w2 times w1 x plus b1 plus b2. And if we simplify, this becomes w2 w1 times x plus w2 b1 plus b2. And it turns out that if I were to set w equals w2 times w1 and set b equals this quantity over here, then what we've just shown is that a2 is equal to wx plus b. So w is just a linear function of the input x. And rather than using a neural network with one hidden layer and one output layer, we might as well have just used a linear regression model. If you're familiar with linear algebra, this result comes from the fact that a linear function of a linear function is itself a linear function. And this is why having multiple layers in a neural network doesn't let the neural network compute any more complex features or learn anything more complex than just a linear function. So in the general case, if you had a neural network with multiple layers like this, and say you were to use a linear activation function for all the hidden layers, and also use a linear activation function for the output layer, then it turns out this model
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5.5 Activation Functions | Why do we need activation functions? --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=fK5YzGIc2u8
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will compute an output that is completely equivalent to linear regression. The output a4 can be expressed as a linear function of the input features x plus b. Or alternatively, if we were to still use a linear activation function for all the hidden layers, for these three hidden layers here, but we were to use a logistic activation function for the output layer, then it turns out you can show that this model becomes equivalent to logistic regression. And a4 in this case can be expressed as 1 over 1 plus e to the negative wx plus b for some values of w and b. And so this big neural network doesn't do anything that you can't also do with logistic regression. That's why a common rule of thumb is don't use the linear activation function in the hidden layers of your neural network. And in fact, I recommend typically using the ReLU activation function should do just fine. So that's why a neural network needs activation functions other than just the linear activation function everywhere. So far, you've learned to build neural networks for binary classification problems, where y is either 0 or 1, as well as for regression problems, where y can take negative or positive values or maybe just positive and non-negative values. In the next video, I'd like to share with you a generalization of what you've seen so far for classification. In particular, when y doesn't just take on two values, but may take on three or four or 10 or even more categorical values. Let's take a look at how you can build a neural network for that type of classification problem.
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5.5 Activation Functions | Why do we need activation functions? --[Machine Learning | Andrew Ng]: will compute an output that is completely equivalent to linear regression. The output a4 can be expressed as a linear function of the input features x plus b. Or alternatively, if we were to still use a linear activation function for all the hidden layers, for these three hidden layers here, but we were to use a logistic activation function for the output layer, then it turns out you can show that this model becomes equivalent to logistic regression. And a4 in this case can be expressed as 1 over 1 plus e to the negative wx plus b for some values of w and b. And so this big neural network doesn't do anything that you can't also do with logistic regression. That's why a common rule of thumb is don't use the linear activation function in the hidden layers of your neural network. And in fact, I recommend typically using the ReLU activation function should do just fine. So that's why a neural network needs activation functions other than just the linear activation function everywhere. So far, you've learned to build neural networks for binary classification problems, where y is either 0 or 1, as well as for regression problems, where y can take negative or positive values or maybe just positive and non-negative values. In the next video, I'd like to share with you a generalization of what you've seen so far for classification. In particular, when y doesn't just take on two values, but may take on three or four or 10 or even more categorical values. Let's take a look at how you can build a neural network for that type of classification problem.
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5.6 Multiclass Classification | Multiclass --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=hfskCwks_X8
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Multi-class classification refers to classification problems where you can have more than just two possible output labels, so not just zero or one. Let's take a look at what that means. For the handwritten digit classification problems we've looked at so far, we were just trying to distinguish between the handwritten digits zero and one. But if you're trying to read postal codes or zip codes on an envelope, well there are actually 10 possible digits you might want to recognize. Or alternatively, in the first course you saw the example if you're trying to classify whether a patient may have any of three or five different possible diseases, that too would be a multi-class classification problem. For one thing I've worked on a lot is visual defect inspection of parts manufactured in the factory, where you might look at a picture of a pill that a pharmaceutical company has manufactured and try to figure out does it have a scratch defect or discoloration defect or a chip defect. And this would again be multiple classes of multiple different types of defects that you could classify this pill as having. So a multi-class classification problem is still a classification problem in that Y can take on only a small number of discrete categories, it's not any number, but now Y can take on more than just two possible values. So whereas previously for binary classification you may have had a data set like this with features X1 and X2, in which case logistic regression would fit a model to estimate what's the probability of Y being 1 given the features X because Y was either 0 or 1. With multi-class classification problems you would instead have a data set that maybe looks like this, where we have four classes where the O's represent one class, the X's represent another class, the triangles represent the third class, and the squares represent the fourth class. And instead of just estimating the chance of Y being equal to 1, we'll now want to estimate what's the chance of Y is equal to 1, or what's the chance of Y is equal to 2, or what's the chance of Y is equal to 3, or the chance of Y being equal to 4. And it turns out that the algorithm you learn about in the next video can learn a decision boundary that maybe looks like this, that divides the space X1 and X2 into four categories rather than just two categories. So that's the definition of the multi-class classification problem. In the next video we'll look at the softmax regression algorithm, which is a generalization of the logistic regression algorithm, and using that you'll be able to carry out multi-class classification problems. And after that we'll take softmax regression and fit it into a new neural network so that you also be able to train a neural network to carry out multi-class classification problems. Let's go on to the next video.
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5.6 Multiclass Classification | Multiclass --[Machine Learning | Andrew Ng]: Multi-class classification refers to classification problems where you can have more than just two possible output labels, so not just zero or one. Let's take a look at what that means. For the handwritten digit classification problems we've looked at so far, we were just trying to distinguish between the handwritten digits zero and one. But if you're trying to read postal codes or zip codes on an envelope, well there are actually 10 possible digits you might want to recognize. Or alternatively, in the first course you saw the example if you're trying to classify whether a patient may have any of three or five different possible diseases, that too would be a multi-class classification problem. For one thing I've worked on a lot is visual defect inspection of parts manufactured in the factory, where you might look at a picture of a pill that a pharmaceutical company has manufactured and try to figure out does it have a scratch defect or discoloration defect or a chip defect. And this would again be multiple classes of multiple different types of defects that you could classify this pill as having. So a multi-class classification problem is still a classification problem in that Y can take on only a small number of discrete categories, it's not any number, but now Y can take on more than just two possible values. So whereas previously for binary classification you may have had a data set like this with features X1 and X2, in which case logistic regression would fit a model to estimate what's the probability of Y being 1 given the features X because Y was either 0 or 1. With multi-class classification problems you would instead have a data set that maybe looks like this, where we have four classes where the O's represent one class, the X's represent another class, the triangles represent the third class, and the squares represent the fourth class. And instead of just estimating the chance of Y being equal to 1, we'll now want to estimate what's the chance of Y is equal to 1, or what's the chance of Y is equal to 2, or what's the chance of Y is equal to 3, or the chance of Y being equal to 4. And it turns out that the algorithm you learn about in the next video can learn a decision boundary that maybe looks like this, that divides the space X1 and X2 into four categories rather than just two categories. So that's the definition of the multi-class classification problem. In the next video we'll look at the softmax regression algorithm, which is a generalization of the logistic regression algorithm, and using that you'll be able to carry out multi-class classification problems. And after that we'll take softmax regression and fit it into a new neural network so that you also be able to train a neural network to carry out multi-class classification problems. Let's go on to the next video.
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5.7 Multiclass Classification | Softmax --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=pzxxgEZkdLM
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The softmax regression algorithm is a generalization of logistic regression, which is a binary classification algorithm to the multi-cost classification context. Let's take a look at how it works. Recall that logistic regression applies when y can take on two possible output values, either 0 or 1. And the way it computes its output is, you would first calculate z equals w dot product with x plus b, and then you would compute what I'm going to call here a equals g of z, which is a sigmoid function applied to z. And we interpreted this as logistic regression's estimate of the probability of y being equal to 1, given those input features x. Now, quick quiz question. If the probability of y equals 1 is 0.71, then what is the probability that y is equal to 0? Well, the chance of y being 1 and the chance of y being 0, they've got to add up to 1, right? So there's a 71% chance of it being 1. There has to be a 29% or a 0.29 chance of it being equal to 0. So to embellish logistic regression a little bit in order to set us up for the generalization to softmax regression, I'm going to think of logistic regression as actually computing two numbers. First, a 1, which is this quantity that we had previously of the chance of y being equal to 1, given x, and second, I'm going to think of logistic regression as also computing a 2, which is 1 minus this, which is just the chance of y being equal to 0, given the input features x. And so a 1 and a 2, of course, have to add up to 1. Let's now generalize this to softmax regression. And I'm going to do this with a concrete example of when y can take on four possible outputs. So y can take on the values 1, 2, 3, or 4. Here's what softmax regression will do. It will compute z1 as w1 dot product with x plus b1, and then z2 equals w2 dot product with x plus b2, and so on for z3 and z4. Here w1, w2, w3, w4, as well as b1, b2, b3, b4, these are the parameters of softmax regression. Next, here's the formula for softmax regression. We'll compute a1 equals e to the z1 divided by e to the z1 plus e to the z2 plus e to the z3 plus e to the z4. And a1 will be interpreted as the average estimate of the chance of y being equal to 1, given the input features x. Then the formula for softmax regression will compute a2 equals e to the z2 divided by the same denominator, e to the z1 plus e to the z2 plus e to the z3 plus e to the z4, and will interpret a2 as the average estimate of the chance that y is equal to 2, given the input features
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5.7 Multiclass Classification | Softmax --[Machine Learning | Andrew Ng]: The softmax regression algorithm is a generalization of logistic regression, which is a binary classification algorithm to the multi-cost classification context. Let's take a look at how it works. Recall that logistic regression applies when y can take on two possible output values, either 0 or 1. And the way it computes its output is, you would first calculate z equals w dot product with x plus b, and then you would compute what I'm going to call here a equals g of z, which is a sigmoid function applied to z. And we interpreted this as logistic regression's estimate of the probability of y being equal to 1, given those input features x. Now, quick quiz question. If the probability of y equals 1 is 0.71, then what is the probability that y is equal to 0? Well, the chance of y being 1 and the chance of y being 0, they've got to add up to 1, right? So there's a 71% chance of it being 1. There has to be a 29% or a 0.29 chance of it being equal to 0. So to embellish logistic regression a little bit in order to set us up for the generalization to softmax regression, I'm going to think of logistic regression as actually computing two numbers. First, a 1, which is this quantity that we had previously of the chance of y being equal to 1, given x, and second, I'm going to think of logistic regression as also computing a 2, which is 1 minus this, which is just the chance of y being equal to 0, given the input features x. And so a 1 and a 2, of course, have to add up to 1. Let's now generalize this to softmax regression. And I'm going to do this with a concrete example of when y can take on four possible outputs. So y can take on the values 1, 2, 3, or 4. Here's what softmax regression will do. It will compute z1 as w1 dot product with x plus b1, and then z2 equals w2 dot product with x plus b2, and so on for z3 and z4. Here w1, w2, w3, w4, as well as b1, b2, b3, b4, these are the parameters of softmax regression. Next, here's the formula for softmax regression. We'll compute a1 equals e to the z1 divided by e to the z1 plus e to the z2 plus e to the z3 plus e to the z4. And a1 will be interpreted as the average estimate of the chance of y being equal to 1, given the input features x. Then the formula for softmax regression will compute a2 equals e to the z2 divided by the same denominator, e to the z1 plus e to the z2 plus e to the z3 plus e to the z4, and will interpret a2 as the average estimate of the chance that y is equal to 2, given the input features
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5.7 Multiclass Classification | Softmax --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=pzxxgEZkdLM
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x. And similarly for a3, where here the numerator is now e to the z3 divided by the same denominator, that's the estimated chance of y being equal to 3, and similarly a4 takes on this expression. Whereas on the left, we wrote down the specification for the logistic regression model, these equations on the right are our specification for the softmax regression model. It has parameters w1 through w4 and b1 through b4, and if you can learn appropriate choices for all these parameters, then this gives you a way of predicting what's the chance of y being 1, 2, 3 or 4, given a set of input features x. Quick quiz, let's say you run softmax regression on a new input x, and you find that a1 is 0.30, a2 is 0.20, a3 is 0.15. What do you think a4 will be? Why don't you take a look at this quiz and see if you can figure out the right answer. So you might have realized that because the chance of y taking on the values of 1, 2, 3 or 4, they have to add up to 1, a4, the chance of y being equal to 4, has to be 0.15. So 0.35, which is 1 minus 0.3 minus 0.2 minus 0.15. So here I wrote down the formulas for softmax regression in the case of 4 possible outputs, and let's now write down the formula for the general case for softmax regression. In the general case, y can take on n possible values, so y can be 1, 2, 3 and so on up to n. In that case, softmax regression will compute zj equals wj dot product with x plus bj, where now the parameters of softmax regression are w1, w2 through wn, as well as b1, b2 through bn. And then finally it will compute aj equals e to the zj divided by sum from k equals 1 to n of e to the z sub k. Well here I'm using another variable k to index the summation, because here j refers to a specific fixed number like j equals 1. aj is interpreted as the model's estimate that y is equal to j given the input features x. And notice that by construction of this formula, if you add up a1, a2 all the way through an, these numbers always will end up adding up to 1. So we specified how you would compute the softmax regression model. And I won't prove it in this video, but it turns out that if you apply softmax regression with n equals 2, so there are only two possible output classes, then softmax regression ends up computing basically the same thing as logistic regression. The parameters end up being a little bit different, but it ends up reducing to a logistic regression model. But that's why the softmax regression model is a generalization of logistic regression. Having defined how softmax regression computes its outputs, let's now
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5.7 Multiclass Classification | Softmax --[Machine Learning | Andrew Ng]: x. And similarly for a3, where here the numerator is now e to the z3 divided by the same denominator, that's the estimated chance of y being equal to 3, and similarly a4 takes on this expression. Whereas on the left, we wrote down the specification for the logistic regression model, these equations on the right are our specification for the softmax regression model. It has parameters w1 through w4 and b1 through b4, and if you can learn appropriate choices for all these parameters, then this gives you a way of predicting what's the chance of y being 1, 2, 3 or 4, given a set of input features x. Quick quiz, let's say you run softmax regression on a new input x, and you find that a1 is 0.30, a2 is 0.20, a3 is 0.15. What do you think a4 will be? Why don't you take a look at this quiz and see if you can figure out the right answer. So you might have realized that because the chance of y taking on the values of 1, 2, 3 or 4, they have to add up to 1, a4, the chance of y being equal to 4, has to be 0.15. So 0.35, which is 1 minus 0.3 minus 0.2 minus 0.15. So here I wrote down the formulas for softmax regression in the case of 4 possible outputs, and let's now write down the formula for the general case for softmax regression. In the general case, y can take on n possible values, so y can be 1, 2, 3 and so on up to n. In that case, softmax regression will compute zj equals wj dot product with x plus bj, where now the parameters of softmax regression are w1, w2 through wn, as well as b1, b2 through bn. And then finally it will compute aj equals e to the zj divided by sum from k equals 1 to n of e to the z sub k. Well here I'm using another variable k to index the summation, because here j refers to a specific fixed number like j equals 1. aj is interpreted as the model's estimate that y is equal to j given the input features x. And notice that by construction of this formula, if you add up a1, a2 all the way through an, these numbers always will end up adding up to 1. So we specified how you would compute the softmax regression model. And I won't prove it in this video, but it turns out that if you apply softmax regression with n equals 2, so there are only two possible output classes, then softmax regression ends up computing basically the same thing as logistic regression. The parameters end up being a little bit different, but it ends up reducing to a logistic regression model. But that's why the softmax regression model is a generalization of logistic regression. Having defined how softmax regression computes its outputs, let's now
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5.7 Multiclass Classification | Softmax --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=pzxxgEZkdLM
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take a look at how to specify the cost function for softmax regression. Recall for logistic regression, this is what we had. We said z is equal to this, and then I wrote earlier that a1 is g of z, it was interpreted as the probability that y is equal to 1. And we also wrote a2 is the probability that y is equal to clause 0. So previously we had written the loss of logistic regression as negative y log a1 minus 1 minus y log 1 minus a1. But 1 minus a1 is also equal to just a2, because a2 is 1 minus a1 according to this expression over here. So I can rewrite or simplify the loss for logistic regression a little bit to be negative y log a1 minus 1 minus y log of a2. And in other words, the loss if y is equal to 1 is negative log a1, and if y is equal to 0, then the loss is negative log a2. And then same as before, the cost function for all the parameters in the model is the average loss, average over the entire training set. So that was the cost function for logistic regression. Let's write down the cost function that is conventionally used for softmax regression. Recall that these are the equations we use for softmax regression. The loss we're going to use for softmax regression is just this. The loss for if the algorithm outputs a1 through an, and the ground truth label is y, is if y equals 1, the loss is negative log a1, so it's negative log of the probability that it thought y was equal to 1. Or if y is equal to 2, then the loss I'm going to define as negative log a2. So if y is equal to 2, the loss of the algorithm on this example is negative log of the probability it thought y was equal to 2. And so on all the way down to if y is equal to n, then the loss is negative log of an. And to illustrate what this is doing, if y is equal to j, then the loss is negative log of aj, and that's what this function looks like. Negative log of aj is a curve that looks like this. And so if aj was very close to 1, then you'd be on this part of the curve and the loss would be very small. But if it thought, say, aj had only a 50% chance, then the loss gets a little bit bigger. And the smaller aj is, the bigger the loss. And so this incentivizes the algorithm to make aj as large as possible, as close to 1 as possible, because whatever the actual value y was, you want the algorithm to say, hopefully, that the chance of y being that value was pretty large. Notice that in this loss function, y in each training example can take on
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5.7 Multiclass Classification | Softmax --[Machine Learning | Andrew Ng]: take a look at how to specify the cost function for softmax regression. Recall for logistic regression, this is what we had. We said z is equal to this, and then I wrote earlier that a1 is g of z, it was interpreted as the probability that y is equal to 1. And we also wrote a2 is the probability that y is equal to clause 0. So previously we had written the loss of logistic regression as negative y log a1 minus 1 minus y log 1 minus a1. But 1 minus a1 is also equal to just a2, because a2 is 1 minus a1 according to this expression over here. So I can rewrite or simplify the loss for logistic regression a little bit to be negative y log a1 minus 1 minus y log of a2. And in other words, the loss if y is equal to 1 is negative log a1, and if y is equal to 0, then the loss is negative log a2. And then same as before, the cost function for all the parameters in the model is the average loss, average over the entire training set. So that was the cost function for logistic regression. Let's write down the cost function that is conventionally used for softmax regression. Recall that these are the equations we use for softmax regression. The loss we're going to use for softmax regression is just this. The loss for if the algorithm outputs a1 through an, and the ground truth label is y, is if y equals 1, the loss is negative log a1, so it's negative log of the probability that it thought y was equal to 1. Or if y is equal to 2, then the loss I'm going to define as negative log a2. So if y is equal to 2, the loss of the algorithm on this example is negative log of the probability it thought y was equal to 2. And so on all the way down to if y is equal to n, then the loss is negative log of an. And to illustrate what this is doing, if y is equal to j, then the loss is negative log of aj, and that's what this function looks like. Negative log of aj is a curve that looks like this. And so if aj was very close to 1, then you'd be on this part of the curve and the loss would be very small. But if it thought, say, aj had only a 50% chance, then the loss gets a little bit bigger. And the smaller aj is, the bigger the loss. And so this incentivizes the algorithm to make aj as large as possible, as close to 1 as possible, because whatever the actual value y was, you want the algorithm to say, hopefully, that the chance of y being that value was pretty large. Notice that in this loss function, y in each training example can take on
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5.7 Multiclass Classification | Softmax --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=pzxxgEZkdLM
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only one value. And so you end up computing this negative log of aj only for one value of aj, which is whatever was the actual value of y equals j in that particular training example. For example, if y was equal to 2, you end up computing negative log of a2, but not any of the other negative log of a1 or the other terms here. So that's the form of the model, as well as the cost function for softmax regression. And if you were to train this model, you can start to build multi-class classification algorithms. And what we'd like to do next is take this softmax regression model and fit it into a neural network so that you're going to do something even better, which is to train a neural network for multi-class classification. Let's go through that in the next video.
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5.7 Multiclass Classification | Softmax --[Machine Learning | Andrew Ng]: only one value. And so you end up computing this negative log of aj only for one value of aj, which is whatever was the actual value of y equals j in that particular training example. For example, if y was equal to 2, you end up computing negative log of a2, but not any of the other negative log of a1 or the other terms here. So that's the form of the model, as well as the cost function for softmax regression. And if you were to train this model, you can start to build multi-class classification algorithms. And what we'd like to do next is take this softmax regression model and fit it into a neural network so that you're going to do something even better, which is to train a neural network for multi-class classification. Let's go through that in the next video.
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5.8 Multiclass Classification | Neural Network with Softmax output --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=24QO9iNXvWs
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In order to build a neural network that can carry out multi-class classification, we're going to take the Softmax regression model and put it into essentially the output layer of a neural network. Let's take a look at how to do that. Previously, when we were doing handwritten digit recognition with just two clauses, we used a neural network with this architecture. If you now want to do handwritten digit classification with 10 clauses, all the digits from 0 to 9, then we're going to change this neural network to have 10 output units, like so. And this new output layer will be a Softmax output layer. So sometimes we'll say this neural network has a Softmax output or that this output layer is a Softmax layer. And the way forward propagation works in this neural network is, given an input x, a1 gets computed exactly the same as before, and then a2, deactivations for the second hidden layer, also get computed exactly the same as before. And we now have to compute deactivations for this output layer. That is a3. This is how it works. If you have 10 output clauses, we will compute z1, z2, through z10 using these expressions. So this is actually very similar to what we had previously for the formula you used to compute z. z1 is w1.product with a2, deactivations from the previous layer, plus b1, and so on, for z1 through z10. Then a1 is equal to e to the z1 divided by e to the z1 plus dot dot dot plus up to e to the z10. And that's our estimate of the chance that y is equal to 1. And similarly for a2, and similarly all the way up to a10. And so this gives you your estimates of the chance of y being equal to 1, 2, and so on up through the 10th possible label for y. And just for completeness, if you want to indicate that these are the quantities associated with layer 3, technically I should add these superscript 3s there. It does make the notation a little bit more cluttered, but this makes explicit that this is, for example, the z31 value and this is the parameters associated with the first unit of layer 3 of this neural network. And with this, your softmax output layer now gives you estimates of the chance of y being any of these 10 possible output labels. I do want to mention that the softmax layer, or sometimes also called the softmax activation function, it is a little bit unusual in one respect compared to the other activation functions we've seen so far like sigmoid, ReLU, and linear, which is that when we're looking at sigmoid or ReLU or linear activation functions, a1 was a function of z1 and a2 was a function of z2 and only z2. In other words, to obtain the activation values, we could apply the activation function g, be it sigmoid or ReLU or
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5.8 Multiclass Classification | Neural Network with Softmax output --[Machine Learning | Andrew Ng]: In order to build a neural network that can carry out multi-class classification, we're going to take the Softmax regression model and put it into essentially the output layer of a neural network. Let's take a look at how to do that. Previously, when we were doing handwritten digit recognition with just two clauses, we used a neural network with this architecture. If you now want to do handwritten digit classification with 10 clauses, all the digits from 0 to 9, then we're going to change this neural network to have 10 output units, like so. And this new output layer will be a Softmax output layer. So sometimes we'll say this neural network has a Softmax output or that this output layer is a Softmax layer. And the way forward propagation works in this neural network is, given an input x, a1 gets computed exactly the same as before, and then a2, deactivations for the second hidden layer, also get computed exactly the same as before. And we now have to compute deactivations for this output layer. That is a3. This is how it works. If you have 10 output clauses, we will compute z1, z2, through z10 using these expressions. So this is actually very similar to what we had previously for the formula you used to compute z. z1 is w1.product with a2, deactivations from the previous layer, plus b1, and so on, for z1 through z10. Then a1 is equal to e to the z1 divided by e to the z1 plus dot dot dot plus up to e to the z10. And that's our estimate of the chance that y is equal to 1. And similarly for a2, and similarly all the way up to a10. And so this gives you your estimates of the chance of y being equal to 1, 2, and so on up through the 10th possible label for y. And just for completeness, if you want to indicate that these are the quantities associated with layer 3, technically I should add these superscript 3s there. It does make the notation a little bit more cluttered, but this makes explicit that this is, for example, the z31 value and this is the parameters associated with the first unit of layer 3 of this neural network. And with this, your softmax output layer now gives you estimates of the chance of y being any of these 10 possible output labels. I do want to mention that the softmax layer, or sometimes also called the softmax activation function, it is a little bit unusual in one respect compared to the other activation functions we've seen so far like sigmoid, ReLU, and linear, which is that when we're looking at sigmoid or ReLU or linear activation functions, a1 was a function of z1 and a2 was a function of z2 and only z2. In other words, to obtain the activation values, we could apply the activation function g, be it sigmoid or ReLU or
|
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5.8 Multiclass Classification | Neural Network with Softmax output --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=24QO9iNXvWs
|
something else, element-wise to z1 and z2 and so on to get any of the a1 and a2 and a3 and a4. But with the softmax activation function, notice that a1 is a function of z1 and z2 and z3 all the way up to z10. So each of these activation values depends on all of the values of z. And this is a property that's a bit unique to the softmax output or the softmax activation function. I'll state it differently, if you want to compute a1 through a10, that is a function of z1 all the way up to z10 simultaneously. And this is unlike the other activation functions we've seen so far. Finally, let's look at how you would implement this in TensorFlow. If you want to implement the neural network that I've shown here on this slide, this is the code to do so. Similar as before, there are three steps to specifying and training the model. The first step is to tell TensorFlow to sequentially string together three layers. First layer is this 25 units with ReLU activation function, second layer, 15 units with ReLU activation function, and then the third layer, because there are now 10 output units, you want to output a1 through a10, so there are now 10 output units, and we'll tell TensorFlow to use the softmax activation function. And the cost function that you saw in the last video, TensorFlow calls that the sparse categorical cross entropy function. So I know this name is a bit of a mouthful, whereas for logistic regression, we had the binary cross entropy function. Here we're using the sparse categorical cross entropy function, and what sparse categorical refers to is that you still classify y into categories, so it's categorical, it takes on values from 1 to 10, and sparse refers to that y can only take on one of these 10 values, so each image is either 0 or 1 or 2 or so on up to 9, you're not going to see a picture that is simultaneously the number 2 and the number 7, so sparse refers to that each digit is only one of these categories. So that's why the loss function that you saw in the last video is called, in TensorFlow, the sparse categorical cross entropy loss function. And then the code for training the model is just the same as before, and if you use this code, you can train a neural network on a multi-class classification problem. Just one important note, if you use this code exactly as I've written here, it will work, but don't actually use this code, because it turns out that in TensorFlow, there's a better version of the code that makes TensorFlow work better. So even though the code shown in this slide works, don't use this code the way I've written it here, because in a later video this week, you'll see a different version, there's actually the recommended version of
| 500
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5.8 Multiclass Classification | Neural Network with Softmax output --[Machine Learning | Andrew Ng]: something else, element-wise to z1 and z2 and so on to get any of the a1 and a2 and a3 and a4. But with the softmax activation function, notice that a1 is a function of z1 and z2 and z3 all the way up to z10. So each of these activation values depends on all of the values of z. And this is a property that's a bit unique to the softmax output or the softmax activation function. I'll state it differently, if you want to compute a1 through a10, that is a function of z1 all the way up to z10 simultaneously. And this is unlike the other activation functions we've seen so far. Finally, let's look at how you would implement this in TensorFlow. If you want to implement the neural network that I've shown here on this slide, this is the code to do so. Similar as before, there are three steps to specifying and training the model. The first step is to tell TensorFlow to sequentially string together three layers. First layer is this 25 units with ReLU activation function, second layer, 15 units with ReLU activation function, and then the third layer, because there are now 10 output units, you want to output a1 through a10, so there are now 10 output units, and we'll tell TensorFlow to use the softmax activation function. And the cost function that you saw in the last video, TensorFlow calls that the sparse categorical cross entropy function. So I know this name is a bit of a mouthful, whereas for logistic regression, we had the binary cross entropy function. Here we're using the sparse categorical cross entropy function, and what sparse categorical refers to is that you still classify y into categories, so it's categorical, it takes on values from 1 to 10, and sparse refers to that y can only take on one of these 10 values, so each image is either 0 or 1 or 2 or so on up to 9, you're not going to see a picture that is simultaneously the number 2 and the number 7, so sparse refers to that each digit is only one of these categories. So that's why the loss function that you saw in the last video is called, in TensorFlow, the sparse categorical cross entropy loss function. And then the code for training the model is just the same as before, and if you use this code, you can train a neural network on a multi-class classification problem. Just one important note, if you use this code exactly as I've written here, it will work, but don't actually use this code, because it turns out that in TensorFlow, there's a better version of the code that makes TensorFlow work better. So even though the code shown in this slide works, don't use this code the way I've written it here, because in a later video this week, you'll see a different version, there's actually the recommended version of
|
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5.8 Multiclass Classification | Neural Network with Softmax output --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=24QO9iNXvWs
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implementing this that will work better, but we'll take a look at that in a later video. So now you know how to train a neural network with a softmax output layer with one caveat. There's a different version of the code that will make TensorFlow able to compute these probabilities much more accurately. Let's take a look at that in the next video, which will also show you the actual code that I recommend you use if you're training a softmax neural network. Let's go on to the next video.
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5.8 Multiclass Classification | Neural Network with Softmax output --[Machine Learning | Andrew Ng]: implementing this that will work better, but we'll take a look at that in a later video. So now you know how to train a neural network with a softmax output layer with one caveat. There's a different version of the code that will make TensorFlow able to compute these probabilities much more accurately. Let's take a look at that in the next video, which will also show you the actual code that I recommend you use if you're training a softmax neural network. Let's go on to the next video.
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5.9 Multiclass Classification | Improved implementation of softmax --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=Izt9Bn8HLUM
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The implementation that you saw in the last video of a neural network with a softmax layer will work okay, but there's an even better way to implement it. Let's take a look at what can go wrong with that implementation and also how to make it better. Let me show you two different ways of computing the same quantity in a computer. Option one, we can set x equals to 2 over 10,000. Option two, we can set x equals 1 plus 1 over 10,000 minus 1 minus 1 over 10,000. I wish you'd first compute this and then compute this and you take the difference. And if you simplify this expression, this turns out to be equal to 2 over 10,000. Let me illustrate this in this notebook. So first, let's set x equals 2 over 10,000 and print the result to a lot of decimal points of accuracy. Okay, that looks pretty good. Second, let me set x equals, I'm going to insist on computing 1 over 1 plus 10,000 and then subtract from that 1 minus 1 over 10,000 and let's print that out. And oh, okay, this looks a little bit off. As if there's some round off error. Because a computer has only a finite amount of memory to store each number, called a floating point number in this case, depending on how you decide to compute the value to over 10,000, the result can have more or less numerical round off error. And it turns out that while the way we have been computing the cost function for softmax is correct, there's a different way of formulating it that reduces these numerical round off errors, leading to more accurate computations within TensorFlow. Let me first explain this a little bit more detail using logistic regression. And then we will show how these ideas apply to improving our implementation of softmax. So first, let me illustrate these ideas using logistic regression. And then we'll move on to show how to improve your implementation of softmax as well. Recall that for logistic regression, if you want to compute the loss function, for a given example, you would first compute this output activation A, which is g of z, or 1 over 1 plus e is negative z. And then you compute the loss using this expression over here. And in fact, this is what the code would look like for a logistic output layer with this binary cross entropy loss. And for logistic regression, this works okay. And usually the numerical round off errors aren't that bad. But it turns out that if you allow TensorFlow to not have to compute A as an intermediate term, but instead, if you tell TensorFlow that the loss is this expression down here, and all I've done is I've taken A and expanded it into this expression down here, then TensorFlow can rearrange terms in this expression and come up with a more numerically accurate way to compute
| 500
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5.9 Multiclass Classification | Improved implementation of softmax --[Machine Learning | Andrew Ng]: The implementation that you saw in the last video of a neural network with a softmax layer will work okay, but there's an even better way to implement it. Let's take a look at what can go wrong with that implementation and also how to make it better. Let me show you two different ways of computing the same quantity in a computer. Option one, we can set x equals to 2 over 10,000. Option two, we can set x equals 1 plus 1 over 10,000 minus 1 minus 1 over 10,000. I wish you'd first compute this and then compute this and you take the difference. And if you simplify this expression, this turns out to be equal to 2 over 10,000. Let me illustrate this in this notebook. So first, let's set x equals 2 over 10,000 and print the result to a lot of decimal points of accuracy. Okay, that looks pretty good. Second, let me set x equals, I'm going to insist on computing 1 over 1 plus 10,000 and then subtract from that 1 minus 1 over 10,000 and let's print that out. And oh, okay, this looks a little bit off. As if there's some round off error. Because a computer has only a finite amount of memory to store each number, called a floating point number in this case, depending on how you decide to compute the value to over 10,000, the result can have more or less numerical round off error. And it turns out that while the way we have been computing the cost function for softmax is correct, there's a different way of formulating it that reduces these numerical round off errors, leading to more accurate computations within TensorFlow. Let me first explain this a little bit more detail using logistic regression. And then we will show how these ideas apply to improving our implementation of softmax. So first, let me illustrate these ideas using logistic regression. And then we'll move on to show how to improve your implementation of softmax as well. Recall that for logistic regression, if you want to compute the loss function, for a given example, you would first compute this output activation A, which is g of z, or 1 over 1 plus e is negative z. And then you compute the loss using this expression over here. And in fact, this is what the code would look like for a logistic output layer with this binary cross entropy loss. And for logistic regression, this works okay. And usually the numerical round off errors aren't that bad. But it turns out that if you allow TensorFlow to not have to compute A as an intermediate term, but instead, if you tell TensorFlow that the loss is this expression down here, and all I've done is I've taken A and expanded it into this expression down here, then TensorFlow can rearrange terms in this expression and come up with a more numerically accurate way to compute
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5.9 Multiclass Classification | Improved implementation of softmax --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=Izt9Bn8HLUM
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this loss function. And so whereas the original procedure was like insisting on computing as an intermediate value, 1 plus 1 over 10,000, and another intermediate value, 1 minus 1 over 10,000, and then manipulating these two to get 2 over 10,000, this original implementation was insisting on explicitly computing A as an intermediate quantity. But instead, by specifying this expression at the bottom directly as a loss function, it gives TensorFlow more flexibility in terms of how to compute this and whether or not it wants to compute A explicitly. And so the code you can use to do this is shown here. And what this does is it sets the output layer to just use a linear activation function, and it puts both the activation function, 1 over 1 plus e to the negative z, as well as this cross entropy loss into the specification of the loss function over here. And that's what this from logits equals true argument causes TensorFlow to do. And in case you're wondering what the logits are, it's basically this number z. So TensorFlow will compute z as an intermediate value, but it can rearrange terms to make this become computed more accurately. One downside of this code is it becomes a little bit less legible, but this causes TensorFlow to have a little bit less numerical round off error. Now in the case of logistic regression, either of these implementations actually works okay. But the numerical round off errors can get worse when it comes to softmax. Now let's take this idea and apply it to softmax regression. Recall what you saw in the last video was you compute the activations as follows. The activations is g of z1 through z10, where a1, for example, is e to the z1 divided by the sum of the e to the zj's. And then the loss was this, depending on what is the actual value of y is negative log of aj for one of the aj's. And so this was the code that we had to do this computation in two separate steps. But once again, if you instead specify that the loss is if y is equal to 1 is negative log of this formula, and so on, if y is equal to 10 is this formula, then this gives TensorFlow the ability to rearrange terms and compute this in a more numerically accurate way. Just to give you some intuition for why TensorFlow might want to do this, it turns out if one of the z's is really small, then e to a negative small number becomes very, very small. Or if one of the z's is a very large number, then e to the z can become a very, very large number. And by rearranging terms, TensorFlow can avoid some of these very small or very large numbers and therefore come up with a more accurate computation for the loss function. So the code for doing this is shown
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5.9 Multiclass Classification | Improved implementation of softmax --[Machine Learning | Andrew Ng]: this loss function. And so whereas the original procedure was like insisting on computing as an intermediate value, 1 plus 1 over 10,000, and another intermediate value, 1 minus 1 over 10,000, and then manipulating these two to get 2 over 10,000, this original implementation was insisting on explicitly computing A as an intermediate quantity. But instead, by specifying this expression at the bottom directly as a loss function, it gives TensorFlow more flexibility in terms of how to compute this and whether or not it wants to compute A explicitly. And so the code you can use to do this is shown here. And what this does is it sets the output layer to just use a linear activation function, and it puts both the activation function, 1 over 1 plus e to the negative z, as well as this cross entropy loss into the specification of the loss function over here. And that's what this from logits equals true argument causes TensorFlow to do. And in case you're wondering what the logits are, it's basically this number z. So TensorFlow will compute z as an intermediate value, but it can rearrange terms to make this become computed more accurately. One downside of this code is it becomes a little bit less legible, but this causes TensorFlow to have a little bit less numerical round off error. Now in the case of logistic regression, either of these implementations actually works okay. But the numerical round off errors can get worse when it comes to softmax. Now let's take this idea and apply it to softmax regression. Recall what you saw in the last video was you compute the activations as follows. The activations is g of z1 through z10, where a1, for example, is e to the z1 divided by the sum of the e to the zj's. And then the loss was this, depending on what is the actual value of y is negative log of aj for one of the aj's. And so this was the code that we had to do this computation in two separate steps. But once again, if you instead specify that the loss is if y is equal to 1 is negative log of this formula, and so on, if y is equal to 10 is this formula, then this gives TensorFlow the ability to rearrange terms and compute this in a more numerically accurate way. Just to give you some intuition for why TensorFlow might want to do this, it turns out if one of the z's is really small, then e to a negative small number becomes very, very small. Or if one of the z's is a very large number, then e to the z can become a very, very large number. And by rearranging terms, TensorFlow can avoid some of these very small or very large numbers and therefore come up with a more accurate computation for the loss function. So the code for doing this is shown
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5.9 Multiclass Classification | Improved implementation of softmax --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=Izt9Bn8HLUM
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here. In the output layer, we're now just using a linear activation function. So the output layer just computes z1 through z10. And this whole computation of the loss is then captured in the loss function over here, where again, we have the firm log of z equals true parameter. So once again, these two pieces of code do pretty much the same thing, except that the version that is recommended is more numerically accurate, although unfortunately, it is a little bit harder to read as well. So if you're reading someone else's code and you see this and you wonder what's going on, it's actually equivalent to the original implementation and these in concept, except that is more numerically accurate. The numerical round off errors for logistic regression aren't that bad, but it is recommended that you use this implementation down at the bottom instead. And conceptually, this code does the same thing as the first version that you had previously, except that it is a little bit more numerically accurate. Although the downside is maybe just a little bit harder to interpret as well. Now, there's just one more detail, which is that we've now changed the neural network to use a linear activation function rather than a softmax activation function. And so the neural network's final layer no longer outputs these probabilities a1 through a10, it is instead outputting z1 through z10. And I didn't talk about it in the case of logistic regression, but if you were combining the output logistic function with the loss function, then for logistic regression, you also have to change the code this way to take the output value and map it through the logistic function in order to actually get the probability. So you now know how to do multiclass classification with a softmax output layer, and also how to do it in a numerically stable way. Before wrapping up multiclass classification, I want to share with you one other type of classification problem called a multi-label classification problem. Let's talk about that in the next video.
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5.9 Multiclass Classification | Improved implementation of softmax --[Machine Learning | Andrew Ng]: here. In the output layer, we're now just using a linear activation function. So the output layer just computes z1 through z10. And this whole computation of the loss is then captured in the loss function over here, where again, we have the firm log of z equals true parameter. So once again, these two pieces of code do pretty much the same thing, except that the version that is recommended is more numerically accurate, although unfortunately, it is a little bit harder to read as well. So if you're reading someone else's code and you see this and you wonder what's going on, it's actually equivalent to the original implementation and these in concept, except that is more numerically accurate. The numerical round off errors for logistic regression aren't that bad, but it is recommended that you use this implementation down at the bottom instead. And conceptually, this code does the same thing as the first version that you had previously, except that it is a little bit more numerically accurate. Although the downside is maybe just a little bit harder to interpret as well. Now, there's just one more detail, which is that we've now changed the neural network to use a linear activation function rather than a softmax activation function. And so the neural network's final layer no longer outputs these probabilities a1 through a10, it is instead outputting z1 through z10. And I didn't talk about it in the case of logistic regression, but if you were combining the output logistic function with the loss function, then for logistic regression, you also have to change the code this way to take the output value and map it through the logistic function in order to actually get the probability. So you now know how to do multiclass classification with a softmax output layer, and also how to do it in a numerically stable way. Before wrapping up multiclass classification, I want to share with you one other type of classification problem called a multi-label classification problem. Let's talk about that in the next video.
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5.10 Mult-label Classification | Classification with multiple outputs (Optional) --[ML | Andrew Ng]
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https://www.youtube.com/watch?v=1yv8_S9Srcg
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You've learned about multi-class classification, where the output label y can be any one of two or potentially many more than two possible categories. There's a different type of classification problem called a multi-label classification problem, which is where, associated with each image, there could be multiple labels. Let me show you what I mean by that. If you're building a self-driving car or maybe a driver assistance system, then given a picture of what's in front of your car, you may want to ask the question, like, is there a car or at least one car, or is there a bus, or is there a pedestrian, or are there any pedestrians? In this case, there is a car, there is no bus, and there is at least one pedestrian, or in this second image, no cars, no buses, and yes pedestrians, and yes car, yes bus, and no pedestrians. So these are examples of multi-label classification problems because associated with a single input image x are three different labels corresponding to whether or not there are any cars, buses, or pedestrians in the image. So in this case, the target output y is actually a vector of three numbers, and this is as distinct from multi-class classification where, for, say, handwritten digit classification, y was just a single number even if that number could take on 10 different possible values. So how do you build a neural network for multi-label classification? One way to go about it is to just treat this as three completely separate machine learning problems. You could build one neural network to decide are there any cars, a second one to detect buses, and a third one to detect pedestrians, and that's actually not an unreasonable approach. Here's the first neural network to detect cars, second one to detect buses, third one to detect pedestrians, but there's another way to do this, which is to train a single neural network to simultaneously detect all three of cars, buses, and pedestrians, which is if your neural network architecture looks like this. That's your input x. First hidden layer outputs a1, second hidden layer outputs a2, and then the final output layer in this case will have three output neurons and will output a3, which is going to be a vector of three numbers. And because we're solving three binary classification problems, so is there a car, is there a bus, is there a pedestrian, you can use a sigmoid activation function for each of these three nodes in the output layer. And so a3 in this case will be a31, a32, and a33 corresponding to whether or not the learning algorithm thinks there's a car and or a bus and or pedestrians in the image. So multi-class classification and multi-label classification are sometimes confused with each other. And that's why in this video I want to share with you just a definition of multi-label classification problems as well so that depending on your application, you could
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5.10 Mult-label Classification | Classification with multiple outputs (Optional) --[ML | Andrew Ng]: You've learned about multi-class classification, where the output label y can be any one of two or potentially many more than two possible categories. There's a different type of classification problem called a multi-label classification problem, which is where, associated with each image, there could be multiple labels. Let me show you what I mean by that. If you're building a self-driving car or maybe a driver assistance system, then given a picture of what's in front of your car, you may want to ask the question, like, is there a car or at least one car, or is there a bus, or is there a pedestrian, or are there any pedestrians? In this case, there is a car, there is no bus, and there is at least one pedestrian, or in this second image, no cars, no buses, and yes pedestrians, and yes car, yes bus, and no pedestrians. So these are examples of multi-label classification problems because associated with a single input image x are three different labels corresponding to whether or not there are any cars, buses, or pedestrians in the image. So in this case, the target output y is actually a vector of three numbers, and this is as distinct from multi-class classification where, for, say, handwritten digit classification, y was just a single number even if that number could take on 10 different possible values. So how do you build a neural network for multi-label classification? One way to go about it is to just treat this as three completely separate machine learning problems. You could build one neural network to decide are there any cars, a second one to detect buses, and a third one to detect pedestrians, and that's actually not an unreasonable approach. Here's the first neural network to detect cars, second one to detect buses, third one to detect pedestrians, but there's another way to do this, which is to train a single neural network to simultaneously detect all three of cars, buses, and pedestrians, which is if your neural network architecture looks like this. That's your input x. First hidden layer outputs a1, second hidden layer outputs a2, and then the final output layer in this case will have three output neurons and will output a3, which is going to be a vector of three numbers. And because we're solving three binary classification problems, so is there a car, is there a bus, is there a pedestrian, you can use a sigmoid activation function for each of these three nodes in the output layer. And so a3 in this case will be a31, a32, and a33 corresponding to whether or not the learning algorithm thinks there's a car and or a bus and or pedestrians in the image. So multi-class classification and multi-label classification are sometimes confused with each other. And that's why in this video I want to share with you just a definition of multi-label classification problems as well so that depending on your application, you could
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5.10 Mult-label Classification | Classification with multiple outputs (Optional) --[ML | Andrew Ng]
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https://www.youtube.com/watch?v=1yv8_S9Srcg
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choose the right one for the job you want to do. So that's it for multi-label classification. I find that sometimes multi-class classification and multi-label classification are confused with each other, which is why I wanted to explicitly in this video share with you what is multi-label classification so that depending on your application, you can choose the right two for the job that you want to do. And that wraps up this section on multi-class and multi-label classification. In the next video, we'll start to look at some more advanced neural network concepts, including an optimization algorithm that is even better than gradient descent. So let's take a look at that algorithm in the next video because it'll help you to get your learning algorithms to learn much faster. So let's go on to the next video.
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5.10 Mult-label Classification | Classification with multiple outputs (Optional) --[ML | Andrew Ng]: choose the right one for the job you want to do. So that's it for multi-label classification. I find that sometimes multi-class classification and multi-label classification are confused with each other, which is why I wanted to explicitly in this video share with you what is multi-label classification so that depending on your application, you can choose the right two for the job that you want to do. And that wraps up this section on multi-class and multi-label classification. In the next video, we'll start to look at some more advanced neural network concepts, including an optimization algorithm that is even better than gradient descent. So let's take a look at that algorithm in the next video because it'll help you to get your learning algorithms to learn much faster. So let's go on to the next video.
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5.11 Additional Neural Network Concepts | Advanced Optimization --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=yo6aW-D7sCM
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Gradient descent is an optimization algorithm that is widely used in machine learning and was the foundation of many algorithms like linear regression and logistic regression and early implementations of neural networks. But it turns out that there are now some other optimization algorithms for minimizing the cost function that are even better than gradient descent. In this video, we'll take a look at an algorithm that can help you train your neural network much faster than gradient descent. Recall that this is the expression for one step of gradient descent. A parameter wj is updated as wj minus the learning rate alpha times this partial derivative term. How can we make this work even better? In this example, I plotted the cost function j using a contour plot comprising these ellipses. And so the minimum of this cost function is at the center of these ellipses down here. Now if you were to start gradient descent down here, one step of gradient descent, if alpha is small, may take you a little bit in that direction, then another step, then another step, then another step, then another step. And you notice that every single step of gradient descent is pretty much going in the same direction. And if you see this to be the case, you might wonder, well, why don't we make alpha bigger? Can we have an algorithm to automatically increase alpha to just make it take bigger steps and get to the minimum faster? There's an algorithm called the Adam algorithm that can do that. If it sees that the learning rate is too small, and we are just taking tiny little steps in a similar direction over and over, we should just make the learning rate alpha bigger. In contrast, here again is the same cost function. If we were starting here and had a relatively big learning rate alpha, then maybe one step of gradient descent takes us here, and the second step takes us here, third step, and the fourth step, and the fifth step, and the sixth step. And if you see gradient descent doing this, it's oscillating back and forth, you'd be tempted to say, well, why don't we make the learning rate smaller? And the Adam algorithm can also do that automatically. And with a smaller learning rate, you can then take a more smooth path toward the minimum of the cost function. So depending on how gradient descent is proceeding, sometimes you wish you had a bigger learning rate alpha, and sometimes you wish you had a smaller learning rate alpha. So the Adam algorithm can adjust the learning rate automatically. Adam stands for Adaptive Moment Estimation, or ADAM. And don't worry too much about what this name means, it's just what the authors had called this algorithm. But interestingly, the Adam algorithm doesn't use a single global learning rate alpha, it uses a different learning rates for every single parameter of your model. So if you have parameters
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5.11 Additional Neural Network Concepts | Advanced Optimization --[Machine Learning | Andrew Ng]: Gradient descent is an optimization algorithm that is widely used in machine learning and was the foundation of many algorithms like linear regression and logistic regression and early implementations of neural networks. But it turns out that there are now some other optimization algorithms for minimizing the cost function that are even better than gradient descent. In this video, we'll take a look at an algorithm that can help you train your neural network much faster than gradient descent. Recall that this is the expression for one step of gradient descent. A parameter wj is updated as wj minus the learning rate alpha times this partial derivative term. How can we make this work even better? In this example, I plotted the cost function j using a contour plot comprising these ellipses. And so the minimum of this cost function is at the center of these ellipses down here. Now if you were to start gradient descent down here, one step of gradient descent, if alpha is small, may take you a little bit in that direction, then another step, then another step, then another step, then another step. And you notice that every single step of gradient descent is pretty much going in the same direction. And if you see this to be the case, you might wonder, well, why don't we make alpha bigger? Can we have an algorithm to automatically increase alpha to just make it take bigger steps and get to the minimum faster? There's an algorithm called the Adam algorithm that can do that. If it sees that the learning rate is too small, and we are just taking tiny little steps in a similar direction over and over, we should just make the learning rate alpha bigger. In contrast, here again is the same cost function. If we were starting here and had a relatively big learning rate alpha, then maybe one step of gradient descent takes us here, and the second step takes us here, third step, and the fourth step, and the fifth step, and the sixth step. And if you see gradient descent doing this, it's oscillating back and forth, you'd be tempted to say, well, why don't we make the learning rate smaller? And the Adam algorithm can also do that automatically. And with a smaller learning rate, you can then take a more smooth path toward the minimum of the cost function. So depending on how gradient descent is proceeding, sometimes you wish you had a bigger learning rate alpha, and sometimes you wish you had a smaller learning rate alpha. So the Adam algorithm can adjust the learning rate automatically. Adam stands for Adaptive Moment Estimation, or ADAM. And don't worry too much about what this name means, it's just what the authors had called this algorithm. But interestingly, the Adam algorithm doesn't use a single global learning rate alpha, it uses a different learning rates for every single parameter of your model. So if you have parameters
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5.11 Additional Neural Network Concepts | Advanced Optimization --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=yo6aW-D7sCM
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w1 through w10, as well as b, then it actually has 11 learning rate parameters, alpha 1, alpha 2, all the way through alpha 10, for w1 through w10, as well as I'll call it alpha 11 for the parameter b. And the intuition behind the Adam algorithm is if a parameter wj or b seems to keep on moving in roughly the same direction, this is what we saw on the first example on the previous slide. But if it seems to keep on moving in roughly the same direction, let's increase the learning rate for that parameter. Let's go faster in that direction. Conversely, if a parameter keeps oscillating back and forth, this is what you saw in the second example on the previous slide, then let's not have it keep on oscillating or bouncing back and forth, let's reduce alpha j for that parameter a little bit. The details of how Adam does this are a bit complicated and beyond the scope of this course. But if you take some more advanced deep learning classes later, you may learn more about the details of this Adam algorithm. But in code, this is how you would implement it. The model is exactly the same as before. And the way you compile the model is very similar to what we had before, except that we now add one extra argument to the compile function, which is that we specify that the optimizer you want to use is tf.keris.optimizers.theAdamOptimizer. So the Adam optimization algorithm does need some default initial learning rate alpha. And in this example, I've set that initial learning rate to be 10 to the negative 3. But when you're using the Adam algorithm in practice, it's worth trying a few values for this initial, this default global learning rate. Try some larger and some smaller values to see what gives you the fastest learning performance. Compared to the original gradient descent algorithm that you had learned in the previous course though, the Adam algorithm, because it can adapt the learning rate a bit automatically, it is more robust to the exact choice of learning rate that you pick. Though it is still worth tuning this parameter a little bit to see if you can get somewhat faster learning. So that's it for the Adam optimization algorithm. It typically works much faster than gradient descent, and it's become a de facto standard in how practitioners train their neural networks. So if you're trying to decide what learning algorithm to use, what optimization algorithm to use to train your neural network, a safe choice would be to just use the Adam optimization algorithm. And most practitioners today will use Adam rather than the original gradient descent algorithm. And with this, I hope that your learning algorithms will be able to learn much more quickly. Now, in the next couple of videos, I'd like to touch on some more advanced concepts for neural networks. And in particular, in the next
| 500
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5.11 Additional Neural Network Concepts | Advanced Optimization --[Machine Learning | Andrew Ng]: w1 through w10, as well as b, then it actually has 11 learning rate parameters, alpha 1, alpha 2, all the way through alpha 10, for w1 through w10, as well as I'll call it alpha 11 for the parameter b. And the intuition behind the Adam algorithm is if a parameter wj or b seems to keep on moving in roughly the same direction, this is what we saw on the first example on the previous slide. But if it seems to keep on moving in roughly the same direction, let's increase the learning rate for that parameter. Let's go faster in that direction. Conversely, if a parameter keeps oscillating back and forth, this is what you saw in the second example on the previous slide, then let's not have it keep on oscillating or bouncing back and forth, let's reduce alpha j for that parameter a little bit. The details of how Adam does this are a bit complicated and beyond the scope of this course. But if you take some more advanced deep learning classes later, you may learn more about the details of this Adam algorithm. But in code, this is how you would implement it. The model is exactly the same as before. And the way you compile the model is very similar to what we had before, except that we now add one extra argument to the compile function, which is that we specify that the optimizer you want to use is tf.keris.optimizers.theAdamOptimizer. So the Adam optimization algorithm does need some default initial learning rate alpha. And in this example, I've set that initial learning rate to be 10 to the negative 3. But when you're using the Adam algorithm in practice, it's worth trying a few values for this initial, this default global learning rate. Try some larger and some smaller values to see what gives you the fastest learning performance. Compared to the original gradient descent algorithm that you had learned in the previous course though, the Adam algorithm, because it can adapt the learning rate a bit automatically, it is more robust to the exact choice of learning rate that you pick. Though it is still worth tuning this parameter a little bit to see if you can get somewhat faster learning. So that's it for the Adam optimization algorithm. It typically works much faster than gradient descent, and it's become a de facto standard in how practitioners train their neural networks. So if you're trying to decide what learning algorithm to use, what optimization algorithm to use to train your neural network, a safe choice would be to just use the Adam optimization algorithm. And most practitioners today will use Adam rather than the original gradient descent algorithm. And with this, I hope that your learning algorithms will be able to learn much more quickly. Now, in the next couple of videos, I'd like to touch on some more advanced concepts for neural networks. And in particular, in the next
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5.12 Additional Neural Network Concepts | Additional Layer Types --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=54TxZZpK5Ok
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All the neural network layers we've used so far have been the dense layer type in which every neuron in a layer gets as its inputs all the activations from the previous layer. And it turns out that just using the dense layer type, you can actually build some pretty powerful learning algorithms. And to help you build further intuition about what neural networks can do, it turns out that there are some other types of layers as well with other properties. In this video, I'd like to briefly touch on this and give you an example of a different type of neural network layer. Let's take a look. To recap, in the dense layer that we've been using, the activation of a neuron in, say, the second hidden layer is a function of every single activation value from the previous layer of A1. But it turns out that for some applications, someone designing a neural network may choose to use a different type of layer. One other layer type that you may see in some work is called a convolutional layer. Let me illustrate this with an example. So what I'm showing on the left is the input X, which is a handwritten digit 9. And what I'm going to do is construct a hidden layer, which will compute different activations as functions of this input image X. But here's something I can do. For the first hidden unit, which I've drawn in blue, rather than saying this neuron can look at all the pixels in this image, I might say this neuron can only look at the pixels in this little rectangular region. The second neuron, which I'm going to illustrate in magenta, is also not going to look at the entire input image X. Instead, it's only going to look at the pixels in a limited region of the image. And so on for the third neuron and the fourth neuron, and so on and so forth, down to the last neuron, which maybe looks only at that region of the image. So why might you want to do this? Why won't you let every neuron look at all the pixels, but instead look at only some of the pixels? Well, some of the benefits are, first, it speeds up computation. And second advantage is that a neural network that uses this type of layer called a convolutional layer can need less training data. Or alternatively, it can also be less prone to overfitting. You'd heard me talk a bit about overfitting in the previous course, but this is something that we'll dive into greater detail on next week as well, when we talk about practical tips for using learning algorithms. And this type of layer, where each neuron only looks at a region of the input image, is called a convolutional layer. It was a researcher, Yang Le Kun, who had figured out a lot of the details of how to get convolutional layers to work
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5.12 Additional Neural Network Concepts | Additional Layer Types --[Machine Learning | Andrew Ng]: All the neural network layers we've used so far have been the dense layer type in which every neuron in a layer gets as its inputs all the activations from the previous layer. And it turns out that just using the dense layer type, you can actually build some pretty powerful learning algorithms. And to help you build further intuition about what neural networks can do, it turns out that there are some other types of layers as well with other properties. In this video, I'd like to briefly touch on this and give you an example of a different type of neural network layer. Let's take a look. To recap, in the dense layer that we've been using, the activation of a neuron in, say, the second hidden layer is a function of every single activation value from the previous layer of A1. But it turns out that for some applications, someone designing a neural network may choose to use a different type of layer. One other layer type that you may see in some work is called a convolutional layer. Let me illustrate this with an example. So what I'm showing on the left is the input X, which is a handwritten digit 9. And what I'm going to do is construct a hidden layer, which will compute different activations as functions of this input image X. But here's something I can do. For the first hidden unit, which I've drawn in blue, rather than saying this neuron can look at all the pixels in this image, I might say this neuron can only look at the pixels in this little rectangular region. The second neuron, which I'm going to illustrate in magenta, is also not going to look at the entire input image X. Instead, it's only going to look at the pixels in a limited region of the image. And so on for the third neuron and the fourth neuron, and so on and so forth, down to the last neuron, which maybe looks only at that region of the image. So why might you want to do this? Why won't you let every neuron look at all the pixels, but instead look at only some of the pixels? Well, some of the benefits are, first, it speeds up computation. And second advantage is that a neural network that uses this type of layer called a convolutional layer can need less training data. Or alternatively, it can also be less prone to overfitting. You'd heard me talk a bit about overfitting in the previous course, but this is something that we'll dive into greater detail on next week as well, when we talk about practical tips for using learning algorithms. And this type of layer, where each neuron only looks at a region of the input image, is called a convolutional layer. It was a researcher, Yang Le Kun, who had figured out a lot of the details of how to get convolutional layers to work
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5.12 Additional Neural Network Concepts | Additional Layer Types --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=54TxZZpK5Ok
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and popularize their use. Let me illustrate in more detail a convolutional layer. And if you have multiple convolutional layers in a neural network, sometimes that's called a convolutional neural network. To illustrate the convolutional layer or convolutional neural network, on this slide, I'm going to use instead of a 2D image input, I'm going to use a one dimensional input. And the motivating example I'm going to use is classification of EKG signals or electrocardiograms. So if you put two electrodes on your chest, you will record voltages that look like this that correspond to your heartbeat. This is actually something that my Stanford research group did research on. We're actually reading EKG signals that actually look like this to try to diagnose if a patient may have a heart issue. So an EKG signal, an electrocardiogram, ECG in some places, EKG in some places, is just a list of numbers corresponding to the height of the surface at different points in time. So you may have, say, 100 numbers corresponding to the height of this curve at 100 different points of time. And the learning task is, given this time series, given this EKG signal, to classify, say whether this patient has a heart disease or some diagnosable heart condition, here's what a convolutional neural network might do. So I'm going to take the EKG signal and rotate it 90 degrees to lay it on the side. And so we have here 100 inputs, X1, X2, all the way through X100, like so. And when I construct the first hidden layer, instead of having the first hidden unit take as input all 100 numbers, let me have the first hidden unit look at only X1 through X20. So that corresponds to looking at just a small window of this EKG signal. The second hidden layer, shown in a different color here, will look at X11 through X30, so it looks at a different window in this EKG signal. And the third hidden layer looks at another window, X21 through X40, and so on. And the final hidden unit in this example will look at X81 through X100, so it looks at a small window toward the end of this EKG time series. So this is a convolutional layer, because each unit in this layer looks at only a limited window of the input. Now this layer of the neural network has nine units. The next layer can also be a convolutional layer. So in the second hidden layer, let me architect my first unit, not to look at all nine activations from the previous layer, but to look at, say, just the first five activations from the previous layer. And then my second unit in this second hidden layer may look at just another five numbers, say A3 to A7. And the third and final hidden unit in this layer will only look at A5 through A9. And then maybe finally, these activations, A2, gets inputs to a
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5.12 Additional Neural Network Concepts | Additional Layer Types --[Machine Learning | Andrew Ng]: and popularize their use. Let me illustrate in more detail a convolutional layer. And if you have multiple convolutional layers in a neural network, sometimes that's called a convolutional neural network. To illustrate the convolutional layer or convolutional neural network, on this slide, I'm going to use instead of a 2D image input, I'm going to use a one dimensional input. And the motivating example I'm going to use is classification of EKG signals or electrocardiograms. So if you put two electrodes on your chest, you will record voltages that look like this that correspond to your heartbeat. This is actually something that my Stanford research group did research on. We're actually reading EKG signals that actually look like this to try to diagnose if a patient may have a heart issue. So an EKG signal, an electrocardiogram, ECG in some places, EKG in some places, is just a list of numbers corresponding to the height of the surface at different points in time. So you may have, say, 100 numbers corresponding to the height of this curve at 100 different points of time. And the learning task is, given this time series, given this EKG signal, to classify, say whether this patient has a heart disease or some diagnosable heart condition, here's what a convolutional neural network might do. So I'm going to take the EKG signal and rotate it 90 degrees to lay it on the side. And so we have here 100 inputs, X1, X2, all the way through X100, like so. And when I construct the first hidden layer, instead of having the first hidden unit take as input all 100 numbers, let me have the first hidden unit look at only X1 through X20. So that corresponds to looking at just a small window of this EKG signal. The second hidden layer, shown in a different color here, will look at X11 through X30, so it looks at a different window in this EKG signal. And the third hidden layer looks at another window, X21 through X40, and so on. And the final hidden unit in this example will look at X81 through X100, so it looks at a small window toward the end of this EKG time series. So this is a convolutional layer, because each unit in this layer looks at only a limited window of the input. Now this layer of the neural network has nine units. The next layer can also be a convolutional layer. So in the second hidden layer, let me architect my first unit, not to look at all nine activations from the previous layer, but to look at, say, just the first five activations from the previous layer. And then my second unit in this second hidden layer may look at just another five numbers, say A3 to A7. And the third and final hidden unit in this layer will only look at A5 through A9. And then maybe finally, these activations, A2, gets inputs to a
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5.12 Additional Neural Network Concepts | Additional Layer Types --[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=54TxZZpK5Ok
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sigmoid unit that does look at all three of these values of A2 in order to make a binary classification regarding presence or absence of heart disease. So this is an example of a neural network with the first hidden layer being a convolutional layer, the second hidden layer also being a convolutional layer, and then the output layer being a sigmoid layer. And it turns out that with convolutional layers, you have many architectural choices, such as how big is the window of inputs that a single neuron should look at, and how many neurons should each layer have. And by choosing those architectural parameters effectively, you can build new versions of neural networks that can be even more effective than the dense layer for some applications. To recap, that's it for the convolutional layer and convolutional neural networks. I'm not going to go deeper into convolutional networks in this class, and you don't need to know anything about them to do the homeworks and finish this class successfully. But I hope that you find this additional intuition that neural networks can have other types of layers as well to be useful. And in fact, if you sometimes hear about the latest cutting edge architectures, like a transformer model or an LSTM or an attention model, a lot of this research in neural networks even today pertains to researchers trying to invent new types of layers for neural networks and plugging these different types of layers together as building blocks to form even more complex and hopefully more powerful neural networks. So that's it for the required videos for this week. Thank you and congrats on sticking with me all the way through this. And I look forward to seeing you next week also, where we'll start to talk about practical advice for how you can build machine learning systems. I hope that the tips you learn next week will help you become much more effective at building useful machine learning systems. And I look forward also to seeing you next week.
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5.12 Additional Neural Network Concepts | Additional Layer Types --[Machine Learning | Andrew Ng]: sigmoid unit that does look at all three of these values of A2 in order to make a binary classification regarding presence or absence of heart disease. So this is an example of a neural network with the first hidden layer being a convolutional layer, the second hidden layer also being a convolutional layer, and then the output layer being a sigmoid layer. And it turns out that with convolutional layers, you have many architectural choices, such as how big is the window of inputs that a single neuron should look at, and how many neurons should each layer have. And by choosing those architectural parameters effectively, you can build new versions of neural networks that can be even more effective than the dense layer for some applications. To recap, that's it for the convolutional layer and convolutional neural networks. I'm not going to go deeper into convolutional networks in this class, and you don't need to know anything about them to do the homeworks and finish this class successfully. But I hope that you find this additional intuition that neural networks can have other types of layers as well to be useful. And in fact, if you sometimes hear about the latest cutting edge architectures, like a transformer model or an LSTM or an attention model, a lot of this research in neural networks even today pertains to researchers trying to invent new types of layers for neural networks and plugging these different types of layers together as building blocks to form even more complex and hopefully more powerful neural networks. So that's it for the required videos for this week. Thank you and congrats on sticking with me all the way through this. And I look forward to seeing you next week also, where we'll start to talk about practical advice for how you can build machine learning systems. I hope that the tips you learn next week will help you become much more effective at building useful machine learning systems. And I look forward also to seeing you next week.
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6.1 Advice for applying machine learning | Deciding what to try next -[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=Y66jLs9ubsY
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Hi and welcome back. By now you've seen a lot of different learning algorithms including linear regression, logistic regression, even deep learning or neural networks, and next week you'll see decision trees as well. So you now have a lot of powerful tools of machine learning, but how do you use these tools effectively? I've seen teams sometimes take six months to build a machine learning system that I think a more skilled team could have taken or done in just a couple of weeks. And the efficiency of how quickly you can get a machine learning system to work well will depend to a large part on how well you can repeatedly make good decisions about what to do next in the course of a machine learning project. So in this week I hope to share with you a number of tips on how to make decisions about what to do next in a machine learning project that I hope will end up saving you a lot of time. So let's take a look at some advice on how to build machine learning systems. Let's start with an example. Say you've implemented regularized linear regression to predict housing prices. So you have the usual cost function for your learning algorithm, squared error plus this regularization term. But if you train the model and find that it makes unacceptably large errors in its predictions, what do you try next? When you're building a machine learning algorithm, there are usually a lot of different things you could try. For example, you could decide to get more training examples since it seems like having more data should help, right? Or maybe you think maybe you have too many features, you could try a smaller set of features. Or maybe you want to get additional features such as find additional properties of the houses to toss into your data. And maybe that will help it to do better. Or you might take the existing features x1, x2, and so on and try adding polynomial features like x1 squared, x2 squared, x1 x2 and so on. Or you might wonder if the value of lambda is chosen well and you might say, maybe it's too big, I want to decrease it. Or you may say, oh, maybe it's too small, I want to try increasing it. So on any given machine learning application, it will often turn out that some of these things could be fruitful and some of these things not fruitful. And a key to being effective at how you build a machine learning algorithm will be if you can find a way to make good choices about where to invest your time. For example, I have seen teams spend literally many, many months collecting more training examples, thinking that more training data has got to help. But it turns out sometimes it helps a lot and sometimes it doesn't. So in this week, you learn about how to carry out a
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6.1 Advice for applying machine learning | Deciding what to try next -[Machine Learning | Andrew Ng]: Hi and welcome back. By now you've seen a lot of different learning algorithms including linear regression, logistic regression, even deep learning or neural networks, and next week you'll see decision trees as well. So you now have a lot of powerful tools of machine learning, but how do you use these tools effectively? I've seen teams sometimes take six months to build a machine learning system that I think a more skilled team could have taken or done in just a couple of weeks. And the efficiency of how quickly you can get a machine learning system to work well will depend to a large part on how well you can repeatedly make good decisions about what to do next in the course of a machine learning project. So in this week I hope to share with you a number of tips on how to make decisions about what to do next in a machine learning project that I hope will end up saving you a lot of time. So let's take a look at some advice on how to build machine learning systems. Let's start with an example. Say you've implemented regularized linear regression to predict housing prices. So you have the usual cost function for your learning algorithm, squared error plus this regularization term. But if you train the model and find that it makes unacceptably large errors in its predictions, what do you try next? When you're building a machine learning algorithm, there are usually a lot of different things you could try. For example, you could decide to get more training examples since it seems like having more data should help, right? Or maybe you think maybe you have too many features, you could try a smaller set of features. Or maybe you want to get additional features such as find additional properties of the houses to toss into your data. And maybe that will help it to do better. Or you might take the existing features x1, x2, and so on and try adding polynomial features like x1 squared, x2 squared, x1 x2 and so on. Or you might wonder if the value of lambda is chosen well and you might say, maybe it's too big, I want to decrease it. Or you may say, oh, maybe it's too small, I want to try increasing it. So on any given machine learning application, it will often turn out that some of these things could be fruitful and some of these things not fruitful. And a key to being effective at how you build a machine learning algorithm will be if you can find a way to make good choices about where to invest your time. For example, I have seen teams spend literally many, many months collecting more training examples, thinking that more training data has got to help. But it turns out sometimes it helps a lot and sometimes it doesn't. So in this week, you learn about how to carry out a
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6.1 Advice for applying machine learning | Deciding what to try next -[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=Y66jLs9ubsY
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set of diagnostics. And by diagnostic, I mean a test you can run to gain insight into what is or isn't working with a learning algorithm to gain guidance into improving performance. And some of these diagnostics will tell you things like, is it worth weeks or even months collecting more training data? Because if it is, then you can then go ahead and make the investment to get more data, which will hopefully lead to improved performance. Or if it isn't, then running that diagnostic could have saved you months of time. And one thing you see this week as well is that diagnostics can take time to implement, but running them can be a very good use of your time. So this week, we'll spend a lot of time talking about different diagnostics you can use to give you guidance on how to improve your learning algorithm's performance. But first, let's take a look at how to evaluate the performance of your learning algorithm. Let's go do that in the next video.
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6.1 Advice for applying machine learning | Deciding what to try next -[Machine Learning | Andrew Ng]: set of diagnostics. And by diagnostic, I mean a test you can run to gain insight into what is or isn't working with a learning algorithm to gain guidance into improving performance. And some of these diagnostics will tell you things like, is it worth weeks or even months collecting more training data? Because if it is, then you can then go ahead and make the investment to get more data, which will hopefully lead to improved performance. Or if it isn't, then running that diagnostic could have saved you months of time. And one thing you see this week as well is that diagnostics can take time to implement, but running them can be a very good use of your time. So this week, we'll spend a lot of time talking about different diagnostics you can use to give you guidance on how to improve your learning algorithm's performance. But first, let's take a look at how to evaluate the performance of your learning algorithm. Let's go do that in the next video.
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6.2 Evaluating and choosing models | Evaluating a model -[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=uNNx1Czrt1w
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Let's say you've trained a machine learning model. How do you evaluate that model's performance? You find that having a systematic way to evaluate performance will also help paint a clearer path for how to then improve this performance. So let's take a look at how to evaluate a model. Let's take the example of learning to predict housing prices as a function of the size. Let's say you've trained a model to predict housing prices as a function of the size x. And for the model that is a fourth order polynomial, so it features x, x squared, x cubed, and x to the fourth. Because we fit a fourth order polynomial to a training set with five data points, this fits the training data really well. But we don't like this model very much because even though the model fits the training data well, we think it will fail to generalize to new examples that aren't in the training set. So when you are predicting prices just a single feature the size of the house, you could plot the model like this and we could see that the curve is very wiggly. So we know this probably isn't a good model. But if you were fitting this model with even more features, say we had x1 the size of the house, number of bedrooms, the number of floors of the house, also the age of the home and years, then it becomes much harder to plot f because f is now a function of x1 through x4. And how do you plot a four dimensional function? So in order to tell if your model is doing well, especially for applications where you have more than one or two features, which makes it difficult to plot f of x, we need some more systematic way to evaluate how well your model is doing. Here's a technique that you can use. If you have a training set, and this is a small training set with just 10 examples listed here, rather than taking all your data to train the parameters w and p of the model, you can instead split the training set into two subsets. I'm going to draw a line here. And let's put 70% of the data into the first part. And I'm going to call that the training set. And the second part of the data, let's say 30% of the data, I'm going to put into a test set. And what we're going to do is train the models parameters on the training sets on this first 70% or so of the data, and then we'll test this performance on this test set. In notation, I'm going to use x1, y1, same as before, to denote the training examples through xm, ym, except that now to make explicit. So in this little example, we would have seven training examples. And to introduce one new piece of notation, I'm going to use m subscript train. m
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6.2 Evaluating and choosing models | Evaluating a model -[Machine Learning | Andrew Ng]: Let's say you've trained a machine learning model. How do you evaluate that model's performance? You find that having a systematic way to evaluate performance will also help paint a clearer path for how to then improve this performance. So let's take a look at how to evaluate a model. Let's take the example of learning to predict housing prices as a function of the size. Let's say you've trained a model to predict housing prices as a function of the size x. And for the model that is a fourth order polynomial, so it features x, x squared, x cubed, and x to the fourth. Because we fit a fourth order polynomial to a training set with five data points, this fits the training data really well. But we don't like this model very much because even though the model fits the training data well, we think it will fail to generalize to new examples that aren't in the training set. So when you are predicting prices just a single feature the size of the house, you could plot the model like this and we could see that the curve is very wiggly. So we know this probably isn't a good model. But if you were fitting this model with even more features, say we had x1 the size of the house, number of bedrooms, the number of floors of the house, also the age of the home and years, then it becomes much harder to plot f because f is now a function of x1 through x4. And how do you plot a four dimensional function? So in order to tell if your model is doing well, especially for applications where you have more than one or two features, which makes it difficult to plot f of x, we need some more systematic way to evaluate how well your model is doing. Here's a technique that you can use. If you have a training set, and this is a small training set with just 10 examples listed here, rather than taking all your data to train the parameters w and p of the model, you can instead split the training set into two subsets. I'm going to draw a line here. And let's put 70% of the data into the first part. And I'm going to call that the training set. And the second part of the data, let's say 30% of the data, I'm going to put into a test set. And what we're going to do is train the models parameters on the training sets on this first 70% or so of the data, and then we'll test this performance on this test set. In notation, I'm going to use x1, y1, same as before, to denote the training examples through xm, ym, except that now to make explicit. So in this little example, we would have seven training examples. And to introduce one new piece of notation, I'm going to use m subscript train. m
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6.2 Evaluating and choosing models | Evaluating a model -[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=uNNx1Czrt1w
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train is a number of training examples, which in this small data set is seven. So the subscript train just emphasizes if we're looking at the training set portion of the data. And for the test sets, I'm going to use the notation x1 subscript test, y1 subscript test to denote the first test example. And this goes all the way to xm test, subscript test, ym test, subscript test. And m test is the number of test examples, which in this case is three. And it's not uncommon to split your data set according to maybe a 70-30 split or 80-20 split with most of your data going into the training set and then a smaller fraction going into the test set. So in order to train a model and evaluate it, this is what it would look like if you're using linear regression with a squared error cost. Start off by fitting the parameters by minimizing the cost function j of wb. So this is a usual cost function. Minimize over wb of this squared error cost plus regularization term, longer over 2m times sum of the wj squared. And then to tell how well this model is doing, you would compute j test of wb, which is equal to the average error on the test set. And that's just equal to 1 over 2 times m test. That's the number of test examples. And then of sum over all the examples from i equals 1 to the number of test examples of the squared error on each of the test examples like so. So it's a prediction on the i-th test example input minus the actual price of the house on the i-th test example squared. And notice that the test error formula j test, it does not include that regularization term. And this will give you a sense of how well your learning algorithm is doing. One other quantity that's often useful to compute as well is the training error, which is a measure of how well your learning algorithm is doing on the training set. So let me define j train of wb to be equal to the average over the training sets 1 over 2m or 1 over 2m subscript train of sum over your training sets of this squared error term. And once again, this does not include the regularization term, unlike the cost function that you were minimizing to fit the parameters. So in a model like what we saw earlier in this video, j train of wb will be low because the average error on your training examples will be zero or very close to zero. So j train will be very close to zero. But if you had a few additional examples in your test set that the algorithm had not trained on then those test examples might look like these. And there's a large gap between what the algorithm is predicting as the estimated housing price and the actual
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6.2 Evaluating and choosing models | Evaluating a model -[Machine Learning | Andrew Ng]: train is a number of training examples, which in this small data set is seven. So the subscript train just emphasizes if we're looking at the training set portion of the data. And for the test sets, I'm going to use the notation x1 subscript test, y1 subscript test to denote the first test example. And this goes all the way to xm test, subscript test, ym test, subscript test. And m test is the number of test examples, which in this case is three. And it's not uncommon to split your data set according to maybe a 70-30 split or 80-20 split with most of your data going into the training set and then a smaller fraction going into the test set. So in order to train a model and evaluate it, this is what it would look like if you're using linear regression with a squared error cost. Start off by fitting the parameters by minimizing the cost function j of wb. So this is a usual cost function. Minimize over wb of this squared error cost plus regularization term, longer over 2m times sum of the wj squared. And then to tell how well this model is doing, you would compute j test of wb, which is equal to the average error on the test set. And that's just equal to 1 over 2 times m test. That's the number of test examples. And then of sum over all the examples from i equals 1 to the number of test examples of the squared error on each of the test examples like so. So it's a prediction on the i-th test example input minus the actual price of the house on the i-th test example squared. And notice that the test error formula j test, it does not include that regularization term. And this will give you a sense of how well your learning algorithm is doing. One other quantity that's often useful to compute as well is the training error, which is a measure of how well your learning algorithm is doing on the training set. So let me define j train of wb to be equal to the average over the training sets 1 over 2m or 1 over 2m subscript train of sum over your training sets of this squared error term. And once again, this does not include the regularization term, unlike the cost function that you were minimizing to fit the parameters. So in a model like what we saw earlier in this video, j train of wb will be low because the average error on your training examples will be zero or very close to zero. So j train will be very close to zero. But if you had a few additional examples in your test set that the algorithm had not trained on then those test examples might look like these. And there's a large gap between what the algorithm is predicting as the estimated housing price and the actual
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6.2 Evaluating and choosing models | Evaluating a model -[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=uNNx1Czrt1w
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value of those housing prices. And so j tests will be high. So seeing that j test is high on this model gives you a way to realize that even though it does great on the training set is actually not so good at generalizing to new examples to new data points that were not in the training set. So that was regression with squared error costs. Now let's take a look at how you'd apply this procedure to a classification problem. For example, if you were classifying between handwritten digits that are either zero or one. So same as before, you fit the parameters by minimizing the cost function to find the parameters wb. For example, if you were training logistic regression, then this would be the cost function j of wb where this is the usual logistic loss function and then plus also the regularization term and to compute the test error. J test is then the average over your test examples. That's that 30% of the data that wasn't in the training set of the logistic loss on your test set. And the training error you could also compute using this formula is the average logistic loss on your training data that the algorithm was using to minimize the cost function j of wb. Well what I describe here will work okay for figuring out if your learning algorithm is doing well. I've seen how well it's doing in terms of test error. When applying machine learning to classification problems, there's actually one other definition of J test and J train that is maybe even more commonly used, which is instead of using the logistic loss to compute the test error and the training error to instead measure what's the fraction of the test set and the fraction of the training set that the algorithm has misclassified. So specifically on the test set, you can have the algorithm make a prediction one or zero on every test example. So recall y hat, we would predict as one if f of x is greater than or equal to 0.5 and zero if it's less than 0.5. And you can then count up in the test set the fraction of examples where y hat is not equal to the actual ground truth label y in the test set. So concretely, if you were classifying handwritten digits zero or one by new classification toss, then J test would be the fraction of that test set where zero was classified as one or one classified as zero. And similarly J train is a fraction of the training set that has been misclassified. Taking a data set and splitting it into a training set and a separate test set gives you a way to systematically evaluate how well your learning algorithm is doing. By computing both J test and J train, you can now measure how it's doing on the test set and on the training set. This procedure is one step
| 500
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6.2 Evaluating and choosing models | Evaluating a model -[Machine Learning | Andrew Ng]: value of those housing prices. And so j tests will be high. So seeing that j test is high on this model gives you a way to realize that even though it does great on the training set is actually not so good at generalizing to new examples to new data points that were not in the training set. So that was regression with squared error costs. Now let's take a look at how you'd apply this procedure to a classification problem. For example, if you were classifying between handwritten digits that are either zero or one. So same as before, you fit the parameters by minimizing the cost function to find the parameters wb. For example, if you were training logistic regression, then this would be the cost function j of wb where this is the usual logistic loss function and then plus also the regularization term and to compute the test error. J test is then the average over your test examples. That's that 30% of the data that wasn't in the training set of the logistic loss on your test set. And the training error you could also compute using this formula is the average logistic loss on your training data that the algorithm was using to minimize the cost function j of wb. Well what I describe here will work okay for figuring out if your learning algorithm is doing well. I've seen how well it's doing in terms of test error. When applying machine learning to classification problems, there's actually one other definition of J test and J train that is maybe even more commonly used, which is instead of using the logistic loss to compute the test error and the training error to instead measure what's the fraction of the test set and the fraction of the training set that the algorithm has misclassified. So specifically on the test set, you can have the algorithm make a prediction one or zero on every test example. So recall y hat, we would predict as one if f of x is greater than or equal to 0.5 and zero if it's less than 0.5. And you can then count up in the test set the fraction of examples where y hat is not equal to the actual ground truth label y in the test set. So concretely, if you were classifying handwritten digits zero or one by new classification toss, then J test would be the fraction of that test set where zero was classified as one or one classified as zero. And similarly J train is a fraction of the training set that has been misclassified. Taking a data set and splitting it into a training set and a separate test set gives you a way to systematically evaluate how well your learning algorithm is doing. By computing both J test and J train, you can now measure how it's doing on the test set and on the training set. This procedure is one step
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6.2 Evaluating and choosing models | Evaluating a model -[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=uNNx1Czrt1w
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to what you being able to automatically choose what model to use for a given machine learning application. For example, if you're trying to predict housing prices, should you fit a straight line to your data or fit a second order polynomial or third order or fourth order polynomial? It turns out that with one further refinement to the idea you saw in this video, you'll be able to have an algorithm help you to automatically make that type of decision well. Let's take a look at how to do that in the next video.
| 95
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6.2 Evaluating and choosing models | Evaluating a model -[Machine Learning | Andrew Ng]: to what you being able to automatically choose what model to use for a given machine learning application. For example, if you're trying to predict housing prices, should you fit a straight line to your data or fit a second order polynomial or third order or fourth order polynomial? It turns out that with one further refinement to the idea you saw in this video, you'll be able to have an algorithm help you to automatically make that type of decision well. Let's take a look at how to do that in the next video.
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6.3 Evaluating and choosing models | Model selection and training/cross validation/test sets-ML Ng
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https://www.youtube.com/watch?v=KwM_IYQ_I-8
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In the last video, you saw how to use a test set to evaluate the performance of a model. Let's make one further refinement to that idea in this video, which will allow you to use a technique to automatically choose a good model for your machine learning algorithm. One thing we've seen is that once the model's parameters w and b have been fit to the training set, the training error may not be a good indicator of how well the algorithm will do or how well it will generalize to new examples that were not in the training set. And in particular, for this example, the training error will be pretty much zero, and that's likely much lower than the actual generalization error. And by that I mean the average error on new examples that were not in the training set. And what you saw in the last video is that J-Test, the performance of the algorithm on examples is not trained on, that that will be a better indicator of how well the model will likely do on new data. And by that I mean other data that's not in the training set. Let's take a look at how this affects how we might use a test set to choose a model for a given machine learning application. So if fitting a function to predict housing prices or some other regression problem, one model you might consider is to fit a linear model like this. And this is a first order polynomial, and I'm going to use D equals one on this slide to denote fitting a one or first order polynomial. If you were to fit a model like this to your training set, you'd get some parameters W and B, and you can then compute J-Test to estimate how well this will generalize to new data. And on this slide, I'm going to use W1, B1, superscript there, to denote that these are the parameters you get if you were to fit a first order polynomial or degree one, D equals one polynomial. Now you might also consider fitting a second order polynomial or quadratic model. So this is the model. And if you were to fit this to your training set, you would get some parameters W2, B2, and you can then similarly evaluate those parameters on your test set and get J-Test W2, B2. And this would give you a sense of how well the second order polynomial does. And you can go on to try D equals three, that's a third order or a degree three polynomial that looks like this, and fit parameters, and similarly get J-Test. And you might keep doing this until, say, you try up to a 10th order polynomial and you end up with J-Test of W10, B10. That gives you a sense of how well the 10th order polynomial is doing. So one procedure you could try, this turns out not to be the best procedure,
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6.3 Evaluating and choosing models | Model selection and training/cross validation/test sets-ML Ng: In the last video, you saw how to use a test set to evaluate the performance of a model. Let's make one further refinement to that idea in this video, which will allow you to use a technique to automatically choose a good model for your machine learning algorithm. One thing we've seen is that once the model's parameters w and b have been fit to the training set, the training error may not be a good indicator of how well the algorithm will do or how well it will generalize to new examples that were not in the training set. And in particular, for this example, the training error will be pretty much zero, and that's likely much lower than the actual generalization error. And by that I mean the average error on new examples that were not in the training set. And what you saw in the last video is that J-Test, the performance of the algorithm on examples is not trained on, that that will be a better indicator of how well the model will likely do on new data. And by that I mean other data that's not in the training set. Let's take a look at how this affects how we might use a test set to choose a model for a given machine learning application. So if fitting a function to predict housing prices or some other regression problem, one model you might consider is to fit a linear model like this. And this is a first order polynomial, and I'm going to use D equals one on this slide to denote fitting a one or first order polynomial. If you were to fit a model like this to your training set, you'd get some parameters W and B, and you can then compute J-Test to estimate how well this will generalize to new data. And on this slide, I'm going to use W1, B1, superscript there, to denote that these are the parameters you get if you were to fit a first order polynomial or degree one, D equals one polynomial. Now you might also consider fitting a second order polynomial or quadratic model. So this is the model. And if you were to fit this to your training set, you would get some parameters W2, B2, and you can then similarly evaluate those parameters on your test set and get J-Test W2, B2. And this would give you a sense of how well the second order polynomial does. And you can go on to try D equals three, that's a third order or a degree three polynomial that looks like this, and fit parameters, and similarly get J-Test. And you might keep doing this until, say, you try up to a 10th order polynomial and you end up with J-Test of W10, B10. That gives you a sense of how well the 10th order polynomial is doing. So one procedure you could try, this turns out not to be the best procedure,
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6.3 Evaluating and choosing models | Model selection and training/cross validation/test sets-ML Ng
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https://www.youtube.com/watch?v=KwM_IYQ_I-8
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but one thing you could try is look at all of these J-Tests and see which one gives you the lowest value. And say you find that J-Test for the fifth order polynomial for W5, B5 turns out to be the lowest. If that's the case, then you might decide that the fifth order polynomial D equals five does best and choose that model for your application. And if you want to estimate how well this model performs, one thing you could do, but this turns out to be a slightly flawed procedure, is to report the test set error J-Test W5, B5. The reason this procedure is flawed is J-Test of W5, B5 is likely to be an optimistic estimate of the generalization error. In other words, it is likely to be lower than the actual generalization error. And the reason is, in the procedure we talked about on the slide, we basically fit one extra parameter, which is D, the degree of polynomial, and we chose this parameter using the test set. So on the previous slide, we saw that if you were to fit WB to the training data, then the training data would be an overly optimistic estimate of generalization error. And it turns out too, that if we were to choose the parameter D using the test set, then the test set J-Test is now an overly optimistic, that is lower than actual estimates of the generalization error. So the procedure on this particular slide is flawed and I don't recommend using this. Instead, if you want to automatically choose a model, such as decide what degree polynomial to use, here's how you modify the training and testing procedure in order to carry out model selection, whereby model selection, I mean, choosing amongst different models, such as these 10 different models that you might contemplate using for your machine learning application. The way we'll modify the procedure is, instead of splitting your data into just two subsets, the training set and the test set, we're going to split your data into three different subsets, which we're going to call the training set, the cross validation set, and then also the test set. So using our example from before of these 10 training examples, we might split it into putting 60% of the data into the training set. And so the notation we'll use for the training set portion will be the same as before, except that now mtrain, the number of training examples will be 6. And we might put 20% of the data into the cross validation set. And the notation I'm going to use is xCV of 1, yCV of 1 for the first cross validation example. So CV stands for cross validation, all the way down to xCV of mCV and yCV of mCV. So here mCV equals 2 in this example is the number of cross validation examples. And then finally, we have the test set same as before. So x1
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6.3 Evaluating and choosing models | Model selection and training/cross validation/test sets-ML Ng: but one thing you could try is look at all of these J-Tests and see which one gives you the lowest value. And say you find that J-Test for the fifth order polynomial for W5, B5 turns out to be the lowest. If that's the case, then you might decide that the fifth order polynomial D equals five does best and choose that model for your application. And if you want to estimate how well this model performs, one thing you could do, but this turns out to be a slightly flawed procedure, is to report the test set error J-Test W5, B5. The reason this procedure is flawed is J-Test of W5, B5 is likely to be an optimistic estimate of the generalization error. In other words, it is likely to be lower than the actual generalization error. And the reason is, in the procedure we talked about on the slide, we basically fit one extra parameter, which is D, the degree of polynomial, and we chose this parameter using the test set. So on the previous slide, we saw that if you were to fit WB to the training data, then the training data would be an overly optimistic estimate of generalization error. And it turns out too, that if we were to choose the parameter D using the test set, then the test set J-Test is now an overly optimistic, that is lower than actual estimates of the generalization error. So the procedure on this particular slide is flawed and I don't recommend using this. Instead, if you want to automatically choose a model, such as decide what degree polynomial to use, here's how you modify the training and testing procedure in order to carry out model selection, whereby model selection, I mean, choosing amongst different models, such as these 10 different models that you might contemplate using for your machine learning application. The way we'll modify the procedure is, instead of splitting your data into just two subsets, the training set and the test set, we're going to split your data into three different subsets, which we're going to call the training set, the cross validation set, and then also the test set. So using our example from before of these 10 training examples, we might split it into putting 60% of the data into the training set. And so the notation we'll use for the training set portion will be the same as before, except that now mtrain, the number of training examples will be 6. And we might put 20% of the data into the cross validation set. And the notation I'm going to use is xCV of 1, yCV of 1 for the first cross validation example. So CV stands for cross validation, all the way down to xCV of mCV and yCV of mCV. So here mCV equals 2 in this example is the number of cross validation examples. And then finally, we have the test set same as before. So x1
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6.3 Evaluating and choosing models | Model selection and training/cross validation/test sets-ML Ng
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https://www.youtube.com/watch?v=KwM_IYQ_I-8
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through xm tests and y1 through ym tests, where m tests here is equal to 2. This is the number of test examples. We'll see on the next slide how to use the cross validation set. So the way we'll modify the procedure is you've already seen the training set and the test set. And we're going to introduce a new subset of the data called the cross validation set. The name cross validation refers to that this is an extra data set that we're going to use to check or cross check the validity or really the accuracy of different models. I don't think it's a great name, but that is what people in machine learning have gotten to call this extra data set. You may also hear people call this the validation set for short, just few syllables then cross validation. Or in some applications, people also call this the development set means basically the same thing. Or for short, sometimes you hear people call this the dev set, but all of these terms mean the same thing as cross validation set. I personally use the term dev set the most often because it's the shortest, fastest way to say it, but cross validation is probably used a little bit more often by machine learning practitioners. So armed with these three subsets of the data, training set, cross validation set and test set, you can then compute the training error, the cross validation error and the test error using these three formulas. Where as usual, none of these terms include the regularization term that is included in the training objective. And this new term in the middle, the cross validation error is just the average over your MCV cross validation examples of the average say squared error. And this term, in addition to being called cross validation error, is also commonly called the validation error for short or even the development set error or the dev error. Armed with these three measures of learning algorithm performance, this is how you can then go about carrying out model selection. You can with the 10 models, same as earlier on the slide with D equals one, D equals two, all the way up to a 10th degree or the 10th order polynomial, you can then fit the parameters W1, B1. But instead of evaluating this on your test set, you would instead evaluate these parameters on your cross validation sets and compute JCV of W1, B1. And similarly for the second model, you get JCV of W2, B2 and all the way down to JCV of W10, B10. Then in order to choose a model, you would look at which model has the lowest cross validation error. And concretely, let's say that JCV of W4, B4 is lowest, then what that means is you would pick this fourth order polynomial as the model you will use for this application. Finally, if you want to report out an estimate of the
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6.3 Evaluating and choosing models | Model selection and training/cross validation/test sets-ML Ng: through xm tests and y1 through ym tests, where m tests here is equal to 2. This is the number of test examples. We'll see on the next slide how to use the cross validation set. So the way we'll modify the procedure is you've already seen the training set and the test set. And we're going to introduce a new subset of the data called the cross validation set. The name cross validation refers to that this is an extra data set that we're going to use to check or cross check the validity or really the accuracy of different models. I don't think it's a great name, but that is what people in machine learning have gotten to call this extra data set. You may also hear people call this the validation set for short, just few syllables then cross validation. Or in some applications, people also call this the development set means basically the same thing. Or for short, sometimes you hear people call this the dev set, but all of these terms mean the same thing as cross validation set. I personally use the term dev set the most often because it's the shortest, fastest way to say it, but cross validation is probably used a little bit more often by machine learning practitioners. So armed with these three subsets of the data, training set, cross validation set and test set, you can then compute the training error, the cross validation error and the test error using these three formulas. Where as usual, none of these terms include the regularization term that is included in the training objective. And this new term in the middle, the cross validation error is just the average over your MCV cross validation examples of the average say squared error. And this term, in addition to being called cross validation error, is also commonly called the validation error for short or even the development set error or the dev error. Armed with these three measures of learning algorithm performance, this is how you can then go about carrying out model selection. You can with the 10 models, same as earlier on the slide with D equals one, D equals two, all the way up to a 10th degree or the 10th order polynomial, you can then fit the parameters W1, B1. But instead of evaluating this on your test set, you would instead evaluate these parameters on your cross validation sets and compute JCV of W1, B1. And similarly for the second model, you get JCV of W2, B2 and all the way down to JCV of W10, B10. Then in order to choose a model, you would look at which model has the lowest cross validation error. And concretely, let's say that JCV of W4, B4 is lowest, then what that means is you would pick this fourth order polynomial as the model you will use for this application. Finally, if you want to report out an estimate of the
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6.3 Evaluating and choosing models | Model selection and training/cross validation/test sets-ML Ng
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https://www.youtube.com/watch?v=KwM_IYQ_I-8
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generalization error of how well this model will do on new data, you would do so using that third subset of your data, the test set and you report out J test of W4, B4. And you notice that throughout this entire procedure, you had fit these parameters using the training set, you then chose the parameter D or chose the degree of polynomial using the cross validation set. And so up until this point, you've not fit any parameters, either W or B or D to the test set. And that's why J test in this example will be a fair estimate of the generalization error of this model that has parameters W4, B4. So this gives a better procedure for model selection. And it lets you automatically make a decision like what order polynomial to choose for your linear regression model. This model selection procedure also works for choosing among other types of models. For example, choosing a neural network architecture. If you are fitting a model for handwritten digit recognition, you might consider three models like these, maybe even a larger set of models than just three, but here are a few different neural networks of small, somewhat larger and then even larger. To help you decide how many layers should your neural network have and how many hidden units per layer should you have. You can then train all three of these models and end up with parameters W1, B1 for the first model, W2, B2 for the second model and W3, B3 for the third model. And you can then evaluate the neural network's performance using JCV using your cross validation set. And with a classification problem, JCV can be the percentage of examples. And since this is a classification problem, JCV, the most common choice would be to compute this as a fraction of cross validation examples that the algorithm has misclassified. And you would compute this using all three models and then pick the model with the lowest cross validation error. So if in this example, this has the lowest cross validation error, you would then pick the second neural network and use parameters trained on this model. And finally, if you want to report out an estimate of the generalization error, you then use the test set to estimate how well the neural network that you just chose will do. So in machine learning practice, it's considered best practice to make all the decisions you want to make regarding your learning algorithm, such as how to choose parameters, what degree polynomial to use, but make decisions only looking at the training set and cross validation set and to not use the test set at all to make decisions about your model. And only after you've made all those decisions, then finally take the model you have designed and evaluated on your test set. And that procedure ensures that you haven't accidentally fit anything to the test set, so that your test set
| 500
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6.3 Evaluating and choosing models | Model selection and training/cross validation/test sets-ML Ng: generalization error of how well this model will do on new data, you would do so using that third subset of your data, the test set and you report out J test of W4, B4. And you notice that throughout this entire procedure, you had fit these parameters using the training set, you then chose the parameter D or chose the degree of polynomial using the cross validation set. And so up until this point, you've not fit any parameters, either W or B or D to the test set. And that's why J test in this example will be a fair estimate of the generalization error of this model that has parameters W4, B4. So this gives a better procedure for model selection. And it lets you automatically make a decision like what order polynomial to choose for your linear regression model. This model selection procedure also works for choosing among other types of models. For example, choosing a neural network architecture. If you are fitting a model for handwritten digit recognition, you might consider three models like these, maybe even a larger set of models than just three, but here are a few different neural networks of small, somewhat larger and then even larger. To help you decide how many layers should your neural network have and how many hidden units per layer should you have. You can then train all three of these models and end up with parameters W1, B1 for the first model, W2, B2 for the second model and W3, B3 for the third model. And you can then evaluate the neural network's performance using JCV using your cross validation set. And with a classification problem, JCV can be the percentage of examples. And since this is a classification problem, JCV, the most common choice would be to compute this as a fraction of cross validation examples that the algorithm has misclassified. And you would compute this using all three models and then pick the model with the lowest cross validation error. So if in this example, this has the lowest cross validation error, you would then pick the second neural network and use parameters trained on this model. And finally, if you want to report out an estimate of the generalization error, you then use the test set to estimate how well the neural network that you just chose will do. So in machine learning practice, it's considered best practice to make all the decisions you want to make regarding your learning algorithm, such as how to choose parameters, what degree polynomial to use, but make decisions only looking at the training set and cross validation set and to not use the test set at all to make decisions about your model. And only after you've made all those decisions, then finally take the model you have designed and evaluated on your test set. And that procedure ensures that you haven't accidentally fit anything to the test set, so that your test set
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6.3 Evaluating and choosing models | Model selection and training/cross validation/test sets-ML Ng
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https://www.youtube.com/watch?v=KwM_IYQ_I-8
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becomes still a fair and not overly optimistic estimate of the generalization error of your algorithm. So it's considered best practice in machine learning that if you have to make decisions about your model, such as fitting parameters or choosing the model architecture, such as neural network architecture or degree of polynomial if you're fitting linear regression, to make all those decisions only using your training set and your cross validation set and to not look at the test set at all while you're still making decisions regarding your learning algorithm. And it's only after you've come up with one model that's your final model to only then evaluate it on the test set. And because you haven't made any decisions using the test set, that ensures that your test set is a fair and not overly optimistic estimate of how well your model will generalize to new data. So that's model selection. And this is actually a very widely used procedure. I use this all the time to automatically choose what model to use for a given machine learning application. Now earlier this week, I mentioned running diagnostics to decide how to improve the performance of a learning algorithm. Now that you have a way to evaluate learning algorithms and even automatically choose a model, let's dive more deeply into examples of some diagnostics. The most powerful diagnostic that I know of and that I use for a lot of machine learning applications is one called bias and variance. Let's take a look at what that means in the next video.
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6.3 Evaluating and choosing models | Model selection and training/cross validation/test sets-ML Ng: becomes still a fair and not overly optimistic estimate of the generalization error of your algorithm. So it's considered best practice in machine learning that if you have to make decisions about your model, such as fitting parameters or choosing the model architecture, such as neural network architecture or degree of polynomial if you're fitting linear regression, to make all those decisions only using your training set and your cross validation set and to not look at the test set at all while you're still making decisions regarding your learning algorithm. And it's only after you've come up with one model that's your final model to only then evaluate it on the test set. And because you haven't made any decisions using the test set, that ensures that your test set is a fair and not overly optimistic estimate of how well your model will generalize to new data. So that's model selection. And this is actually a very widely used procedure. I use this all the time to automatically choose what model to use for a given machine learning application. Now earlier this week, I mentioned running diagnostics to decide how to improve the performance of a learning algorithm. Now that you have a way to evaluate learning algorithms and even automatically choose a model, let's dive more deeply into examples of some diagnostics. The most powerful diagnostic that I know of and that I use for a lot of machine learning applications is one called bias and variance. Let's take a look at what that means in the next video.
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6.4 Bias and variance | Diagnosing bias and variance -[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=YB61HDL7EzE
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The typical workflow of developing a machine learning system is that you have an idea and you train a model and you almost always find that it doesn't work as well as you wish yet. When I'm training machine learning model, it pretty much never works that well the first time. And so key to the process of building machine learning system is how to decide what to do next in order to improve this performance. I've found across many different applications that looking at the bias and variance of a learning algorithm gives you very good guidance on what to try next. Let's take a look at what this means. You might remember this example from the first course on linear regression, where given this data set, if you were to fill a straight line to it, it doesn't do that well. And we said that this algorithm has high bias or that it underfits this data set. Or if you were to fit a four-folder polynomial, then it has high variance or it overfits. And in the middle, if you fit a quadratic polynomial, then it looks pretty good. And I said that was just right. Because this is a problem with just a single feature x, we could plot the function f and look at it like this. But if you had more features, you can't plot f and visualize whether it's doing well as easily. So instead of trying to look at plots like this, a more systematic way to diagnose or to find out if your algorithm has high bias or high variance will be to look at the performance of your algorithm on the training set and on the cross validation set. In particular, let's look at the example on the left. If you were to compute J train, how well does the algorithm do on the training set? Not that well. So let's say J train here would be high because there are actually pretty large errors between the examples and the actual predictions of the model. And how about Jcv? So Jcv would be if you had a few new examples, maybe examples like that, that the algorithm had not previously seen. And here, the algorithm also doesn't do that well on examples that it had not previously seen. So Jcv would also be high. And one characteristic of an algorithm with high bias, something that is underfitting is that it's not even doing that well on the training set. And so when J train is high, that gives you a strong indicator that this algorithm has high bias. Let's now look at the example on the right. If you were to compute J train, how well is this doing on the training set? Well, it's actually doing great on the training set, fits the training data really well. So J train here will be low. But if you were to evaluate this model on other houses not in the training set,
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6.4 Bias and variance | Diagnosing bias and variance -[Machine Learning | Andrew Ng]: The typical workflow of developing a machine learning system is that you have an idea and you train a model and you almost always find that it doesn't work as well as you wish yet. When I'm training machine learning model, it pretty much never works that well the first time. And so key to the process of building machine learning system is how to decide what to do next in order to improve this performance. I've found across many different applications that looking at the bias and variance of a learning algorithm gives you very good guidance on what to try next. Let's take a look at what this means. You might remember this example from the first course on linear regression, where given this data set, if you were to fill a straight line to it, it doesn't do that well. And we said that this algorithm has high bias or that it underfits this data set. Or if you were to fit a four-folder polynomial, then it has high variance or it overfits. And in the middle, if you fit a quadratic polynomial, then it looks pretty good. And I said that was just right. Because this is a problem with just a single feature x, we could plot the function f and look at it like this. But if you had more features, you can't plot f and visualize whether it's doing well as easily. So instead of trying to look at plots like this, a more systematic way to diagnose or to find out if your algorithm has high bias or high variance will be to look at the performance of your algorithm on the training set and on the cross validation set. In particular, let's look at the example on the left. If you were to compute J train, how well does the algorithm do on the training set? Not that well. So let's say J train here would be high because there are actually pretty large errors between the examples and the actual predictions of the model. And how about Jcv? So Jcv would be if you had a few new examples, maybe examples like that, that the algorithm had not previously seen. And here, the algorithm also doesn't do that well on examples that it had not previously seen. So Jcv would also be high. And one characteristic of an algorithm with high bias, something that is underfitting is that it's not even doing that well on the training set. And so when J train is high, that gives you a strong indicator that this algorithm has high bias. Let's now look at the example on the right. If you were to compute J train, how well is this doing on the training set? Well, it's actually doing great on the training set, fits the training data really well. So J train here will be low. But if you were to evaluate this model on other houses not in the training set,
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6.4 Bias and variance | Diagnosing bias and variance -[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=YB61HDL7EzE
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then you find that Jcv, the cross validation error will be quite high. And so a characteristic signature or characteristic cue that your algorithm has high variance will be of Jcv is much higher than J train. In other words, it does much better on data it has seen than on data it has not seen. And this turns out to be a strong indicator that your algorithm has high variance. And again, the point of what we're doing is that by computing J train and Jcv and seeing if J train is high, or if Jcv is much higher than J train, this gives you a sense even if you can't plot to function f of whether your algorithm has high bias or high variance. And finally, the case in the middle, if you look at J train is pretty low since it's doing quite well on the training set. And if you were to look at a few new examples like those from say your cross validation set, you find that Jcv is also pretty low. And so J train not being too high, indicates this doesn't have a high bias problem. And Jcv not being much worse than J train, this indicates that it doesn't have a high variance problem either, which is why this model, the quadratic model seems to be a pretty good one for this application. Let me share with you another view of bias and variance. So to summarize when d equals one for a linear polynomial, J train was high and Jcv was high. When d equals four, J train was low, but Jcv is high. And when d equals two, both were pretty low. Let's now take a different view on bias and variance. And in particular, on the next slide, I'd like to show you how J train and Jcv vary as a function of the degree of the polynomial you're fitting. So let me draw a figure where the horizontal axis of this figure will be the degree of polynomial that we're fitting to the data. Over on the left will correspond to a small value of d, like d equals one, which corresponds to a fitting straight line. And over to the right will correspond to say d equals four or even higher values of d, where we're fitting this high order polynomial. So if you were to plot J train of Wb as a function of degree of polynomial, what you find is that as you fit a higher and higher degree polynomial, here I'm assuming we're not using regularization, but as you fit a higher and higher order polynomial, the training error will tend to go down because when you have a very simple linear function, it doesn't fit the training data that well. When you fit a quadratic function or a third order polynomial or a fourth order polynomial, it fits the training data better and better. So as the degree of polynomial increases, J
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6.4 Bias and variance | Diagnosing bias and variance -[Machine Learning | Andrew Ng]: then you find that Jcv, the cross validation error will be quite high. And so a characteristic signature or characteristic cue that your algorithm has high variance will be of Jcv is much higher than J train. In other words, it does much better on data it has seen than on data it has not seen. And this turns out to be a strong indicator that your algorithm has high variance. And again, the point of what we're doing is that by computing J train and Jcv and seeing if J train is high, or if Jcv is much higher than J train, this gives you a sense even if you can't plot to function f of whether your algorithm has high bias or high variance. And finally, the case in the middle, if you look at J train is pretty low since it's doing quite well on the training set. And if you were to look at a few new examples like those from say your cross validation set, you find that Jcv is also pretty low. And so J train not being too high, indicates this doesn't have a high bias problem. And Jcv not being much worse than J train, this indicates that it doesn't have a high variance problem either, which is why this model, the quadratic model seems to be a pretty good one for this application. Let me share with you another view of bias and variance. So to summarize when d equals one for a linear polynomial, J train was high and Jcv was high. When d equals four, J train was low, but Jcv is high. And when d equals two, both were pretty low. Let's now take a different view on bias and variance. And in particular, on the next slide, I'd like to show you how J train and Jcv vary as a function of the degree of the polynomial you're fitting. So let me draw a figure where the horizontal axis of this figure will be the degree of polynomial that we're fitting to the data. Over on the left will correspond to a small value of d, like d equals one, which corresponds to a fitting straight line. And over to the right will correspond to say d equals four or even higher values of d, where we're fitting this high order polynomial. So if you were to plot J train of Wb as a function of degree of polynomial, what you find is that as you fit a higher and higher degree polynomial, here I'm assuming we're not using regularization, but as you fit a higher and higher order polynomial, the training error will tend to go down because when you have a very simple linear function, it doesn't fit the training data that well. When you fit a quadratic function or a third order polynomial or a fourth order polynomial, it fits the training data better and better. So as the degree of polynomial increases, J
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6.4 Bias and variance | Diagnosing bias and variance -[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=YB61HDL7EzE
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train will typically go down. Next, let's look at Jcv, which is how well does it do on data that it did not get to fit to? What we saw was when d equals one, when the degree of polynomial was very low, Jcv was pretty high because it underfit, so it didn't do well on the cross validation set. And here on the right as well, when the degree of polynomial is very large, say four, it doesn't do well on the cross validation set either and so is also high. But if d was in between, say a second order polynomial, then it actually did much better. And so if you were to vary the degree of polynomial, you'd actually get a curve that looks like this, which comes down and then goes back up. Where if the degree of polynomial is too low, it underfits and so doesn't do well on the cross validation set. If it is too high, it overfits and also doesn't do well on the cross validation set. And there's only if it's somewhere in the middle that is just right, which is why the second order polynomial in our example ends up with a lower cross validation error and neither high bias nor high variance. So to summarize, how do you diagnose bias and variance in your learning algorithm? If your learning algorithm has high bias or has underfit data, the key indicator will be if Jtrain is high. And so that corresponds to this leftmost portion of the curve, which is where Jtrain is high. And usually you have Jtrain and JCV will be close to each other. And how do you diagnose if you have high variance? Well the key indicator for high variance will be if JCV is much greater than Jtrain. This double greater than sign in math refers to much greater than. So this is greater and this means much greater. And this rightmost portion of the plot is where JCV is much greater than Jtrain. And usually Jtrain will be pretty low, but the key indicator is whether JCV is much greater than Jtrain. And that's what happens when we had fit a very high order polynomial to this small dataset. And even though we've just seen bias and variance, it turns out in some cases it's possible to simultaneously have high bias and have high variance. You won't see this happen that much for linear regression, but it turns out that if you're training a neural network, there are some applications where unfortunately you have high bias and high variance. And one way to recognize that situation will be if Jtrain is high, so you're not doing that well on the training set, but even worse the cross validation error is again even much larger than the training set. The notion of high bias and high variance, it doesn't really happen for linear models applied to 1D, but to give intuition about what it looks
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6.4 Bias and variance | Diagnosing bias and variance -[Machine Learning | Andrew Ng]: train will typically go down. Next, let's look at Jcv, which is how well does it do on data that it did not get to fit to? What we saw was when d equals one, when the degree of polynomial was very low, Jcv was pretty high because it underfit, so it didn't do well on the cross validation set. And here on the right as well, when the degree of polynomial is very large, say four, it doesn't do well on the cross validation set either and so is also high. But if d was in between, say a second order polynomial, then it actually did much better. And so if you were to vary the degree of polynomial, you'd actually get a curve that looks like this, which comes down and then goes back up. Where if the degree of polynomial is too low, it underfits and so doesn't do well on the cross validation set. If it is too high, it overfits and also doesn't do well on the cross validation set. And there's only if it's somewhere in the middle that is just right, which is why the second order polynomial in our example ends up with a lower cross validation error and neither high bias nor high variance. So to summarize, how do you diagnose bias and variance in your learning algorithm? If your learning algorithm has high bias or has underfit data, the key indicator will be if Jtrain is high. And so that corresponds to this leftmost portion of the curve, which is where Jtrain is high. And usually you have Jtrain and JCV will be close to each other. And how do you diagnose if you have high variance? Well the key indicator for high variance will be if JCV is much greater than Jtrain. This double greater than sign in math refers to much greater than. So this is greater and this means much greater. And this rightmost portion of the plot is where JCV is much greater than Jtrain. And usually Jtrain will be pretty low, but the key indicator is whether JCV is much greater than Jtrain. And that's what happens when we had fit a very high order polynomial to this small dataset. And even though we've just seen bias and variance, it turns out in some cases it's possible to simultaneously have high bias and have high variance. You won't see this happen that much for linear regression, but it turns out that if you're training a neural network, there are some applications where unfortunately you have high bias and high variance. And one way to recognize that situation will be if Jtrain is high, so you're not doing that well on the training set, but even worse the cross validation error is again even much larger than the training set. The notion of high bias and high variance, it doesn't really happen for linear models applied to 1D, but to give intuition about what it looks
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6.4 Bias and variance | Diagnosing bias and variance -[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=YB61HDL7EzE
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like, it would be as if for part of the input you had a very complicated model that overfit, so it overfits to part of the input. But then for some reason for other parts of the input, it doesn't even fit the training data well, and so it underfits for part of the input. In this example, which looks artificial because it's a single feature input, we fit the training set really well and we overfit in part of the input, and we don't even fit the training data well and we underfit in part of the input. And that's how in some applications you can unfortunately end up with both high bias and high variance. And the indicator for that will be if the algorithm does poorly on the training set and it even does much worse than on the training set. For most learning applications, you probably have primarily a high bias or a high variance problem rather than both at the same time, but it is possible sometimes that both are the same time. So I know that there's a lot to process, there are a lot of concepts on the slides, but the key takeaways are high bias means it's not even doing well on the training set, and high variance means it does much worse on the cross-validation set than the training set. Whenever I'm training a machine learning algorithm, I will almost always try to figure out to what extent the algorithm has a high bias or underfitting versus a high variance or an overfitting problem. And this will give good guidance, as we'll see later this week, on how you can improve the performance of the algorithm. But first, let's take a look at how regularization affects the bias and variance of a learning algorithm, because that will help you better understand when you should use regularization. Let's take a look at that in the next video.
| 325
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6.4 Bias and variance | Diagnosing bias and variance -[Machine Learning | Andrew Ng]: like, it would be as if for part of the input you had a very complicated model that overfit, so it overfits to part of the input. But then for some reason for other parts of the input, it doesn't even fit the training data well, and so it underfits for part of the input. In this example, which looks artificial because it's a single feature input, we fit the training set really well and we overfit in part of the input, and we don't even fit the training data well and we underfit in part of the input. And that's how in some applications you can unfortunately end up with both high bias and high variance. And the indicator for that will be if the algorithm does poorly on the training set and it even does much worse than on the training set. For most learning applications, you probably have primarily a high bias or a high variance problem rather than both at the same time, but it is possible sometimes that both are the same time. So I know that there's a lot to process, there are a lot of concepts on the slides, but the key takeaways are high bias means it's not even doing well on the training set, and high variance means it does much worse on the cross-validation set than the training set. Whenever I'm training a machine learning algorithm, I will almost always try to figure out to what extent the algorithm has a high bias or underfitting versus a high variance or an overfitting problem. And this will give good guidance, as we'll see later this week, on how you can improve the performance of the algorithm. But first, let's take a look at how regularization affects the bias and variance of a learning algorithm, because that will help you better understand when you should use regularization. Let's take a look at that in the next video.
|
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6.5 Bias and variance | Regularization and bias/variance -[Machine Learning | Andrew Ng]
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https://www.youtube.com/watch?v=2Ji4Upc606c
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You saw in the last video how different choices of the degree of polynomial d affects the bias and variance of your learning algorithm and therefore its overall performance. In this video, let's take a look at how regularization, specifically the choice of the regularization parameter lambda, affects the bias and variance and therefore the overall performance of the algorithm. This, it turns out, will be helpful for when you want to choose a good value of lambda of the regularization parameter for your algorithm. Let's take a look. In this example, I'm going to use a fourth order polynomial, but we're going to fit this model using regularization, where here the value of lambda is the regularization parameter that controls how much you trade off keeping the parameters w small versus fitting the training data well. Let's start with the example of setting lambda to be a very large value. Say lambda is equal to 10,000. If you were to do so, you would end up fitting a model that looks roughly like this. Because if lambda were very, very large, then the algorithm is highly motivated to keep these parameters w very small, and so you end up with w1, w2, really all of these parameters would be very close to zero. The model ends up being f of x is just approximately b, a constant value, which is why you end up with a model like this. This model clearly has high bias and it underfits the training data because it doesn't even do well on the training set, and J train is large. Let's take a look at the other extreme. Let's say you set lambda to be a very small value. So with a small value of lambda, in fact, let's go to extreme of setting lambda equals zero. With that choice of lambda, there is no regularization, and so we're just fitting a four-folder polynomial with no regularization, and you end up with that curve that you saw previously that overfits the data. What we saw previously was when you have a model like this, J train is small, but JCV is much larger than J train. JCV is large, and so this indicates we have high variance and it overfits this data. It would be if you have some intermediate value of lambda, not really large, 10,000, but not so small as zero, that hopefully you get a model that looks like this, that is just right and fits the data well with small J train and small JCV. If you are trying to decide what is a good value of lambda to use for the regularization parameter, cross-validation gives you a way to do so as well. Let's take a look at how we could do so, and just as a reminder, the problem we're addressing is if you're fitting a four-folder polynomial, so that's the model, and you're using regularization, how can you choose a good value of lambda? This would
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6.5 Bias and variance | Regularization and bias/variance -[Machine Learning | Andrew Ng]: You saw in the last video how different choices of the degree of polynomial d affects the bias and variance of your learning algorithm and therefore its overall performance. In this video, let's take a look at how regularization, specifically the choice of the regularization parameter lambda, affects the bias and variance and therefore the overall performance of the algorithm. This, it turns out, will be helpful for when you want to choose a good value of lambda of the regularization parameter for your algorithm. Let's take a look. In this example, I'm going to use a fourth order polynomial, but we're going to fit this model using regularization, where here the value of lambda is the regularization parameter that controls how much you trade off keeping the parameters w small versus fitting the training data well. Let's start with the example of setting lambda to be a very large value. Say lambda is equal to 10,000. If you were to do so, you would end up fitting a model that looks roughly like this. Because if lambda were very, very large, then the algorithm is highly motivated to keep these parameters w very small, and so you end up with w1, w2, really all of these parameters would be very close to zero. The model ends up being f of x is just approximately b, a constant value, which is why you end up with a model like this. This model clearly has high bias and it underfits the training data because it doesn't even do well on the training set, and J train is large. Let's take a look at the other extreme. Let's say you set lambda to be a very small value. So with a small value of lambda, in fact, let's go to extreme of setting lambda equals zero. With that choice of lambda, there is no regularization, and so we're just fitting a four-folder polynomial with no regularization, and you end up with that curve that you saw previously that overfits the data. What we saw previously was when you have a model like this, J train is small, but JCV is much larger than J train. JCV is large, and so this indicates we have high variance and it overfits this data. It would be if you have some intermediate value of lambda, not really large, 10,000, but not so small as zero, that hopefully you get a model that looks like this, that is just right and fits the data well with small J train and small JCV. If you are trying to decide what is a good value of lambda to use for the regularization parameter, cross-validation gives you a way to do so as well. Let's take a look at how we could do so, and just as a reminder, the problem we're addressing is if you're fitting a four-folder polynomial, so that's the model, and you're using regularization, how can you choose a good value of lambda? This would
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