In Andrew Ng's machine learning course, he introduces linear regression and logistic regression, and shows how to fit the model parameters using gradient descent and Newton's method.

I know gradient descent can be useful in some applications of machine learning (e.g., backpropogation), but in the more general case is there any reason why you wouldn't solve for the parameters in closed form-- i.e., by taking the derivative of the cost function and solving via Calculus?

What is the advantage of using an iterative algorithm like gradient descent over a closed-form solution in general, when one is available?

Unless the closed form solution is extremely expensive to compute, it generally is the way to go when it is available.  However,


For most nonlinear regression problems there is no closed form solution.  
Even in linear regression (one of the few cases where a closed form solution is available), it may be impractical to use the formula.  The following example shows one way in which this can happen.  


For linear regression on a model of the form $y=X\beta$, where $X$ is a matrix with full column rank, the least squares solution, 

$\hat{\beta} = \arg \min \| X \beta -y \|_{2}$ 

is given by 

$\hat{\beta}=(X^{T}X)^{-1}X^{T}y$

Now, imagine that $X$ is a very large but sparse matrix.  e.g. $X$ might have 100,000 columns and 1,000,000 rows, but only 0.001% of the entries in $X$ are nonzero.  There are specialized data structures for storing only the nonzero entries of such sparse matrices.  

Also imagine that we're unlucky, and $X^{T}X$ is a fairly dense matrix with a much higher percentage of nonzero entries.  Storing a dense 100,000 by 100,000 element $X^{T}X$ matrix would then require $1 \times 10^{10}$ floating point numbers (at 8 bytes per number, this comes to 80 gigabytes.)  This would be impractical to store on anything but a supercomputer.  Furthermore, the inverse of this matrix (or more commonly a Cholesky factor) would also tend to have mostly nonzero entries.  

However, there are iterative methods for solving the least squares problem that require no more storage than $X$, $y$, and $\hat{\beta}$ and never explicitly form the matrix product $X^{T}X$.  

In this situation, using an iterative method is much more computationally efficient than using the closed form solution to the least squares problem.     

This example might seem absurdly large.  However, large sparse least squares problems of this size are routinely solved by iterative methods on desktop computers in seismic tomography research.  
