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.
Natural language processing come to mind. For instance, you might predict the amount of money someone spends on your website by their review. The review is text, encoded by an n-gram model. The ith, jth element of your training matrix is the ith customer and counts of jth n-gram. An n-gram is a string of contiguous words like, "the product was excellent."
Typically n-gram data is encoded in a sparse matrix. There are several data-structures suitable. One of the easiest to explain is a coordinate list. A coordinate list is a list of tuples [(i, j, value)]. Since the matrix is mostly zero, this is much more efficient than allocating a dense array.
Best Answer
Gradient descent is actually a pretty poor way of solving a linear regression problem. The
lm()function in R internally uses a form of QR decomposition, which is considerably more efficient. However, gradient descent is a generally useful technique, and worth introducing in this simple context, so that it's clearer how to apply it in more complex problems. If you want to implement your own version as a learning exercise, it's a worthwhile thing to do, butlm()is a better choice if all you want is a tool to do linear regression.