For some models, linear regression included, there may be a closed form solution, but it might just be too expensive to compute. In particular, high dimensions can screw everything and sparsity can allow much cheaper solutions, so sometimes you just revert to gradient descent or SGD.
For completeness I would include the closed form solution in a discussion of linear regression. In matrix notation is fairly simple and it's easy to follow how it falls apart when the assumptions underlying linear regression are not present in the data you're working with (homoskedasticity, independence of errors etc.)
