Gradient Descent
Constructor
GradientDescent(; alphaguess = LineSearches.InitialPrevious(),
linesearch = LineSearches.HagerZhang(),
P = nothing,
precondprep = (P, x) -> nothing)Description
Gradient Descent a common name for a quasi-Newton solver. This means that it takes steps according to
\[x_{n+1} = x_n - P^{-1}\nabla f(x_n)\]
where $P$ is a positive definite matrix. If $P$ is the Hessian, we get Newton's method. In Gradient Descent, $P$ is simply an appropriately dimensioned identity matrix, such that we go in the exact opposite direction of the gradient. This means that we do not use the curvature information from the Hessian, or an approximation of it. While it does seem quite logical to go in the opposite direction of the fastest increase in objective value, the procedure can be very slow if the problem is ill-conditioned. See the section on preconditioners for ways to remedy this when using Gradient Descent.
As with the other quasi-Newton solvers in this package, a scalar $\alpha$ is introduced as follows
\[x_{n+1} = x_n - \alpha P^{-1}\nabla f(x_n)\]
and is chosen by a linesearch algorithm such that each step gives sufficient descent.