Stochastic Proximal Gradient Methods for Nonconvex Problems in Hilbert Spaces

Abstract

For finite-dimensional problems, stochastic approximation methods have long been used to solve stochastic optimization problems. Their application to infinite dimensional problems is less understood, particularly for nonconvex objectives. This paper presents convergence results for the stochastic proximal gradient method applied to Hilbert spaces, motivated by optimization problems with partial differential equation (PDE) constraints with random inputs and coefficients. We study stochastic algorithms for nonconvex and nonsmooth problems, where the nonsmooth part is convex and the nonconvex part is the expectation, which is assumed to have a Lipschitz continuous gradient. The optimization variable is an element of a Hilbert space. We show almost sure convergence of strong limit points of the random sequence generated by the algorithm to stationary points. We demonstrate the stochastic proximal gradient algorithm on a tracking-type functional with a L 1 -penalty term constrained by a semilinear PDE and box constraints, where input terms and coefficients are subject to uncertainty. We verify conditions for ensuring convergence of the algorithm and show a simulation.

Taylor approximation for chance constrained optimization problems governed by partial differential equations with high-dimensional random parameters

“…It is critical to incorporate this uncertainty in the optimization problem to make the optimal solution more reliable and robust. Optimization under uncertainty has become an important research area and received increasing attentions in recent years [74,10,45,41,70,76,49,26,78,19,52,63,20,51,48,8,3,5,84,85,42,69,53,50,55,33,59,47,79,80,81,27,18,86,82,35,54,61,6,38,39,31,37,36]. To account for the uncertainty in the optimization problem, different statistical measures of the objective function have been studied, e.g., mean, variance, conditional value-at-risk, worst case scenario, etc., [70,48,85,3,…”

Some convergent results for Backtracking Gradient Descent method on Banach spaces

“…Indeed, it seems that most of the (convergence) results in this direction so far is based on Banach's fixed point theorem for contracting mappings, and applies only to maps which have some types of monotonicity. For systematic development in GD, one can see for example the very recent work [6], where the problem is of stochastic nature, the space is a Hilbert space, and f is either in C 1,1 L (in which case only the results lim n→∞ ||x n+1 − x n || = 0 is proven); or when with additional assumption on learning rates n δ n = ∞ and n δ 2 n < ∞, where convergence to 0 of {∇f (x n )} can be proven. Some other references are [4] (where the function is assumed to be strongly convex) and [1] (where the VMPT method is used), where again only lim n→∞ ||∇f (z n )|| = 0 is considered, and no discussion of (weak or strong) convergence of {x n } itself or avoidance of saddle points are given.…”