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 $$L^1$$ 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.

A proximal gradient method for control problems with non-smooth and non-convex control cost

“…There are results for projected gradient methods, see, e.g., [14,17]. Recently, a stochastic version of the algorithm was analyzed in [16]. However, in these papers no convergence results for weakly converging subsequences of iterates are given.…”

A Stochastic Gradient Method With Mesh Refinement for PDE-Constrained Optimization Under Uncertainty

“…In this paper, we pursued our investigation of [7] about stochastic optimization problems in Hilbert space with focus on asymptotic convergence for stochastic gradient methods. In contrast to the first paper, which considered nonconvex non-smooth optimization problems, we analyze here nonconvex, but smooth, stochastic optimization problems.…”

“…In contrast to the first paper, which considered nonconvex non-smooth optimization problems, we analyze here nonconvex, but smooth, stochastic optimization problems. Unlike [7], where a control constraint was present and the proposed algorithm included a proximal step, the convergence analysis here is considerably simpler. The semilinear elliptic PDE problem is subject to Neumann boundary conditions and is slightly more general than in [7].…”

“…This paper aims to extend the convergence analysis of stochastic approximation methods in the case where H is an infinite-dimensional space and the optimization problem is nonconvex but smooth. The convergence theory here is easier than in [7] on account of the smoothness but is surprisingly missing from the literature. We refer the reader to the recent manuscripts [1,15] for a discussion about the application of stochastic methods in optimal control with randomness and for some numerical issues.…”

“…The convergence of the method on a Hilbert space in the convex case has been shown in [4] and with bias for the projected stochastic gradient method in [6]. Recently, the nonconvex infinite-dimensional case was also analyzed for the stochastic proximal gradient method in [7]. This paper aims to extend the convergence analysis of stochastic approximation methods in the case where H is an infinite-dimensional space and the optimization problem is nonconvex but smooth.…”