Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming

TL;DR

This paper introduces a new stochastic approximation algorithm, RSG, for nonconvex stochastic programming, providing complexity analysis, convergence rates, and a variant with improved large-deviation properties, applicable to zeroth-order problems.

Contribution

The paper presents the RSG algorithm for nonconvex stochastic programming, including complexity bounds, convergence analysis, and a variant with enhanced large-deviation performance.

Findings

RSG achieves nearly optimal convergence rates in convex cases.

The variant with post-optimization improves large-deviation properties.

Applicable to zeroth-order stochastic optimization problems.

Abstract

In this paper, we introduce a new stochastic approximation (SA) type algorithm, namely the randomized stochastic gradient (RSG) method, for solving an important class of nonlinear (possibly nonconvex) stochastic programming (SP) problems. We establish the complexity of this method for computing an approximate stationary point of a nonlinear programming problem. We also show that this method possesses a nearly optimal rate of convergence if the problem is convex. We discuss a variant of the algorithm which consists of applying a post-optimization phase to evaluate a short list of solutions generated by several independent runs of the RSG method, and show that such modification allows to improve significantly the large-deviation properties of the algorithm. These methods are then specialized for solving a class of simulation-based optimization problems in which only stochastic zeroth-order information is available.