Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming
This work addresses optimization challenges in stochastic programming for researchers and practitioners, but it is incremental as it builds on existing stochastic approximation methods.
The authors tackled the problem of solving nonconvex stochastic programming by introducing the randomized stochastic gradient (RSG) method, establishing its complexity for approximate stationary points and showing a nearly optimal convergence rate for convex cases, with a post-optimization variant improving large-deviation properties.
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.