ZEROTH-ORDER STOCHASTIC PROJECTED GRADIENT DESCENT FOR NONCONVEX OPTIMIZATION

Abstract

In this paper, we analyze the convergence of the zeroth-order stochastic projected gradient descent (ZO-SPGD) method for constrained convex and nonconvex optimization scenarios where only objective function values (not gradients) are directly available. We show statistical properties of a new random gradient estimator, constructed through random direction samples drawn from a bounded uniform distribution. We prove that ZO-SPGD yields a O(d/(bq√T) + 1/√T) convergence rate for convex but non-smooth optimization, where d is the number of optimization variables, b is the minibatch size, q is the number of random direction samples for gradient estimation, and T is the number of iterations. For nonconvex optimization, we show that ZO-SPGD achieves O(1/√T) convergence rate but suffers an additional O((d + q)/(bq)) error. Our the oretical investigation on ZO-SPGD provides a general framework to study the convergence rate of zeroth-order algorithms.