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 $O\left( {\frac{d}{{bq\sqrt T }} + \frac{1}{{\sqrt T }}} \right)$ 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 $O\left( {\frac{1}{{\sqrt T }}} \right)$ convergence rate but suffers an additional $O\left( {\frac{{d + q}}{{bq}}} \right)$ error. Our the oretical investigation on ZO-SPGD provides a general framework to study the convergence rate of zeroth-order algorithms.