High Probability Complexity Bounds for Adaptive Step Search Based on Stochastic Oracles

Abstract

SIAM Journal on Optimization, Volume 34, Issue 3, Page 2411-2439, September 2024.
Abstract. We consider a step search method for continuous optimization under a stochastic setting where the function values and gradients are available only through inexact probabilistic zeroth- and first-order oracles. (We introduce the term step search for a class of methods, similar to line search, but where step direction can change during the back-tracking procedure.) Unlike the stochastic gradient method and its many variants, the algorithm does not use a prespecified sequence of step sizes but increases or decreases the step size adaptively according to the estimated progress of the algorithm. These oracles capture multiple standard settings including expected loss minimization and zeroth-order optimization. Moreover, our framework is very general and allows the function and gradient estimates to be biased. The proposed algorithm is simple to describe and easy to implement. Under fairly general conditions on the oracles, we derive a high probability tail bound on the iteration complexity of the algorithm when it is applied to nonconvex, convex, and strongly convex (more generally, those satisfying the Polyak-Łojasiewicz (PL) condition) functions. Our analysis strengthens and extends prior results for stochastic step and line search methods.