Stochastic Zeroth-Order Optimization under Strongly Convexity and Lipschitz Hessian: Minimax Sample Complexity

TL;DR

This paper studies the problem of optimizing strongly convex functions with Lipschitz Hessian using only noisy function evaluations, providing tight bounds on the minimax regret and proposing an optimal algorithm.

Contribution

It introduces the first tight characterization of minimax simple regret for stochastic zeroth-order optimization of second-order smooth convex functions, with a novel algorithm and analysis.

Findings

Established matching upper and lower bounds for minimax regret.

Developed a new gradient estimator leveraging higher-order smoothness.

Designed an algorithm combining bootstrapping and mirror descent stages.

Abstract

Optimization of convex functions under stochastic zeroth-order feedback has been a major and challenging question in online learning. In this work, we consider the problem of optimizing second-order smooth and strongly convex functions where the algorithm is only accessible to noisy evaluations of the objective function it queries. We provide the first tight characterization for the rate of the minimax simple regret by developing matching upper and lower bounds. We propose an algorithm that features a combination of a bootstrapping stage and a mirror-descent stage. Our main technical innovation consists of a sharp characterization for the spherical-sampling gradient estimator under higher-order smoothness conditions, which allows the algorithm to optimally balance the bias-variance tradeoff, and a new iterative method for the bootstrapping stage, which maintains the performance for unbounded Hessian.