Stochastic Zeroth-Order Optimization under Strongly Convexity and Lipschitz Hessian: Minimax Sample Complexity
This work addresses a major challenge in online learning for convex optimization under limited feedback, offering foundational theoretical guarantees.
The paper tackles the problem of optimizing strongly convex functions with Lipschitz Hessian using only noisy zeroth-order feedback, providing the first tight minimax sample complexity bounds for simple regret. They propose an algorithm combining bootstrapping and mirror-descent stages, achieving matching upper and lower bounds.
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.