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

Qian Yu, Yining Wang, Baihe Huang, Qi Lei, Jason D. Lee

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. The main innovation of our approach is the usage of a gradient estimation scheme that exploits the local geometry of the objective function, and we provide sharp analysis for the corresponding estimation bounds.

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