Sample Complexity for Quadratic Bandits: Hessian Dependent Bounds and Optimal Algorithms
Abstract
In stochastic zeroth-order optimization, a problem of practical relevance is understanding how to fully exploit the local geometry of the underlying objective function. We consider a fundamental setting in which the objective function is quadratic, and provide the first tight characterization of the optimal Hessian-dependent sample complexity. Our contribution is twofold. First, from an information-theoretic point of view, we prove tight lower bounds on Hessian-dependent complexities by introducing a concept called energy allocation, which captures the interaction between the searching algorithm and the geometry of objective functions. A matching upper bound is obtained by solving the optimal energy spectrum. Then, algorithmically, we show the existence of a Hessian-independent algorithm that universally achieves the asymptotic optimal sample complexities for all Hessian instances. The optimal sample complexities achieved by our algorithm remain valid for heavy-tailed noise distributions, which are enabled by a truncation method.
Cite
@article{arxiv.2306.12383,
title = {Sample Complexity for Quadratic Bandits: Hessian Dependent Bounds and Optimal Algorithms},
author = {Qian Yu and Yining Wang and Baihe Huang and Qi Lei and Jason D. Lee},
journal= {arXiv preprint arXiv:2306.12383},
year = {2023}
}