Sample Complexity for Quadratic Bandits: Hessian Dependent Bounds and Optimal Algorithms

• URL: http://arxiv.org/abs/2306.12383v3
• Date: Mon, 25 Dec 2023 06:41:11 GMT
• Title: Sample Complexity for Quadratic Bandits: Hessian Dependent Bounds and Optimal Algorithms
• Authors: Qian Yu, Yining Wang, Baihe Huang, Qi Lei, Jason D. Lee
• Abstract summary: We show the first tight characterization of the optimal Hessian-dependent sample complexity. A Hessian-independent algorithm universally achieves the optimal sample complexities for all Hessian instances. The optimal sample complexities achieved by our algorithm remain valid for heavy-tailed noise distributions.
• Score: 64.10576998630981
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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