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Sample Complexity for Quadratic Bandits: Hessian Dependent Bounds and Optimal Algorithms by Yu, Qian; Wang, Yining; Huang, Baihe; Lei, Qi; Lee, Jason D. is a scholarly article available to read on EtoBox.

What is Sample Complexity for Quadratic Bandits: Hessian Dependent Bounds and Optimal Algorithms about?

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.

Author

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

Published

2023

Language

EN