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

TL;DR

This paper characterizes the optimal sample complexity for quadratic bandits in stochastic zeroth-order optimization, revealing Hessian-dependent bounds and proposing a universal algorithm that achieves these bounds.

Contribution

It provides the first tight Hessian-dependent sample complexity bounds and introduces a Hessian-independent algorithm that is asymptotically optimal across all instances.

Findings

Established tight lower bounds on Hessian-dependent complexities.

Designed a Hessian-independent algorithm achieving optimal sample complexity.

Validated the algorithm's effectiveness under heavy-tailed noise distributions.

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.