Regret analysis of the Piyavskii-Shubert algorithm for global Lipschitz optimization

TL;DR

This paper analyzes the Piyavskii-Shubert algorithm for global Lipschitz optimization, providing new bounds on evaluation complexity using a bandit-optimization perspective, and addresses an open problem from prior research.

Contribution

It offers novel theoretical bounds on the number of function evaluations needed for the Piyavskii-Shubert algorithm, solving an open problem in the field.

Findings

Derived new bounds on evaluation complexity.

Applied bandit-optimization framework to analyze the algorithm.

Solved an open problem from Hansen et al. (1991).

Abstract

We consider the problem of maximizing a non-concave Lipschitz multivariate function over a compact domain by sequentially querying its (possibly perturbed) values. We study a natural algorithm designed originally by Piyavskii and Shubert in 1972, for which we prove new bounds on the number of evaluations of the function needed to reach or certify a given optimization accuracy. Our analysis uses a bandit-optimization viewpoint and solves an open problem from Hansen et al.\ (1991) by bounding the number of evaluations to certify a given accuracy with a near-optimal sum of packing numbers.