Regret analysis of the Piyavskii-Shubert algorithm for global Lipschitz optimization
This work addresses a theoretical optimization problem for researchers in global optimization and bandit algorithms, providing incremental improvements by solving a specific open problem.
The paper tackles the problem of maximizing a non-concave Lipschitz multivariate function over a compact domain by sequentially querying its values, and it proves new bounds on the number of evaluations needed to reach or certify a given optimization accuracy for the Piyavskii-Shubert algorithm, solving an open problem from 1991 with near-optimal sum of packing numbers.
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.