[Paper Review] Regret analysis of the Piyavskii-Shubert algorithm for global Lipschitz optimization

Clément Bouttier, Tommaso Cesari+2 more|arXiv (Cornell University)|Feb 6, 2020

Advanced Bandit Algorithms Research34 references4 citations

TL;DR

This paper provides the first regret analysis of the Piyavskii-Shubert algorithm for global Lipschitz optimization, establishing tight sample complexity bounds for both non-certified and certified settings. It proves that the algorithm achieves near-optimal performance by leveraging packing numbers of function layers, resolving an open problem from Hansen et al. (1991) on certification complexity.

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.

Motivation & Objective

• To analyze the sample complexity of the Piyavskii-Shubert algorithm in global Lipschitz optimization under sequential black-box access.
• To resolve an open problem from Hansen et al. (1991) on the number of evaluations needed to certify a given optimization accuracy.
• To establish bounds on the number of function evaluations required to achieve ε-optimality in both non-certified and certified variants of the algorithm.
• To demonstrate that the algorithm adapts to multiple shapes and scales through a novel analysis involving layers of the objective function.

Proposed method

• The algorithm uses a piecewise-conic proxy function that upper-bounds the unknown function based on Lipschitz continuity and past evaluations.
• At each round, the next query point is chosen as a maximizer of the proxy function, ensuring progressive refinement of the search.
• For certified algorithms, an error certificate is computed as the maximum difference between the proxy function and the current recommendation.
• The analysis introduces a novel integral representation of the sum of packing numbers over function layers, replacing near-optimal sets to improve adaptivity.
• The method leverages a bandit-optimization perspective to derive bounds on simple regret and certification error.
• It establishes that the sample complexity scales with the sum of packing numbers of function layers, achieving near-optimality.

Experimental results

Research questions

• RQ1 What is the sample complexity of the Piyavskii-Shubert algorithm in the non-certified setting for achieving ε-optimality?
• RQ2 How many evaluations are required to certify that a recommendation is ε-optimal in the certified setting?
• RQ3 Can the algorithm’s performance be bounded using packing numbers of function layers rather than near-optimal sets?
• RQ4 Does the Piyavskii-Shubert algorithm achieve near-optimal sample complexity in both certified and non-certified settings?
• RQ5 How does the algorithm adapt to different shapes and scales of the objective function under Lipschitz constraints?

Key findings

• The paper establishes that the number of evaluations needed to achieve ε-optimality in the non-certified setting is bounded by a sum of packing numbers of function layers.
• For the certified setting, the number of evaluations required to certify ε-optimality is bounded by a near-optimal sum of packing numbers, resolving an open problem from Hansen et al. (1991).
• The analysis shows that the algorithm's performance adapts to the intrinsic geometry of the objective function through layer-based packing numbers.
• The bounds are tight up to logarithmic factors and improve upon previous worst-case bounds that scale as n⁻¹/d.
• The certified version of the algorithm achieves a sample complexity that matches the instance-dependent lower bound up to a constant factor, as shown in related work by Bachoc et al. [2].
• The method provides an integral representation of the sum of packing numbers, improving multiplicative constants in the representation compared to prior work.