Global optimization of Lipschitz functions

TL;DR

This paper develops and analyzes algorithms for efficiently optimizing unknown Lipschitz functions, providing theoretical guarantees and demonstrating their effectiveness compared to existing methods.

Contribution

It introduces the LIPO algorithm for known Lipschitz constants and an adaptive version for unknown constants, with proven consistency and optimal rates.

Findings

LIPO achieves minimax optimal rates under Lipschitz assumptions.

The adaptive LIPO performs well when the Lipschitz constant is unknown.

Numerical experiments show competitive performance against state-of-the-art methods.

Abstract

The goal of the paper is to design sequential strategies which lead to efficient optimization of an unknown function under the only assumption that it has a finite Lipschitz constant. We first identify sufficient conditions for the consistency of generic sequential algorithms and formulate the expected minimax rate for their performance. We introduce and analyze a first algorithm called LIPO which assumes the Lipschitz constant to be known. Consistency, minimax rates for LIPO are proved, as well as fast rates under an additional H\"older like condition. An adaptive version of LIPO is also introduced for the more realistic setup where the Lipschitz constant is unknown and has to be estimated along with the optimization. Similar theoretical guarantees are shown to hold for the adaptive LIPO algorithm and a numerical assessment is provided at the end of the paper to illustrate the potential of this strategy with respect to state-of-the-art methods over typical benchmark problems for global optimization.