[Paper Review] Global optimization of Lipschitz functions

Cédric Malherbe, Nicolas Vayatis|arXiv (Cornell University)|Mar 7, 2017

Advanced Bandit Algorithms Research19 references53 citations

TL;DR

This paper designs sequential strategies for optimizing unknown Lipschitz functions, proves consistency and minimax rates, and introduces adaptive and fixed-Lipschitz algorithms with theoretical guarantees.

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 Hölder 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.

Motivation & Objective

• Motivate the problem of optimizing an unknown function with a finite Lipschitz constant under costly evaluations.
• Characterize what is achievable by any sequential optimization algorithm over Lipschitz function classes.
• Develop and analyze an algorithm (LIPO) that leverages a known Lipschitz constant for efficient global optimization.
• Extend LIPO to the unknown-Lipschitz setting with an adaptive version (AdaLIPO) that estimates the Lipschitz constant online.
• Provide theoretical guarantees (consistency, minimax rates, fast rates) and empirical demonstrations against state-of-the-art methods.

Proposed method

• Define the optimization framework with a compact domain and Lipschitz function class Lip(k).
• Introduce LIPO: sample X_t+1 uniformly and evaluate only if an upper bound UB_k,t(x) derived from previously observed values could improve the current best.
• Prove consistency of LIPO for Lip(k) and derive finite-time bounds and fast rates under a Decreasing rate around the maximum (κ) condition.
• Extend to AdaLIPO: an adaptive scheme that estimates the Lipschitz constant via a mesh grid and alternates between exploration and exploitation.
• Provide comparisons with Piyavskii/DOO and Pure Random Search, including a discussion of rates and constants.

Experimental results

Research questions

• RQ1 What is the minimal performance achievable by any algorithm over the class of Lipschitz functions in global optimization?
• RQ2 How can one design efficient sequential strategies that exploit global Lipschitz smoothness to improve convergence rates?
• RQ3 Can a fixed-Lipschitz-constant method (LIPO) attain minimax optimality and exponential fast rates on globally smooth problems?
• RQ4 Can one develop a consistent adaptive method (AdaLIPO) that learns the Lipschitz constant during optimization without sacrificing guarantees?
• RQ5 How do the proposed methods compare theoretically and empirically to existing approaches like Piyavskii, DOO, and PURE RANDOM SEARCH?

Key findings

• The paper establishes a necessary and sufficient condition for consistency of sequential optimization over Lipschitz function classes.
• For Lip(k), the minimax rate of convergence is of order n^{-1/d} and LIPO achieves this with matching upper bounds under standard conditions.
• Under a Decreasing rate around the maximum condition (κ ≥ 1), LIPO attains fast rates, including exponential decay for κ = 1 and κ > 1 scenarios.
• AdaLIPO extends LIPO to unknown Lipschitz constants and retains similar theoretical guarantees, with consistent recovery of the relevant Lipschitz class.
• Compared to DOO and Piyavskii, LIPO leverages global smoothness to outperform pure random search in theory and shows favorable empirical performance.
• The results include finite-time bounds, consistency proofs, and illustrative examples (sphere, linear, largest-coordinate functions) demonstrating the rates.