Hesitant Adaptive Search with Estimation and Quantile Adaptive Search for Global Optimization with Noise

TL;DR

This paper introduces a new adaptive search framework for noisy global optimization problems, providing finite-time performance bounds and emphasizing the importance of sampling over estimation refinement.

Contribution

It proposes Hesitant Adaptive Search with Estimation and extends it to Quantile Adaptive Search, offering the first finite-time analysis for noisy objective functions in adaptive random search.

Findings

Performance bound is cubic in dimension under certain conditions.

Sampling improving points is more effective than refining estimates.

Framework applicable to simulation-based optimization problems.

Abstract

Adaptive random search approaches have been shown to be effective for global optimization problems, where under certain conditions, the expected performance time increases only linearly with dimension. However, previous analyses assume that the objective function can be observed directly. We consider the case where the objective function must be estimated, often using a noisy function, as in simulation. We present a finite-time analysis of algorithm performance that combines estimation with a sampling distribution. We present a framework called Hesitant Adaptive Search with Estimation, and derive an upper bound on function evaluations that is cubic in dimension, under certain conditions. We extend the framework to Quantile Adaptive Search with Estimation, which focuses sampling points from a series of nested quantile level sets. The analyses suggest that computational effort is better expended on sampling improving points than refining estimates of objective function values during the progress of an adaptive search algorithm.