A divergence-based condition to ensure quantile improvement in black-box global optimization

TL;DR

This paper introduces a divergence-based framework for analyzing and designing black-box global optimization algorithms, providing theoretical guarantees for quantile improvement and applying it to novel methods including mixture models and heavy-tailed proposals.

Contribution

The paper proposes a new divergence-decrease condition framework for black-box optimization, enabling theoretical analysis of proposal improvements and introducing two novel algorithms.

Findings

Framework quantifies proposal improvement at each iteration.

Information-geometric optimization fits within the new framework.

Proposes two new algorithms with proven proposal improvement.

Abstract

Black-box global optimization aims at minimizing an objective function whose analytical form is not known. To do so, many state-of-the-art methods rely on sampling-based strategies, where sampling distributions are built in an iterative fashion, so that their mass concentrate where the objective function is low. Despite empirical success, the theoretical study of these methods remains difficult. In this work, we introduce a new framework, based on divergence-decrease conditions, to study and design black-box global optimization algorithms. Our approach allows to establish and quantify the improvement of proposals at each iteration, in terms of expected value or quantile of the objective. We show that the information-geometric optimization approach fits within our framework, yielding a new approach for its analysis. We also establish proposal improvement results for two novel algorithms, one related with the cross-entropy approach with mixture models, and another one using heavy-tailed sampling proposal distributions.