Enhancing Exploration in Global Optimization by Noise Injection in the Probability Measures Space

TL;DR

This paper introduces two novel noise injection methods into the probability law dynamics of McKean-Vlasov systems to improve exploration and convergence in global optimization tasks, especially for multimodal functions.

Contribution

The paper proposes two principled noise injection strategies into MKV systems, enhancing exploration capabilities in global optimization frameworks.

Findings

Both methods improve exploration in multimodal landscapes.

Enhanced convergence observed across various MKV-based algorithms.

Framework is versatile and applicable to multiple optimization dynamics.

Abstract

McKean-Vlasov (MKV) systems provide a unifying framework for recent state-of-the-art particlebased methods for global optimization. While individual particles follow stochastic trajectories, the probability law evolves deterministically in the mean-field limit, potentially limiting exploration in multimodal landscapes. We introduce two principled approaches to inject noise directly into the probability law dynamics: a perturbative method based on conditional MKV theory, and a geometric approach leveraging tangent space structure. While these approaches are of independent interest, the aim of this work is to apply them to global optimization. Our framework applies generically to any method that can be formulated as a MKV system. Extensive experiments on multimodal objective functions demonstrate that both our noise injection strategies enhance consistently the exploration and convergence across different configurations of dynamics, such as Langevin, Consensus-Based Optimization, and Stein Boltzmann Sampling, providing a versatile toolkit for global optimization.