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

Gaëtan Serré, Pierre Germain, Samuel Gruffaz, Argyris Kalogeratos

TL;DR

This work targets exploration-exploitation in MKV-based global optimization by injecting noise directly into the probability-law dynamics. It introduces two plug-in rho-noise strategies: Stochastic Moment Dynamics (SMD), which perturbs macroscopic observables such as mean, second moment, and variance, and Geometric Common Noise (GCN), which injects noise in the tangent space of the probability-measure manifold via RKHS kernels. Across multiple MKV dynamics (Langevin, CBO, SBS, MSGD) and seven multimodal benchmarks, both approaches enhance exploration and convergence, with SMD (Mean+Var) offering a favorable trade-off between performance and computational cost. These methods broaden the MKV toolkit for global optimization and open paths for adaptive noise strategies and extensions to sampling tasks.

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.

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

TL;DR

This work targets exploration-exploitation in MKV-based global optimization by injecting noise directly into the probability-law dynamics. It introduces two plug-in rho-noise strategies: Stochastic Moment Dynamics (SMD), which perturbs macroscopic observables such as mean, second moment, and variance, and Geometric Common Noise (GCN), which injects noise in the tangent space of the probability-measure manifold via RKHS kernels. Across multiple MKV dynamics (Langevin, CBO, SBS, MSGD) and seven multimodal benchmarks, both approaches enhance exploration and convergence, with SMD (Mean+Var) offering a favorable trade-off between performance and computational cost. These methods broaden the MKV toolkit for global optimization and open paths for adaptive noise strategies and extensions to sampling tasks.

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.
Paper Structure (46 sections, 5 theorems, 79 equations, 4 figures, 2 tables)

This paper contains 46 sections, 5 theorems, 79 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. McKean-Vlasov systems.
  3. Noise as a plug-in component.
  4. Contributions.
  5. Notations.
  6. Preliminaries
  7. Methodology
  8. Noise through moment perturbation
  9. Mean-field optimization and particle systems.
  10. Common noise and conditional MKV theory.
  11. Macroscopic perturbations via stochastic moments.
  12. Macroscopic observable of interest $F$.
  13. SMD General Recipe.
  14. Positivity constraints and Bessel dynamics.
  15. Noise using tangent space
  16. ...and 31 more sections

Key Result

Lemma 3.1

SMD-Mean $F=F_{\text{mean}}$, $(a,s)=(0_d,I_d)$. SMD Second-order moment $F=F_{\text{s.m}}$, $(a,s)=(a_+^\delta,s_+^\delta)$. SMD Variance $F=F_{\text{var}}$, $(a,s)=(a_+^\delta,s_+^\delta)$. SMD (Mean + variance) $F=[F_{\text{mean}},F_{\text{var}}]$, $(a,s)=((0_d,a_+^\delta),(I_d,s_+^\delta))$. where $\tilde{b}(x,\rho)$ is the same of the variance perturbation case.

Figures (4)

  • Figure 1: Impact of perturbed observables on CBO (left) and multi-start gradient descent dynamics (right) on Ackley function in dimension $4$ with $30$ particles. The mean (line) and standard deviation (shaded area) of the best found value are reported over $5$ runs.
  • Figure 2: Impact of $\sigma$ on the RKHS noise for the SBS dynamics (left) and the CBO dynamics (right) on the Levy function in dimension $10$ with $10$ particles. The mean (line) and standard deviation (shaded area) of the best found value are reported over $5$ runs.
  • Figure 3: Illustration of the problem-dependent effects of noise injection methods in the multimodal setting. The mean (line) and standard deviation (shaded area) of the best found value are reported over $5$ runs. In low dimension with sufficient particles (left), the effect of SMD is negligible, while in very high dimension with scarce particles (right), SMD yields modest improvements.
  • Figure 4: Impact of perturbed observables on MSGD (left) and SBS dynamics (right) on the Square function in dimension $20$ with $2$ particles. The mean (line) and standard deviation (shaded area) of the best found value are reported over $5$ runs. In this unimodal setting, the noise injection slightly degrades performance by introducing unnecessary perturbations.

Theorems & Definitions (9)

  • Lemma 3.1
  • Definition 3.2: $\mathcal{H}_d$-Gaussian random field
  • Lemma 3.3
  • Remark 3.4
  • Lemma 3.5
  • Theorem 3.6
  • Theorem 3.7
  • proof
  • proof