[論文レビュー] Enhancing Exploration in Global Optimization by Noise Injection in the Probability Measures Space

Gaëtan Serré, Pierre Germain+2 more|arXiv (Cornell University)|Jan 30, 2026

Stochastic Gradient Optimization Techniques被引用数 0

ひとこと要約

The paper introduces two principled ways to inject noise into McKean–Vlasov (MKV) probability-law dynamics to boost exploration in multimodal global optimization, and demonstrates improved performance across multiple MKV-based methods.

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.

研究の動機と目的

• Motivate exploration challenges in MKV-based global optimization and the risk of trapping in multimodal landscapes.
• Propose two modular noise-injection mechanisms (SMD and GCN) that perturb the probability-law dynamics without changing the underlying MKV framework.
• Demonstrate through extensive experiments that rho-noise improves exploration and convergence across Langevin, CBO, SBS, and MSGD methods on multimodal benchmarks.

提案手法

• Two noise-injection schemes into MKV dynamics: Stochastic Moment Dynamics (SMD) using perturbations of macroscopic observables, and Geometric Common Noise (GCN) injecting noise in the tangent space of the probability law manifold.
• SMD adds a beta-weighted perturbation to the particle dynamics that targets moments like mean, second-order moment, and variance (with delta-Bessel process safeguards for positivity).
• GCN injects infinite-dimensional noise via RKHS-valued Gaussian fields aligned with the SVGD tangent space, yielding a Langevin-type stochastic flow on the space of probability laws.
• The framework is demonstrated as a plug-in module applicable to any MKV-based method (e.g., MSGD, Langevin, SBS, CBO).
• Theoretical grounding links GCN to a stochastic partial differential equation on rho and shows its relation to SMD through kernel bandwidth.

実験結果

リサーチクエスチョン

• RQ1 直接に確率法則ダイナミクス（rho-noise）にノイズを注入することが、MKVベースのグローバル最適化における探索を強化するか？
• RQ2 提案された二つのrho-noiseアプローチ（SMDとGCN）は、さまざまなMKVダイナミクスで探索と収束をどのように改善するか？
• RQ3 ノイズ注入をモジュール化して、さまざまな粒子ベースの最適化スキームへのプラグインとして機能させられるか？
• RQ4 SVGDベース設定におけるカーネル帯域と観測可能の選択がrho-noiseの有効性に与える影響は？
• RQ5 提案手法は標準的な多峰ベンチマークで素のMKVダイナミクスを一貫して上回るか？

主な発見

Benchmark	CBO	SMD-CBO Mean	SMD-CBO M^2	SMD-CBO Var	SMD-CBO Mean+Var	GCN-CBO	p-value	Benchmark	SBS	SMD-SBS Mean	SMD-SBS M^2	SMD-SBS Var	SMD-SBS Mean+Var	GCN-SBS	p-value	Benchmark	Langevin	SMD-Langevin Mean	SMD-Langevin M^2	SMD-Langevin Var	SMD-Langevin Mean+Var	GCN-Langevin	p-value	Benchmark	MSGD	SMD-MSGD Mean	SMD-MSGD M^2	SMD-MSGD Var	SMD-MSGD Mean+Var	GCN-MSGD	p-value
Ackley	21.097	20.615	21.083	21.088	20.610	20.591	0.000	Deb N.1	-1.000	-1.000	-1.000	-1.000	-1.000	-1.000	0.000	Griewank	20.279	19.562	19.657	19.683	20.073	19.506	0.237	Levy	102.351	86.857	95.816	95.945	81.912	85.146	0.000	Rastrigin	253.358	238.943	250.021	250.463	240.406	238.768	0.000	Schwefel	5157.189	5127.111	5139.564	5146.530	5199.557	5111.855	0.387	Styblinski-Tang	-19.513	-20.798	-20.006	-19.881	-20.091	-20.191	0.000	Avg Rank	5.14	2.00	3.29	4.14	3.00	1.29	ECR	1.0648	1.0097	1.0424	1.0443	1.0132	1.0104

• rho-noiseは複数のMKVダイナミクス（Langevin、CBO、SBS、MSGD）に対して探索と収束を大幅に改善する。
• SMDは有限次元の観測可能ベースの摂動を提供し、マクロ的な確率性を駆動してベース間の跳躍挙動を改善する。
• GCNは無限次元の接空間ノイズを導入し、確率法則上にLangevin型ダイナミクスを生み出し、カーネル帯域幅によって白色ノイズとSMD平均の間を補間できる。
• 次元20、粒子数150の7つの多峰関数での経験的結果は、ノイズ強化手法の一貫した改善と有利なランクを示す。
• rho-noiseフレームワークはグローバル最適化とサンプリングのための汎用ツールキットであり、ベンチマーク関数で顕著な性能向上を示す。