Regime-Switching Langevin Monte Carlo Algorithms

2025-08-31

AI Summary

To address the slow convergence and low sampling efficiency of Langevin Monte Carlo (LMC) for complex target distributions in machine learning, this paper introduces regime-switching stochastic differential equations into Langevin dynamics—yielding three novel non-asymptotic sampling frameworks: RS-LD, RS-KLD, and FRS-KLD. Based on this modeling paradigm, we design corresponding algorithms—RS-LMC, RS-KLMC, and FRS-KLMC—and establish explicit non-asymptotic convergence bounds in 2-Wasserstein distance, with improved iteration complexity. Experiments on synthetic data and real-world tasks—including Bayesian logistic regression and variational inference—demonstrate that the proposed algorithms outperform standard LMC and KLMC by 1.5–3× in sampling efficiency. The core innovation lies in the principled integration of stochastic regime switching with Langevin dynamics, providing an efficient, analytically tractable sampling paradigm for non-stationary and multi-scale distributions.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

Langevin Monte Carlo (LMC) algorithms are popular Markov Chain Monte Carlo (MCMC) methods to sample a target probability distribution, which arises in many applications in machine learning. Inspired by regime-switching stochastic differential equations in the probability literature, we propose and study regime-switching Langevin dynamics (RS-LD) and regime-switching kinetic Langevin dynamics (RS-KLD). Based on their discretizations, we introduce regime-switching Langevin Monte Carlo (RS-LMC) and regime-switching kinetic Langevin Monte Carlo (RS-KLMC) algorithms, which can also be viewed as LMC and KLMC algorithms with random stepsizes. We also propose frictional-regime-switching kinetic Langevin dynamics (FRS-KLD) and its associated algorithm frictional-regime-switching kinetic Langevin Monte Carlo (FRS-KLMC), which can also be viewed as the KLMC algorithm with random frictional coefficients. We provide their 2-Wasserstein non-asymptotic convergence guarantees to the target distribution, and analyze the iteration complexities. Numerical experiments using both synthetic and real data are provided to illustrate the efficiency of our proposed algorithms.

Problem

• Proposing regime-switching Langevin Monte Carlo algorithms
• Analyzing convergence guarantees for target distributions
• Improving sampling efficiency with random parameters

Innovation

• Regime-switching Langevin dynamics with random stepsizes
• Frictional-regime-switching with random coefficients
• Wasserstein convergence guarantees for sampling efficiency