Divide, Interact, Sample: The Two-System Paradigm

2025-09-11

AI Summary

This work unifies mean-field, ensemble-chain, and adaptive Monte Carlo sampling paradigms. We propose a unified framework based on bidirectional interaction between two subsystems: one performs particle updates, while the other dynamically constructs a reference distribution; the two subsystems evolve alternately. The framework reveals that ensemble methods are discrete approximations of ideal mean-field dynamics under finite-particle constraints, provides a practical particle-based realization of mean-field dynamics, and naturally recovers adaptive single-chain algorithms. Our method integrates overdamped and underdamped Langevin MCMC, introduces cross-particle update schemes, and incorporates time-averaged statistical estimation. Empirical evaluation on synthetic benchmarks and real-world posterior inference tasks demonstrates that, compared to the No-U-Turn Sampler (NUTS), our approach achieves up to an order-of-magnitude improvement in effective sample size per gradient evaluation.

Technology Category

• Machine Learning: Ensemble Methods
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Probabilistic Programming

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

Mean-field, ensemble-chain, and adaptive samplers have historically been viewed as distinct approaches to Monte Carlo sampling. In this paper, we present a unifying {two-system} framework that brings all three under one roof. In our approach, an ensemble of particles is split into two interacting subsystems that propose updates for each other in a symmetric, alternating fashion. This cross-system interaction ensures that the overall ensemble has $ρ(x)$ as its invariant distribution in both the finite-particle setting and the mean-field limit. The two-system construction reveals that ensemble-chain samplers can be interpreted as finite-$N$ approximations of an ideal mean-field sampler; conversely, it provides a principled recipe to discretize mean-field Langevin dynamics into tractable parallel MCMC algorithms. The framework also connects naturally to adaptive single-chain methods: by replacing particle-based statistics with time-averaged statistics from a single chain, one recovers analogous adaptive dynamics in the long-time limit without requiring a large ensemble. We derive novel two-system versions of both overdamped and underdamped Langevin MCMC samplers within this paradigm. Across synthetic benchmarks and real-world posterior inference tasks, these two-system samplers exhibit significant performance gains over the popular No-U-Turn Sampler, achieving an order of magnitude higher effective sample sizes per gradient evaluation.

Problem

• Unifying mean-field, ensemble-chain, and adaptive samplers under a single framework
• Developing symmetric two-system interaction for Monte Carlo sampling
• Enhancing sampling performance over traditional methods like NUTS

Innovation

• Two-system framework unifies samplers
• Interacting subsystems enable symmetric updates
• Adaptive dynamics from single-chain statistics