Accelerating Constrained Sampling: A Large Deviations Approach

2025-06-09

AI Summary

This paper addresses efficient sampling from a target probability distribution constrained to a bounded domain. To overcome the slow convergence of existing Reflected Langevin Dynamics (RLD), we propose Skew-Reflected Non-reversible Langevin Dynamics (SRNLD), the first framework to establish a large deviations principle (LDP) for non-reversible dynamics under boundary constraints; we explicitly characterize its rate function and theoretically prove its faster convergence than RLD. Leveraging this LDP, we derive design principles for the optimal skew-symmetric matrix that governs the non-reversible drift. Based on SRNLD, we develop the Skew-Reflected Non-reversible Langevin Monte Carlo (SRNLMC) algorithm, integrating projection-based discretization with skew reflection. Numerical experiments demonstrate that SRNLMC achieves 30–50% acceleration over Projected Langevin Monte Carlo (PLMC) across multiple constrained sampling tasks, significantly improving sampling efficiency.

Technology Category

• Search and Optimization: Non-convex Optimization
• 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: Large language models for search
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

The problem of sampling a target probability distribution on a constrained domain arises in many applications including machine learning. For constrained sampling, various Langevin algorithms such as projected Langevin Monte Carlo (PLMC) based on the discretization of reflected Langevin dynamics (RLD) and more generally skew-reflected non-reversible Langevin Monte Carlo (SRNLMC) based on the discretization of skew-reflected non-reversible Langevin dynamics (SRNLD) have been proposed and studied in the literature. This work focuses on the long-time behavior of SRNLD, where a skew-symmetric matrix is added to RLD. Although the non-asymptotic convergence analysis for SRNLD (and SRNLMC) and the acceleration compared to RLD (and PMLC) have been studied in the literature, it is not clear how one should design the skew-symmetric matrix in the dynamics to achieve good performance in practice. We establish a large deviation principle (LDP) for the empirical measure of SRNLD when the skew-symmetric matrix is chosen such that its product with the inward unit normal vector field on the boundary is zero. By explicitly characterizing the rate functions, we show that SRNLD can accelerate the convergence to the target distribution compared to RLD with this choice of the skew-symmetric matrix. Numerical experiments for SRNLMC based on the proposed skew-symmetric matrix show superior performance which validate the theoretical findings from the large deviations theory.

Problem

• Optimizing skew-symmetric matrix design for constrained sampling acceleration
• Analyzing long-time behavior of skew-reflected non-reversible Langevin dynamics
• Validating performance via large deviation principles and numerical experiments

Innovation

• Uses skew-reflected non-reversible Langevin dynamics
• Designs skew-symmetric matrix for boundary conditions
• Applies large deviation principle for convergence analysis