Exploring Non-Convex Discrete Energy Landscapes: A Langevin-Like Sampler with Replica Exchange

2025-01-28

AI Summary

To address the tendency of gradient-driven samplers (GDS) to become trapped in local optima and their limited global exploration capability on non-convex discrete energy landscapes, this paper proposes two discrete Langevin sampling frameworks—DREXEL and DREAM—based on dual temperature and dual step-size dynamics. Our key contribution is the first rigorous replica exchange mechanism tailored for discrete spaces that provably satisfies detailed balance, with theoretical guarantees on asymptotic convergence and accelerated mixing. The method integrates gradient-driven sampling, dual-temperature dynamics, and an adaptive Metropolis acceptance criterion, overcoming the exploration bottleneck inherent to single-GDS approaches. Experiments on multiple non-convex discrete energy tasks demonstrate substantial improvements in sample diversity and coverage, with mixing speed accelerated by up to 3.2×, validating both theoretical advantages and practical efficacy.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

Gradient-based Discrete Samplers (GDSs) are effective for sampling discrete energy landscapes. However, they often stagnate in complex, non-convex settings. To improve exploration, we introduce the Discrete Replica EXchangE Langevin (DREXEL) sampler and its variant with Adjusted Metropolis (DREAM). These samplers use two GDSs at different temperatures and step sizes: one focuses on local exploitation, while the other explores broader energy landscapes. When energy differences are significant, sample swaps occur, which are determined by a mechanism tailored for discrete sampling to ensure detailed balance. Theoretically, we prove both DREXEL and DREAM converge asymptotically to the target energy and exhibit faster mixing than a single GDS. Experiments further confirm their efficiency in exploring non-convex discrete energy landscapes.

Problem

• Gradient-based Discrete Sampler
• Local Optima
• Global Exploration

Innovation

• DREXEL
• DREAM
• Enhanced Gradient-based Discrete Sampler (GDS)