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

TL;DR

This paper introduces DREXEL and DREAM, novel Langevin-like samplers with replica exchange for efficient exploration of complex non-convex discrete energy landscapes, ensuring convergence and improved sampling performance.

Contribution

The paper proposes the DREXEL and DREAM samplers that incorporate replica exchange and temperature differences to enhance exploration in discrete energy landscapes, with theoretical guarantees.

Findings

Proven to satisfy detailed balance and converge to the target distribution.

Demonstrated efficiency in synthetic simulations, Ising models, and deep energy-based models.

Outperformed existing methods in exploring complex energy landscapes.

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 that the proposed samplers satisfy detailed balance and converge to the target distribution under mild conditions. Experiments across 2d synthetic simulations, sampling from Ising models and restricted Boltzmann machines, and training deep energy-based models further confirm their efficiency in exploring non-convex discrete energy landscapes.