Enhancing Gradient-based Discrete Sampling via Parallel Tempering

TL;DR

This paper introduces PTDLP, a novel parallel tempering method combined with discrete Langevin proposals, improving sampling efficiency and convergence in complex, multimodal discrete distributions.

Contribution

We develop PTDLP, integrating parallel tempering with discrete Langevin proposals, and introduce an automatic temperature scheme for better sampling in high-dimensional discrete spaces.

Findings

Faster mixing compared to single-chain methods

Effective in synthetic, RBM, and deep energy models

Converges non-asymptotically to target distribution

Abstract

While gradient-based discrete samplers are effective in sampling from complex distributions, they are susceptible to getting trapped in local minima, particularly in high-dimensional, multimodal discrete distributions, owing to the discontinuities inherent in these landscapes. To circumvent this issue, we combine parallel tempering, also known as replica exchange, with the discrete Langevin proposal and develop the Parallel Tempering enhanced Discrete Langevin Proposal (PTDLP), which are simulated at a series of temperatures. Significant energy differences prompt sample swaps, which are governed by a Metropolis criterion specifically designed for discrete sampling to ensure detailed balance is maintained. Additionally, we introduce an automatic scheme to determine the optimal temperature schedule and the number of chains, ensuring adaptability across diverse tasks with minimal tuning. Theoretically, we establish that our algorithm converges non-asymptotically to the target energy and exhibits faster mixing compared to a single chain. Empirical results further emphasize the superiority of our method in sampling from complex, multimodal discrete distributions, including synthetic problems, restricted Boltzmann machines, and deep energy-based models.