On the Infinite Swapping Limit for Parallel Tempering

TL;DR

This paper analyzes the infinite swapping limit in parallel tempering algorithms, introducing a new theoretical framework and practical variations that improve understanding and efficiency of sampling complex systems.

Contribution

It develops a large deviation theory-based performance criterion and constructs a limit process for infinite swap rates, offering new insights into parallel tempering methods.

Findings

The swap rate increases convergence speed.

A limit process without actual swaps is constructed.

Practical near-optimal variations are proposed.

Abstract

Parallel tempering, also known as replica exchange sampling, is an important method for simulating complex systems. In this algorithm simulations are conducted in parallel at a series of temperatures, and the key feature of the algorithm is a swap mechanism that exchanges configurations between the parallel simulations at a given rate. The mechanism is designed to allow the low temperature system of interest to escape from deep local energy minima where it might otherwise be trapped, via those swaps with the higher temperature components. In this paper we introduce a performance criteria for such schemes based on large deviation theory, and argue that the rate of convergence is a monotone increasing function of the swap rate. This motivates the study of the limit process as the swap rate goes to infinity. We construct a scheme which is equivalent to this limit in a distributional sense, but which involves no swapping at all. Instead, the effect of the swapping is captured by a collection of weights that influence both the dynamics and the empirical measure. While theoretically optimal, this limit is not computationally feasible when the number of temperatures is large, and so variations that are easy to implement and nearly optimal are also developed.