Strengths and Weaknesses of Parallel Tempering

TL;DR

This paper analyzes the effectiveness of parallel tempering in two simple energy landscapes, revealing its strengths in certain scenarios and limitations in others, with implications for spin glass studies.

Contribution

It provides a detailed analysis of parallel tempering's performance on specific free energy landscapes, highlighting conditions for exponential speedup and potential biases.

Findings

Parallel tempering scales as the square root of barrier height for exponential speedup.

Marginal speed-up observed in golf course potential.

Large free energy differences slow down equilibration significantly.

Abstract

Parallel tempering, also known as replica exchange Monte Carlo, is studied in the context of two simple free energy landscapes. The first is a double well potential defined by two macrostates separated by a barrier. The second is a `golf course' potential defined by microstates having two possible energies with exponentially more high energy states than low energy states. The equilibration time for replica exchange is analyzed for both systems. For the double well system, parallel tempering with a number of replicas that scales as the square root of the barrier height yields exponential speedup of the equilibration time. On the other hand, replica exchange yields only marginal speed-up for the golf course system. For the double well system, the free energy difference between the two wells has a large effect on the equilibration time. Nearly degenerate wells equilibrate much more slowly than strongly asymmetric wells. It is proposed that this difference in equilibration time may lead to a bias in measuring overlaps in spin glasses. These examples illustrate the strengths and weaknesses of replica exchange and may serve as a guide for understanding and improving the method in various applications.