# Effects of setting the temperatures in the parallel tempering Monte Carlo algorithm

## TL;DR

This paper compares different temperature-setting methods in parallel tempering Monte Carlo, finding that feedback-optimized techniques significantly improve performance on certain problems with first-order phase transitions.

## Contribution

It provides a comprehensive comparison of temperature-setting strategies, highlighting the effectiveness of feedback-optimized methods for specific complex problems.

## Key findings

- Feedback-optimized method speeds up solutions on Wishart problems with first-order transitions.
- No performance gain observed for nonuniform swapping methods on spin-glass problems with smooth transitions.
- Different problem types require tailored temperature-setting approaches for optimal efficiency.

## Abstract

Parallel tempering Monte Carlo has proven to be an efficient method in optimization and sampling applications. Having an optimized temperature set enhances the efficiency of the algorithm through more-frequent replica visits to the temperature limits. The approaches for finding an optimal temperature set can be divided into two main categories. The methods of the first category distribute the replicas such that the swapping ratio between neighbouring replicas is constant and independent of the temperature values. The second-category techniques including the feedback-optimized method, on the other hand, aim for a temperature distribution that has higher density at simulation bottlenecks, resulting in temperature-dependent replica-exchange probabilities. In this paper, we compare the performance of various temperature setting methods on both sparse and fully-connected spin-glass problems as well as fully-connected Wishart problems that have planted solutions. These include two classes of problems that have either continuous or discontinuous phase transitions in the order parameter. Our results demonstrate that there is no performance advantage for the methods that promote nonuniform swapping probabilities on spin-glass problems where the order parameter has a smooth transition between phases at the critical temperature. However, on Wishart problems that have a first-order phase transition at low temperatures, the feedback-optimized method exhibits a time-to-solution speedup of at least a factor of two over the other approaches.