Efficient assignment of the temperature set for Parallel Tempering

Abstract

We propose a simple algorithm able to identify a set of temperatures for a Parallel Tempering Monte Carlo simulation, that maximizes the probability that the configurations drift across all temperature values, from the coldest to the hottest ones, and vice versa. The proposed algorithm starts from data gathered from relatively short Monte Carlo simulations and is straightforward to implement. We assess its effectiveness on a test case simulation of an Edwards–Anderson spin glass on a lattice of 12³ sites.

Introduction

The Parallel Tempering (PT) method was first introduced in literature by Swendsen and Wang [1] in order to reduce the long correlation time characteristic of the Monte Carlo (MC) simulation of complex systems, and then further developed by several authors [2], [3].

Dealing with spin glasses, proteins or neural networks means handling rough landscapes of the free energy and facing problems such as the stall of the system in a metastable state. PT tries to overcome these problems by simulating many copies of the system in parallel (hence the name) at different temperatures T_i (and correspondingly, inverse temperatures β_i) and allowing copies, whose inverse temperature (energy) difference is Δβ(ΔE), to exchange their temperatures with probability min {1, e^ΔβΔE}. A given configuration will perform many times a walk in temperature space, wandering from high temperatures, where equilibration in fast, to low temperatures, where relaxation times can be long, exploring efficiently the complex energy landscape [4] with correct statistical weights, so the system can more easily overcome barriers in the free energy landscape. In spite of the simplicity and the power of this algorithm, its application can backfire in all cases in which the walk of the configuration in temperature space gets stuck.

Choosing a good temperature ensemble for a PT-enabled Monte Carlo simulation is not an easy task, as one must in principle satisfy many conflicting requirements, such as: (i) fast equilibration (sufficient time spent at “hot” temperatures); (ii) able to efficiently manage the study of “cold” enough temperatures (typically where the interesting physics is); (iii) good acceptance ratio (a large enough set of temperatures allowing a good exchange efficiency, but not too large, kicking the system away from its position, and preventing equilibration); (iv) an acceptance ratio independent of the temperature and (v) limited computational requests (a small set of temperatures).

Over the years, several methods have been proposed to define an optimal (or, at least, a “good”) set of temperatures. A simple approach might be choosing a geometric progression in the desired range of N temperatures [β₁, β_N]; however, if the system presents a divergent specific heat, this approach does not give the desired effect [5]. For this reason more complex methods [6], [7], [8] aim to obtain a better temperature set tuning values according to the theoretical acceptance, expressing the acceptance itself as a weighted function of the specific heat. Recently, a new method has been proposed, the so called iterative feedback-optimized, in which one considers the diffusion of a configuration in temperature space, and tries to keep its drift velocity constant, by appropriately choosing the specific temperature values (which are fixed in number) [9], [10], [11]. In this case one defines a flux function for every temperature T_k:

$$f(T_k)=\frac{n_{\mathit{up}}(T_k)}{n_{\mathit{up}}(T_k)+n_{\mathit{down}}(T_k)}\text{,}$$

where n_up(n_down) is the normalized fraction of configurations which have visited for at least one MC sweep the lowest (highest) temperature and are traveling away from it, and tries to maximize a current defined as

$$j=D(T)\eta (T)\frac{\mathit{df}}{\mathit{dT}}$$

in which D(T) is the so called diffusivity; this is done by appropriately adjusting the temperature distribution density η(T) (for a better explanation of this method see [9], [10], [11]). Once f(T_k) is known, an optimal solution for η can be derived; however, measuring f(T_k) (or equivalently n_up and n_down) takes very large computational resources. Other approaches have been suggested, a review can be found in [12].

In this paper, we present a very simple algorithm, whose tuning requires limited computational resources, that determines a set of temperatures β_k that equalizes the exchange probability of all pairs of copies of the system sitting at nearby temperatures: in this way we try to increase the probability for a configuration to move across all available temperatures back and forth between the two (fixed) extremal ones. Our algorithm is based on the analysis of the energy distribution of the MC configurations at some given temperatures and we calculate their overlap to extract the acceptance ratio of a given exchange proposal. In our approach the number of available temperatures N and the lowest and highest temperatures are fixed: this is appropriate if one wants to explore some given physically interesting temperature range and has a given amount of available computational resources. This paper proceeds in Section 2 introducing our optimization algorithm, followed (Section 3) by a review of performance, measured in terms of a few specific metrics in a test case that would be difficult to handle for the usual PT algorithm, in order to better assess the advantages of our approach. The paper ends with some concluding remarks (Section 4).