Exchanging Replicas with Unequal Cost, Infinitely and Permanently

Abstract

We developed a replica exchange method that is effectively parallelizable even if the computational cost of the Monte Carlo moves in the parallel replicas are considerably different, for instance, because the replicas run on different types of processor units or because of the algorithmic complexity. To prove detailed-balance, we make a paradigm shift from the common conceptual viewpoint in which the set of parallel replicas represents a high-dimensional superstate, to an ensemble-based criterion in which the other ensembles represent an environment that might or might not participate in the Monte Carlo move. In addition, based on a recent algorithm for computing permanents, we effectively increase the exchange rate to infinite without the steep factorial scaling as a function of the number of replicas. We illustrate the effectiveness of this replica exchange methodology by combining it with a quantitative path sampling method, replica exchange transition interface sampling (RETIS), in which the costs for a Monte Carlo move can vary enormously as paths in a RETIS algorithm do not have the same length and the average path lengths tend to vary considerably for the different path ensembles that run in parallel. This combination, coined ∞RETIS, was tested on three model systems.

PyRETIS 3: Conquering rare and slow events without boundaries

“…These approaches are provided by the scikit-learn python package 48 The new code structure allows for effective parallel simulations to be performed in different ensembles, potentially speeding up sampling performance by optimising cluster usage. An effective simulation handler procedure to advantageously execute multiple parallel simulations has been recently published by Roet et al 49 and it can be expected to be included in forthcoming software releases.…”

How far can we stretch the timescale with RETIS?

“…To effectively compute ensembles in parallel without idle time, we recently developed an asynchronous replica exchange method [34] that executes multiple swaps upon completion of each path by a worker. Previous asynchronous replica exchange methods for parallel tempering or Hamiltonian exchange [35][36][37] differ in that they maintain a fast state generation move, typically comprising a single MD step, while aiming to solve the issue of hardware diversity (slow vs. fast workers), rather than addressing inherent CPU cost imbalances of the states (long vs. short paths).…”

“…The algorithm of ref. [34] can be proven in a fundamental different way based on a twisted balance relation in which the superstate view is replaced by an ensemble plus environment based view. A crucial aspect of the method is that, like ref.…”

“…Furthermore, it is possible to perfectly replicate the hypothetical outcome of performing an infinite number of swapping moves by calculating the sum of all probabilities for every potential permutation between free ensembles and free states [34]. The idea of doing an infinite number of swaps was proposed before [38], but the steep factorial increase in the number of permutations drastically restricts the number of replicas that can be involved.…”

“…Addressing this problem, ref. [34] formulates the summations as permanents of weight matrices. This approach provides a versatile solution that outperforms the factorial method [39] and enables even faster computation by exploiting specific characteristics of the weight matrix.…”