A method to reduce the rejection rate in Monte Carlo Markov Chains

TL;DR

The paper introduces the reduced-rejection-rate (RRR) method, a novel Monte Carlo sampling technique that significantly lowers rejection rates in systems with discrete variables, enhancing efficiency especially in complex or dense graph scenarios.

Contribution

It proposes a new prior-based approach within the Metropolis-Hastings scheme that extends rejection-free methods to a broader class of models, including non-sparse graphs.

Findings

Achieves near rejection-free sampling in various models

Demonstrates efficiency in quantum spin models with transverse fields

Provides publicly available, extensible code implementation

Abstract

We present a method for Monte Carlo sampling on systems with discrete variables (focusing in the Ising case), introducing a prior on the candidate moves in a Metropolis-Hastings scheme which can significantly reduce the rejection rate, called the reduced-rejection-rate (RRR) method. The method employs same probability distribution for the choice of the moves as rejection-free schemes such as the method proposed by Bortz, Kalos and Lebowitz (BKL) [Bortz et al. J.Comput.Phys. 1975]; however, it uses it as a prior in an otherwise standard Metropolis scheme: it is thus not fully rejection-free, but in a wide range of scenarios it is nearly so. This allows to extend the method to cases for which rejection-free schemes become inefficient, in particular when the graph connectivity is not sparse, but the energy can nevertheless be expressed as a sum of two components, one of which is computed on a sparse graph and dominates the measure. As examples of such instances, we demonstrate that the method yields excellent results when performing Monte Carlo simulations of quantum spin models in presence of a transverse field in the Suzuki-Trotter formalism, and when exploring the so-called robust ensemble which was recently introduced in Baldassi et al. [PNAS 2016]. Our code for the Ising case is publicly available [https://github.com/carlobaldassi/RRRMC.jl], and extensible to user-defined models: it provides efficient implementations of standard Metropolis, the RRR method, the BKL method (extended to the case of continuous energy specra), and the waiting time method [Dall and Sibani Comput.Phys.Commun. 2001].