Performance of machine-learning-assisted Monte Carlo in sampling from simple statistical physics models

TL;DR

This paper provides a comprehensive theoretical analysis of machine-learning-assisted Monte Carlo sampling, focusing on Sequential Tempering with a shallow MADE architecture for the Curie-Weiss model, and compares its performance with and without local Metropolis steps.

Contribution

It offers the first complete analytic study of Sequential Tempering with a shallow MADE, detailing optimal weights, training dynamics, and performance predictions.

Findings

Optimal weights and training dynamics are characterized.

Sequential Tempering with local Metropolis steps outperforms without.

Theoretical predictions guide best practices for ML-assisted Monte Carlo sampling.

Abstract

Recent years have seen a rise in the application of machine learning techniques to aid the simulation of hard-to-sample systems that cannot be studied using traditional methods. Despite the introduction of many different architectures and procedures, a wide theoretical understanding is still lacking, with the risk of suboptimal implementations. As a first step to address this gap, we provide here a complete analytic study of the widely-used Sequential Tempering procedure applied to a shallow MADE architecture for the Curie-Weiss model. The contribution of this work is twofold: firstly, we give a description of the optimal weights and of the training under Gradient Descent optimization. Secondly, we compare what happens in Sequential Tempering with and without the addition of local Metropolis Monte Carlo steps. We are thus able to give theoretical predictions on the best procedure to apply in this case. This work establishes a clear theoretical basis for the integration of machine learning techniques into Monte Carlo sampling and optimization.