Automatic Differentiable Monte Carlo: Theory and Application

TL;DR

This paper introduces a novel framework called automatic differentiable Monte Carlo (ADMC) that enables infinite-order automatic differentiation on Monte Carlo expectations, enhancing applications in physics and statistics.

Contribution

It develops the theoretical foundation for ADMC, allowing integration with machine learning frameworks to improve Monte Carlo methods in various scientific fields.

Findings

Enables automatic differentiation of Monte Carlo expectations.

Facilitates applications like phase transition search and ground state computation.

Potential to address quantum many-body sign problems.

Abstract

Differentiable programming has emerged as a key programming paradigm empowering rapid developments of deep learning while its applications to important computational methods such as Monte Carlo remain largely unexplored. Here we present the general theory enabling infinite-order automatic differentiation on expectations computed by Monte Carlo with unnormalized probability distributions, which we call "automatic differentiable Monte Carlo" (ADMC). By implementing ADMC algorithms on computational graphs, one can also leverage state-of-the-art machine learning frameworks and techniques to traditional Monte Carlo applications in statistics and physics. We illustrate the versatility of ADMC by showing some applications: fast search of phase transitions and accurately finding ground states of interacting many-body models in two dimensions. ADMC paves a promising way to innovate Monte Carlo in various aspects to achieve higher accuracy and efficiency, e.g. easing or solving the sign problem of quantum many-body models through ADMC.