Free-Energy Machine for Combinatorial Optimization

TL;DR

The paper introduces the Free-Energy Machine (FEM), a versatile and efficient method for solving large-scale combinatorial optimization problems by combining principles from statistical physics and machine learning, leveraging modern hardware.

Contribution

It presents a unified, flexible framework based on free-energy minimization that outperforms existing specialized algorithms in speed and effectiveness across diverse problem types.

Findings

FEM scales to millions of variables efficiently.

Outperforms state-of-the-art algorithms on benchmark problems.

Works effectively on synthetic, real-world, and competition instances.

Abstract

Finding optimal solutions to combinatorial optimization problems is pivotal in both scientific and technological domains, within academic research and industrial applications. A considerable amount of effort has been invested in the development of accelerated methods that leverage sophisticated models and harness the power of advanced computational hardware. Despite the advancements, a critical challenge persists, the dual demand for both high efficiency and broad generality in solving problems. In this work, we propose a general method, Free-Energy Machine (FEM), based on the ideas of free-energy minimization in statistical physics, combined with automatic differentiation and gradient-based optimization in machine learning. The algorithm is flexible, solving various combinatorial optimization problems using a unified framework, and is efficient, naturally utilizing massive parallel computational devices such as graph processing units (GPUs) and field-programmable gate arrays (FPGAs). We benchmark our algorithm on various problems including the maximum cut problems, balanced minimum cut problems, and maximum $k$-satisfiability problems, scaled to millions of variables, across both synthetic, real-world, and competition problem instances. The findings indicate that our algorithm not only exhibits exceptional speed but also surpasses the performance of state-of-the-art algorithms tailored for individual problems. This highlights that the interdisciplinary fusion of statistical physics and machine learning opens the door to delivering cutting-edge methodologies that will have broad implications across various scientific and industrial landscapes.