Learning with Local Search MCMC Layers

TL;DR

This paper introduces a theoretically grounded method for integrating local search heuristics into neural networks using MCMC, enabling differentiable combinatorial layers and efficient learning for complex NP-hard problems.

Contribution

It proposes transforming local search heuristics into proposal distributions for MCMC, providing a principled way to learn with inexact combinatorial solvers.

Findings

Reduces computational burden in learning for combinatorial problems

Demonstrates effectiveness on large-scale vehicle routing problem

Provides a differentiable layer for combinatorial optimization

Abstract

Integrating combinatorial optimization layers into neural networks has recently attracted significant research interest. However, many existing approaches lack theoretical guarantees or fail to perform adequately when relying on inexact solvers. This is a critical limitation, as many operations research problems are NP-hard, often necessitating the use of neighborhood-based local search heuristics. These heuristics iteratively generate and evaluate candidate solutions based on an acceptance rule. In this paper, we introduce a theoretically-principled approach for learning with such inexact combinatorial solvers. Inspired by the connection between simulated annealing and Metropolis-Hastings, we propose to transform problem-specific neighborhood systems used in local search heuristics into proposal distributions, implementing MCMC on the combinatorial space of feasible solutions. This allows us to construct differentiable combinatorial layers and associated loss functions. Replacing an exact solver by a local search strongly reduces the computational burden of learning on many applications. We demonstrate our approach on a large-scale dynamic vehicle routing problem with time windows.