Local entropy as a measure for sampling solutions in Constraint Satisfaction Problems

TL;DR

This paper introduces a local entropy-based Monte Carlo method for efficiently sampling solutions in constraint satisfaction problems, outperforming traditional methods like simulated annealing in finding optimal solutions quickly.

Contribution

It develops a novel entropy-driven Monte Carlo strategy that leverages local entropy estimates to efficiently explore solution spaces of CSPs, including the Binary Perceptron and K-Satisfiability problems.

Findings

EdMC efficiently finds solutions in CSPs with fewer steps.

Local entropy landscape guides the search more effectively than traditional methods.

Standard simulated annealing often fails or requires long procedures.

Abstract

We introduce a novel Entropy-driven Monte Carlo (EdMC) strategy to efficiently sample solutions of random Constraint Satisfaction Problems (CSPs). First, we extend a recent result that, using a large-deviation analysis, shows that the geometry of the space of solutions of the Binary Perceptron Learning Problem (a prototypical CSP), contains regions of very high-density of solutions. Despite being sub-dominant, these regions can be found by optimizing a local entropy measure. Building on these results, we construct a fast solver that relies exclusively on a local entropy estimate, and can be applied to general CSPs. We describe its performance not only for the Perceptron Learning Problem but also for the random $K$-Satisfiabilty Problem (another prototypical CSP with a radically different structure), and show numerically that a simple zero-temperature Metropolis search in the smooth local entropy landscape can reach sub-dominant clusters of optimal solutions in a small number of steps, while standard Simulated Annealing either requires extremely long cooling procedures or just fails. We also discuss how the EdMC can heuristically be made even more efficient for the cases we studied.