Stochastic optimization: Glauber dynamics versus stochastic cellular automata

TL;DR

This paper compares Glauber dynamics and stochastic cellular automata (SCA) algorithms for minimizing Hamiltonian functions in Ising models, demonstrating that SCA, especially the ε-SCA variant, often outperforms Glauber dynamics in solving combinatorial optimization problems.

Contribution

The paper provides rigorous justification for using SCA in simulated annealing and compares its performance with Glauber dynamics across multiple optimization problems.

Findings

SCA outperforms Glauber dynamics in certain cases.

ε-SCA consistently shows the best performance.

SCA is effective for max-cut, TSP, and spin glass Hamiltonians.

Abstract

The topic we address in this paper concerns the minimization of a Hamiltonian function for an Ising model through the application of simulated annealing algorithms based on (single-site) Glauber dynamics and stochastic cellular automata (SCA). Some rigorous results are presented in order to justify the application of simulated annealing for a particular kind of SCA. After that, we compare the SCA algorithm and its variation, namely the $\varepsilon$-SCA algorithm, studied in this paper with the Glauber dynamics by analyzing their accuracy in obtaining optimal solutions for the max-cut problem on Erd\H{o}s-R\'enyi random graphs, the traveling salesman problem (TSP), and the minimization of Gaussian and Bernoulli spin glass Hamiltonians. We observed that the SCA performed better than the Glauber dynamics in some special cases, while the $\varepsilon$-SCA showed the highest performance in all scenarios.