Empirical universality and non-universality of local dynamics in the Sherrington-Kirkpatrick model

TL;DR

This paper investigates how local search algorithms perform on the SK model, revealing that greedy search is universally efficient while reluctant search's performance varies with the distribution of couplings.

Contribution

It provides empirical evidence of universality for greedy search and non-universality for reluctant search in the SK model, highlighting the impact of coupling distribution.

Findings

Greedy search runtime is universal across distributions.

Reluctant search runtime is sensitive to coupling distribution.

Discrete coupling support affects reluctant search behavior.

Abstract

Several recent works have aimed to design algorithms for optimizing the Hamiltonians of spin glass models from statistical physics. While Montanari (2018) eventually gave a sophisticated message-passing algorithm to do this nearly optimally for the Sherrington-Kirkpatrick (SK) model, the recent work of Erba, Behrens, Krzakala, and Zdeborov\'a (2024) also observed that a simple yet unusual algorithm first proposed by Parisi (2003) seems to perform just as well: perform local reluctant search, repeatedly making the local adjustment improving the objective function by the smallest possible amount. This is in contrast to the more intuitive local greedy search that repeatedly makes the local adjustment improving the objective by the largest possible amount. We study empirically how the performance of these algorithms depends on the distribution of entries of the coupling matrix in the SK model. We find evidence that, while the runtime of greedy search enjoys universality over a broad range of distributions, the runtime of reluctant search surprisingly is not universal, sometimes depending quite sensitively on the entry distribution. We propose that one mechanism leading to this non-universality is a change in the behavior of reluctant search when the couplings have discrete support on an evenly-spaced grid, and give experimental results supporting this proposal and investigating other properties of a distribution that might affect the performance of reluctant search.