Hysteretic Optimization For Spin Glasses

TL;DR

Hysteretic Optimization (HO) is effective for infinite-range spin systems but performs poorly for finite-dimensional lattices, with its success linked to the system's connectivity and avalanche behavior during optimization.

Contribution

This study evaluates HO's performance on 1D long-range Ising spin chains and identifies the critical role of interaction range and avalanche dynamics in its effectiveness.

Findings

HO performs well for infinite-range interactions (σ<0.5)

Performance sharply declines as interactions become more short-range

System connectivity influences the efficiency of the hysteretic optimization process

Abstract

The recently proposed Hysteretic Optimization (HO) procedure is applied to the 1D Ising spin chain with long range interactions. To study its effectiveness, the quality of ground state energies found as a function of the distance dependence exponent, $\sigma$, is assessed. It is found that the transition from an infinite-range to a long-range interaction at $\sigma=0.5$ is accompanied by a sharp decrease in the performance . The transition is signaled by a change in the scaling behavior of the average avalanche size observed during the hysteresis process. This indicates that HO requires the system to be infinite-range, with a high degree of interconnectivity between variables leading to large avalanches, in order to function properly. An analysis of the way auto-correlations evolve during the optimization procedure confirm that the search of phase space is less efficient, with the system becoming effectively stuck in suboptimal configurations much earlier. These observations explain the poor performance that HO obtained for the Edwards-Anderson spin glass on finite-dimensional lattices, and suggest that its usefulness might be limited in many combinatorial optimization problems.