Local minima in disordered mean-field ferromagnets

TL;DR

This paper investigates the complexity of disordered ferromagnetic landscapes on the hypercube, revealing a diverging number of local minima as system size increases and analyzing their properties through modified algorithms.

Contribution

The authors develop modified algorithms to explore near-zero magnetization regions and numerically verify the increasing complexity of local minima in random ferromagnetic models.

Findings

Number of local minima diverges as system size grows

Local minima exhibit specific energy and magnetization characteristics

Heavy-tailed disorder affects landscape properties

Abstract

We consider the complexity of random ferromagnetic landscapes on the hypercube $\{\pm 1\}^N$ given by Ising models on the complete graph with i.i.d. non-negative edge-weights. This includes, in particular, the case of Bernoulli disorder corresponding to the Ising model on a dense random graph $\mathcal G(N,p)$. Previous results had shown that, with high probability as $N\to\infty$, the gradient search (energy-lowering) algorithm, initialized uniformly at random, converges to one of the homogeneous global minima (all-plus or all-minus). Here, we devise two modified algorithms tailored to explore the landscape at near-zero magnetizations (where the effect of the ferromagnetic drift is minimized). With these, we numerically verify the landscape complexity of random ferromagnets, finding a diverging number of (1-spin-flip-stable) local minima as $N\to\infty$. We then investigate some of the properties of these local minima (e.g., typical energy and magnetization) and compare to the situation where the edge-weights are drawn from a heavy-tailed distribution.