Learning the Sherrington-Kirkpatrick Model Even at Low Temperature

TL;DR

This paper presents a simple, polynomial-time algorithm for learning the parameters of the SK model at low temperatures, extending beyond previous high-temperature limitations using concentration techniques.

Contribution

It introduces a straightforward algorithm that learns the SK model parameters at low temperatures, surpassing prior methods limited to high-temperature regimes.

Findings

Successfully learns SK model parameters at inverse temperature up to √log n

Extends learning algorithms to higher-order MRFs like p-spin models

Uses subgaussian concentration for analysis, simplifying previous approaches

Abstract

We consider the fundamental problem of learning the parameters of an undirected graphical model or Markov Random Field (MRF) in the setting where the edge weights are chosen at random. For Ising models, we show that a multiplicative-weight update algorithm due to Klivans and Meka learns the parameters in polynomial time for any inverse temperature $\beta \leq \sqrt{\log n}$. This immediately yields an algorithm for learning the Sherrington-Kirkpatrick (SK) model beyond the high-temperature regime of $\beta < 1$. Prior work breaks down at $\beta = 1$ and requires heavy machinery from statistical physics or functional inequalities. In contrast, our analysis is relatively simple and uses only subgaussian concentration. Our results extend to MRFs of higher order (such as pure $p$-spin models), where even results in the high-temperature regime were not known.