Sharp threshold sequence and universality for Ising perceptron models

TL;DR

This paper establishes the existence of a sharp threshold sequence and universality of free energy in a broad class of Ising perceptron models with ,1-valued activation functions, extending previous results with new methods.

Contribution

It introduces new 'add one constraint' estimates that generalize Talagrand's work, applying to more general Ising perceptron models and demonstrating universality and sharp thresholds.

Findings

Free energy is self-averaging across models.

Existence of a sharp threshold sequence for the models.

Free energy universality with respect to disorder.

Abstract

We study a family of Ising perceptron models with $\{0,1\}$-valued activation functions. This includes the classical half-space models, as well as some of the symmetric models considered in recent works. For each of these models we show that the free energy is self-averaging, there is a sharp threshold sequence, and the free energy is universal with respect to the disorder. A prior work of Xu (2019) used very different methods to show a sharp threshold sequence in the half-space Ising perceptron with Bernoulli disorder. Recent works of Perkins--Xu (2021) and Abbe--Li--Sly (2021) determined the sharp threshold and limiting free energy in a symmetric perceptron model. The results of this paper apply in more general settings, and are based on new "add one constraint" estimates extending Talagrand's estimates for the half-space model (1999, 2011).