Sharp threshold sequence and universality for Ising perceptron models

Abstract

We study a family of Ising perceptron models with t0, 1u-valued activation functions. is 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 C. Xu (2019) used very di erent methods to show a sharp threshold sequence in the halfspace 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. e results of this paper apply in more general se ings, and are based on new "add one constraint" estimates extending Talagrand's estimates for the half-space model (1999, 2011).

Critical Window of The Symmetric Perceptron

“…Perkins and Xu adapted and improved a technique of Talagrand [23] for the asymmetric perceptron. Sun and Nakajima [15] recently extended the method of Talagrand to give a sharp threshold for wide class of random matrices, yielding comparable results to [2,16] in a broader setting (including the asymmetric problem).…”