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A note on the capacity of the binary perceptron

Dylan J. Altschuler, Konstantin Tikhomirov

TL;DR

The paper addresses the capacity of the binary perceptron in a Gaussian setting and proves a rigorous upper bound $\alpha_c<0.847$ by linking the binary problem to the better understood spherical perceptron. The approach combines a conditional first-moment argument with the Gardner– Derrida free-energy formalism for the spherical model, leveraging the known GD expression and its rigorous properties to bound satisfiability probabilities. A key step is evaluating the GD bound at a fixed order parameter $q$ (notably $q=1/2$) to obtain a concrete numerical gap, and a decoupling argument via a Haar rotation translates this bound into a high-probability upper bound for the binary problem. Together, these elements yield a complete, self-contained proof that sharpens the prior upper bounds and clarifies the role of spherical-perceptron results in constraining binary constraint-satisfaction problems.

Abstract

Determining the capacity $α_c$ of the Binary Perceptron is a long-standing problem. Krauth and Mezard (1989) conjectured an explicit value of $α_c$, approximately equal to .833, and a rigorous lower bound matching this prediction was recently established by Ding and Sun (2019). Regarding the upper bound, Kim and Roche (1998) and Talagrand (1999) independently showed that $α_c$ < .996, while Krauth and Mezard outlined an argument which can be used to show that $α_c$ < .847. The purpose of this expository note is to record a complete proof of the bound $α_c$ < .847. The proof is a conditional first moment method combined with known results on the spherical perceptron

A note on the capacity of the binary perceptron

TL;DR

The paper addresses the capacity of the binary perceptron in a Gaussian setting and proves a rigorous upper bound by linking the binary problem to the better understood spherical perceptron. The approach combines a conditional first-moment argument with the Gardner– Derrida free-energy formalism for the spherical model, leveraging the known GD expression and its rigorous properties to bound satisfiability probabilities. A key step is evaluating the GD bound at a fixed order parameter (notably ) to obtain a concrete numerical gap, and a decoupling argument via a Haar rotation translates this bound into a high-probability upper bound for the binary problem. Together, these elements yield a complete, self-contained proof that sharpens the prior upper bounds and clarifies the role of spherical-perceptron results in constraining binary constraint-satisfaction problems.

Abstract

Determining the capacity of the Binary Perceptron is a long-standing problem. Krauth and Mezard (1989) conjectured an explicit value of , approximately equal to .833, and a rigorous lower bound matching this prediction was recently established by Ding and Sun (2019). Regarding the upper bound, Kim and Roche (1998) and Talagrand (1999) independently showed that < .996, while Krauth and Mezard outlined an argument which can be used to show that < .847. The purpose of this expository note is to record a complete proof of the bound < .847. The proof is a conditional first moment method combined with known results on the spherical perceptron
Paper Structure (3 sections, 4 theorems, 10 equations)

This paper contains 3 sections, 4 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Spherical Perceptron
  3. Proof of Main Result

Key Result

Theorem 1

With high probability, the capacity of the binary perceptron is at least $\alpha_c$ (i.e. $.833...$) and at most $.9963$.

Theorems & Definitions (6)

  • Conjecture 1: Krauth and Mezard k-m, 1989
  • Theorem 1: Ding and Sun ding-sun; Kim and Roche k-r; Talagrand tala-rsa; Xu xu
  • Theorem 2
  • Theorem 3: Gardner-Derrida formula for the Spherical Perceptron sch-tir
  • Proposition 1
  • proof : Proof of Proposition