Discrepancy Algorithms for the Binary Perceptron

TL;DR

This paper analyzes discrepancy minimization algorithms for the binary perceptron problem, providing new algorithmic results and bounds in different regimes of the intercept parameter, and characterizing storage capacity.

Contribution

It offers novel algorithmic insights and bounds for the asymmetric binary perceptron, especially in extreme intercept cases, extending previous understanding.

Findings

New algorithms for   cases of  perceptron

Characterization of storage capacity in   cases

Overlap-gap lower bounds matching algorithms

Abstract

The binary perceptron problem asks us to find a sign vector in the intersection of independently chosen random halfspaces with intercept $-\kappa$. We analyze the performance of the canonical discrepancy minimization algorithms of Lovett-Meka and Rothvoss/Eldan-Singh for the asymmetric binary perceptron problem. We obtain new algorithmic results in the $\kappa = 0$ case and in the large-$|\kappa|$ case. In the $\kappa\to-\infty$ case, we additionally characterize the storage capacity and complement our algorithmic results with an almost-matching overlap-gap lower bound.