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Discrepancy Algorithms for the Binary Perceptron

Shuangping Li, Tselil Schramm, Kangjie Zhou

TL;DR

The paper analyzes discrepancy-minimization algorithms applied to the asymmetric binary perceptron, reframing the problem as finding a sign vector in the intersection of random halfspaces. It introduces a two-stage algorithm (LP relaxation followed by Edge-Walk rounding) and leverages Gaussian comparison techniques to obtain sharp, regime-dependent guarantees across large positive, large negative, and zero margins. The results yield precise algorithmic thresholds that closely track storage capacity in some regimes (nearly no information–computational gap for large positive margins) while revealing substantial gaps and an m-OGP hardness barrier for others (notably as $\kappa\to-\infty$). The work advances understanding of the information–computational landscape for random CSPs, showing how convex relaxations paired with discrepancy-based rounding can approach, and in some regimes match, the statistical limits, while also highlighting fundamental algorithmic barriers via m-OGP. Overall, the findings illuminate when efficient algorithms can succeed against random halfspace constraints and quantify the hardness in regimes where the solution geometry is highly clustered or fragile to input perturbations.

Abstract

The binary perceptron problem asks us to find a sign vector in the intersection of independently chosen random halfspaces with intercept $-κ$. 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 $κ= 0$ case and in the large-$|κ|$ case. In the $κ\to-\infty$ case, we additionally characterize the storage capacity and complement our algorithmic results with an almost-matching overlap-gap lower bound.

Discrepancy Algorithms for the Binary Perceptron

TL;DR

The paper analyzes discrepancy-minimization algorithms applied to the asymmetric binary perceptron, reframing the problem as finding a sign vector in the intersection of random halfspaces. It introduces a two-stage algorithm (LP relaxation followed by Edge-Walk rounding) and leverages Gaussian comparison techniques to obtain sharp, regime-dependent guarantees across large positive, large negative, and zero margins. The results yield precise algorithmic thresholds that closely track storage capacity in some regimes (nearly no information–computational gap for large positive margins) while revealing substantial gaps and an m-OGP hardness barrier for others (notably as ). The work advances understanding of the information–computational landscape for random CSPs, showing how convex relaxations paired with discrepancy-based rounding can approach, and in some regimes match, the statistical limits, while also highlighting fundamental algorithmic barriers via m-OGP. Overall, the findings illuminate when efficient algorithms can succeed against random halfspace constraints and quantify the hardness in regimes where the solution geometry is highly clustered or fragile to input perturbations.

Abstract

The binary perceptron problem asks us to find a sign vector in the intersection of independently chosen random halfspaces with intercept . 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 case and in the large- case. In the case, we additionally characterize the storage capacity and complement our algorithmic results with an almost-matching overlap-gap lower bound.
Paper Structure (50 sections, 46 theorems, 271 equations, 1 figure, 2 algorithms)

This paper contains 50 sections, 46 theorems, 271 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. The Binary Perceptron Model
  3. Main Results
  4. Our Algorithm.
  5. Results for Large Positive $\kappa$.
  6. Results for Large Negative $\kappa$.
  7. Results for $\kappa = 0$.
  8. Some remarks on our results.
  9. Related Work
  10. Algorithms from Discrepancy Minimization
  11. Computational Complexity
  12. Other Algorithms for the Perceptron Model
  13. Kim-Roche Algorithm.
  14. AMP-Type Algorithms for Spherical Perceptron.
  15. Heuristic Algorithms Based on Belief Propagation.
  16. ...and 35 more sections

Key Result

Theorem 1.1

[Informal, see thm:alg_kappa_positive] When $\kappa \to \infty$ and $\alpha = \frac{2}{\pi \kappa^2} (1+o_\kappa(1))$, our algorithm finds a solution for the binary perceptron model with margin $\kappa$ with high probability.

Figures (1)

  • Figure 1: The information-computation landscape for the Binary Perceptron. Our results appear in blue. When $\kappa \to \infty$, the upper bound $\alpha_{\mathrm{up}}$ on the storage capacity is due to stojnic2013discrete. In the case $\kappa = 0$, $\alpha_{\mathrm{KR}}$ is the density at which the Kim-Roche algorithm provably succeeds KR98; in simulation, belief-propagation algorithms succeed up to density $\alpha_{\mathrm{BP}}$BBBZ07BZ06, and the satisfiability threshold $\alpha_{\star}$ was conjectured in KM89 and rigorously established in ding2019capacityHuang24.

Theorems & Definitions (84)

  • Theorem 1.1
  • Theorem 1.2: Informal, see \ref{['sec:sat']}
  • Theorem 1.3: Informal, see \ref{['thm:kappaverynegative']}
  • Theorem 1.4: Informal, see \ref{['thm:mogp']} and \ref{['thm:hardness']}
  • Theorem 1.5: Informal, see \ref{['thm:fullcoloring0']} and \ref{['thm:fullcoloring02']}
  • Theorem 2.1: Upper bound on $\alpha_{\star} (\kappa)$
  • proof
  • Definition 2.1: Lower bound on $\alpha_{\star} (\kappa)$
  • Theorem 2.2: Lower bound on $\alpha_{\star} (\kappa)$
  • proof
  • ...and 74 more