The Symmetric Perceptron: a Teacher-Student Scenario

2026-03-26

AI Summary

This work reformulates the traditional storage-oriented symmetric perceptron as a solvable teacher–student inference model to address the feasibility of learning across arbitrary sample densities. By introducing a two-region potential function, the study analyzes Bayes-optimal learning behavior under both noiseless and noisy conditions, establishing—through annealed and quenched free entropy analyses, high-dimensional asymptotics, and Monte Carlo optimization—the first teacher–student framework for the symmetric perceptron. The research fully characterizes the three-parameter phase diagram governed by sample density α, hyperplane offset κ, and temperature T, revealing the dynamical mechanisms between suboptimal metastable states induced by second-order instabilities and full alignment driven by first-order phase transitions. It further elucidates the critical role of potential function choice in shaping learning trajectories and identifies robust learning regimes alongside their melting pathways toward planted solutions.

Technology Category

• Machine Learning: Bayesian Learning
• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Social networks and social learning

Abstract

We introduce and solve a teacher-student formulation of the symmetric binary Perceptron, turning a traditionally storage-oriented model into a planted inference problem with a guaranteed solution at any sample density. We adapt the formulation of the symmetric Perceptron which traditionally considers either the u-shaped potential or the rectangular one, by including labels in both regions. With this formulation, we analyze both the Bayes-optimal regime at for noise-less examples and the effect of thermal noise under two different potential/classification rules. Using annealed and quenched free-entropy calculations in the high-dimensional limit, we map the phase diagram in the three control parameters, namely the sample density $α$, the distance between the origin and one of the symmetric hyperplanes $κ$ and temperature $T$, and identify a robust scenario where learning is organized by a second-order instability that creates teacher-correlated suboptimal states, followed by a first-order transition to full alignment. We show how this structure depends on the choice of potential, the interplay between metastability of the suboptimal solution and its melting towards the planted configuration, which is relevant for Monte Carlo-based optimization algorithms.

Problem

• Symmetric Perceptron
• Teacher-Student
• Planted Inference
• Phase Transition
• Statistical Physics

Innovation

• teacher-student model
• symmetric perceptron
• phase transition
• free-entropy calculation
• planted inference