Clustering of solutions in the symmetric binary perceptron

TL;DR

This paper investigates the solution landscape of the symmetric binary perceptron, revealing the existence of dense solution clusters and their geometric properties, which are important for understanding neural network optimization and generalization.

Contribution

It provides the first rigorous bounds on the existence of dense solution clusters in the symmetric perceptron model, linking landscape geometry to optimization success.

Findings

Existence of dense solution clusters in certain parameter regimes.

First and second moment bounds for pairs of close solutions.

Non-rigorous derivation of bounds for sets of solutions at fixed distances.

Abstract

The geometrical features of the (non-convex) loss landscape of neural network models are crucial in ensuring successful optimization and, most importantly, the capability to generalize well. While minimizers' flatness consistently correlates with good generalization, there has been little rigorous work in exploring the condition of existence of such minimizers, even in toy models. Here we consider a simple neural network model, the symmetric perceptron, with binary weights. Phrasing the learning problem as a constraint satisfaction problem, the analogous of a flat minimizer becomes a large and dense cluster of solutions, while the narrowest minimizers are isolated solutions. We perform the first steps toward the rigorous proof of the existence of a dense cluster in certain regimes of the parameters, by computing the first and second moment upper bounds for the existence of pairs of arbitrarily close solutions. Moreover, we present a non rigorous derivation of the same bounds for sets of $y$ solutions at fixed pairwise distances.