Hardness of sampling solutions from the Symmetric Binary Perceptron

TL;DR

This paper demonstrates that certain classes of algorithms cannot efficiently sample solutions from the symmetric binary perceptron at any density, highlighting a fundamental computational hardness in approximate sampling despite solution existence.

Contribution

It establishes a hardness result for approximate sampling from the symmetric binary perceptron, contrasting with known efficient algorithms for finding solutions at low density.

Findings

Stable algorithms cannot sample solutions at any density.

Bounded-depth Boolean circuits cannot sample solutions at any density.

Solution sampling is computationally hard despite solution existence.

Abstract

We show that two related classes of algorithms, stable algorithms and Boolean circuits with bounded depth, cannot produce an approximate sample from the uniform measure over the set of solutions to the symmetric binary perceptron model at any constraint-to-variable density. This result is in contrast to the question of finding \emph{a} solution to the same problem, where efficient (and stable) algorithms are known to succeed at sufficiently low density. This result suggests that the solutions found efficiently -- whenever this task is possible -- must be highly atypical, and therefore provides an example of a problem where search is efficiently possible but approximate sampling from the set of solutions is not, at least within these two classes of algorithms.