Generative diffusion for perceptron problems: statistical physics analysis and efficient algorithms

TL;DR

This paper uses statistical physics and diffusion algorithms to analyze the sampling of solutions in high-dimensional perceptron problems, revealing conditions for efficient sampling and proposing a new MCMC method for binary perceptrons.

Contribution

It introduces a replica theory-based formalism to predict sampling limits and develops an efficient diffusion-based algorithm for binary perceptron solutions.

Findings

Uniform sampling is efficient in the spherical perceptron within the Replica Symmetric region.

Sampling the uniform distribution for binary weights is generally intractable.

A potential U(s) = -log(s) enables efficient sampling via diffusion, leading to a robust MCMC algorithm.

Abstract

We consider random instances of non-convex perceptron problems in the high-dimensional limit of a large number of examples $M$ and weights $N$, with finite load $\alpha = M/N$. We develop a formalism based on replica theory to predict the fundamental limits of efficiently sampling the solution space using generative diffusion algorithms, conjectured to be saturated when the score function is provided by Approximate Message Passing. For the spherical perceptron with negative margin $\kappa$, we find that the uniform distribution over solutions can be efficiently sampled in most of the Replica Symmetric region of the $\alpha$-$\kappa$ plane. In contrast, for binary weights, sampling from the uniform distribution remains intractable. A theoretical analysis of this obstruction leads us to identify a potential $U(s) = -\log(s)$, under which the corresponding tilted distribution becomes efficiently samplable via diffusion. Moreover, we show numerically that an annealing procedure over the shape of this potential yields a fast and robust Markov Chain Monte Carlo algorithm for sampling the solution space of the binary perceptron.