Exact full-RSB SAT/UNSAT transition in infinitely wide two-layer neural networks

TL;DR

This paper provides an exact analysis of the SAT/UNSAT transition in infinitely wide two-layer neural networks, revealing the limitations of gradient descent and AMP algorithms in reaching maximal capacity due to complex state overlaps.

Contribution

It introduces a full-RSB analysis for the capacity transition in continuous neural network models and uncovers the impact of overlap gaps on algorithmic performance.

Findings

Exact SAT/UNSAT transition computed using full-RSB ansatz

Overlap gap in the negative perceptron model affects algorithm convergence

Gradient Descent cannot reach maximal capacity, independent of overlap gaps

Abstract

We analyze the problem of storing random pattern-label associations using two classes of continuous non-convex weights models, namely the perceptron with negative margin and an infinite-width two-layer neural network with non-overlapping receptive fields and generic activation function. Using a full-RSB ansatz we compute the exact value of the SAT/UNSAT transition. Furthermore, in the case of the negative perceptron we show that the overlap distribution of typical states displays an overlap gap (a disconnected support) in certain regions of the phase diagram defined by the value of the margin and the density of patterns to be stored. This implies that some recent theorems that ensure convergence of Approximate Message Passing (AMP) based algorithms to capacity are not applicable. Finally, we show that Gradient Descent is not able to reach the maximal capacity, irrespectively of the presence of an overlap gap for typical states. This finding, similarly to what occurs in binary weight models, suggests that gradient-based algorithms are biased towards highly atypical states, whose inaccessibility determines the algorithmic threshold.