Fixed width treelike neural networks capacity analysis -- generic activations

TL;DR

This paper extends capacity analysis of treelike neural networks to more general activation functions, revealing that maximum capacity occurs with just two hidden neurons and that capacity converges as layer width increases.

Contribution

It generalizes existing capacity analysis frameworks to quadratic and ReLU activations, showing capacity bounds decrease with width and peak at two hidden neurons.

Findings

Capacity bounds decrease with increasing hidden layer width.

Maximum capacity is achieved with exactly two hidden neurons.

Results align with statistical physics predictions.

Abstract

We consider the capacity of \emph{treelike committee machines} (TCM) neural networks. Relying on Random Duality Theory (RDT), \cite{Stojnictcmspnncaprdt23} recently introduced a generic framework for their capacity analysis. An upgrade based on the so-called \emph{partially lifted} RDT (pl RDT) was then presented in \cite{Stojnictcmspnncapliftedrdt23}. Both lines of work focused on the networks with the most typical, \emph{sign}, activations. Here, on the other hand, we focus on networks with other, more general, types of activations and show that the frameworks of \cite{Stojnictcmspnncaprdt23,Stojnictcmspnncapliftedrdt23} are sufficiently powerful to enable handling of such scenarios as well. In addition to the standard \emph{linear} activations, we uncover that particularly convenient results can be obtained for two very commonly used activations, namely, the \emph{quadratic} and \emph{rectified linear unit (ReLU)} ones. In more concrete terms, for each of these activations, we obtain both the RDT and pl RDT based memory capacities upper bound characterization for \emph{any} given (even) number of the hidden layer neurons, $d$. In the process, we also uncover the following two, rather remarkable, facts: 1) contrary to the common wisdom, both sets of results show that the bounding capacity decreases for large $d$ (the width of the hidden layer) while converging to a constant value; and 2) the maximum bounding capacity is achieved for the networks with precisely \textbf{\emph{two}} hidden layer neurons! Moreover, the large $d$ converging values are observed to be in excellent agrement with the statistical physics replica theory based predictions.