Capacity of the treelike sign perceptrons neural networks with one hidden layer -- RDT based upper bounds

TL;DR

This paper rigorously determines the capacity bounds of treelike sign perceptron neural networks with one hidden layer using Random Duality Theory, confirming predictions from statistical physics and improving previous bounds for small network sizes.

Contribution

It introduces a mathematical framework to establish upper bounds on the capacity of 1-hidden layer treelike sign perceptrons, matching physics predictions and improving known bounds for small networks.

Findings

Exact capacity bounds match replica symmetry predictions.

Improved bounds for networks with up to 5 neurons.

First rigorous progress in over 30 years for small network capacities.

Abstract

We study the capacity of \emph{sign} perceptrons neural networks (SPNN) and particularly focus on 1-hidden layer \emph{treelike committee machine} (TCM) architectures. Similarly to what happens in the case of a single perceptron neuron, it turns out that, in a statistical sense, the capacity of a corresponding multilayered network architecture consisting of multiple \emph{sign} perceptrons also undergoes the so-called phase transition (PT) phenomenon. This means: (i) for certain range of system parameters (size of data, number of neurons), the network can be properly trained to accurately memorize \emph{all} elements of the input dataset; and (ii) outside the region such a training does not exist. Clearly, determining the corresponding phase transition curve that separates these regions is an extraordinary task and among the most fundamental questions related to the performance of any network. Utilizing powerful mathematical engine called Random Duality Theory (RDT), we establish a generic framework for determining the upper bounds on the 1-hidden layer TCM SPNN capacity. Moreover, we do so for \emph{any} given (odd) number of neurons. We further show that the obtained results \emph{exactly} match the replica symmetry predictions of \cite{EKTVZ92,BHS92}, thereby proving that the statistical physics based results are not only nice estimates but also mathematically rigorous bounds as well. Moreover, for $d\leq 5$, we obtain the capacity values that improve on the best known rigorous ones of \cite{MitchDurb89}, thereby establishing a first, mathematically rigorous, progress in well over 30 years.