# The capacity of feedforward neural networks

## TL;DR

This paper introduces a quantitative measure of neural network capacity based on the number of functions they can compute, providing formulas for layered architectures and insights into their expressive power.

## Contribution

It defines the capacity of layered neural networks, develops new techniques for capacity bounds, and analyzes how architecture influences function complexity and regularization.

## Key findings

- Capacity is a cubic polynomial in layer sizes.
- Bottleneck layers limit capacity.
- Deep networks produce more regular, interesting functions.

## Abstract

A long standing open problem in the theory of neural networks is the development of quantitative methods to estimate and compare the capabilities of different architectures. Here we define the capacity of an architecture by the binary logarithm of the number of functions it can compute, as the synaptic weights are varied. The capacity provides an upper bound on the number of bits that can be extracted from the training data and stored in the architecture during learning. We study the capacity of layered, fully-connected, architectures of linear threshold neurons with $L$ layers of size $n_1,n_2, \ldots, n_L$ and show that in essence the capacity is given by a cubic polynomial in the layer sizes: $C(n_1,\ldots, n_L)=\sum_{k=1}^{L-1} \min(n_1,\ldots,n_k)n_kn_{k+1}$, where layers that are smaller than all previous layers act as bottlenecks. In proving the main result, we also develop new techniques (multiplexing, enrichment, and stacking) as well as new bounds on the capacity of finite sets. We use the main result to identify architectures with maximal or minimal capacity under a number of natural constraints. This leads to the notion of structural regularization for deep architectures. While in general, everything else being equal, shallow networks compute more functions than deep networks, the functions computed by deep networks are more regular and "interesting".