Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networks

TL;DR

This paper establishes nearly tight bounds on the VC-dimension of deep neural networks with ReLU and other piecewise linear activations, revealing how complexity scales with network parameters.

Contribution

It provides the first tight bounds on VC-dimension for piecewise linear neural networks, improving upon previous bounds and generalizing to various activation functions.

Findings

VC-dimension is O(W L log(W)) for networks with W weights and L layers

Examples show VC-dimension is at least Omega(W L log(W/L))

Bounds extend to pseudodimensions and arbitrary piecewise linear activations

Abstract

We prove new upper and lower bounds on the VC-dimension of deep neural networks with the ReLU activation function. These bounds are tight for almost the entire range of parameters. Letting $W$ be the number of weights and $L$ be the number of layers, we prove that the VC-dimension is $O(W L \log(W))$, and provide examples with VC-dimension $\Omega( W L \log(W/L) )$. This improves both the previously known upper bounds and lower bounds. In terms of the number $U$ of non-linear units, we prove a tight bound $\Theta(W U)$ on the VC-dimension. All of these bounds generalize to arbitrary piecewise linear activation functions, and also hold for the pseudodimensions of these function classes. Combined with previous results, this gives an intriguing range of dependencies of the VC-dimension on depth for networks with different non-linearities: there is no dependence for piecewise-constant, linear dependence for piecewise-linear, and no more than quadratic dependence for general piecewise-polynomial.