  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.
