  We prove sharp dimension-free representation results for neural networks with
$D$ ReLU layers under square loss for a class of functions $\mathcal{G}_D$
defined in the paper. These results capture the precise benefits of depth in
the following sense:
  1. The rates for representing the class of functions $\mathcal{G}_D$ via $D$
ReLU layers is sharp up to constants, as shown by matching lower bounds.
  2. For each $D$, $\mathcal{G}_{D} \subseteq \mathcal{G}_{D+1}$ and as $D$
grows the class of functions $\mathcal{G}_{D}$ contains progressively less
smooth functions.
  3. If $D^{\prime} < D$, then the approximation rate for the class
$\mathcal{G}_D$ achieved by depth $D^{\prime}$ networks is strictly worse than
that achieved by depth $D$ networks.
  This constitutes a fine-grained characterization of the representation power
of feedforward networks of arbitrary depth $D$ and number of neurons $N$, in
contrast to existing representation results which either require $D$ growing
quickly with $N$ or assume that the function being represented is highly
smooth. In the latter case similar rates can be obtained with a single
nonlinear layer. Our results confirm the prevailing hypothesis that deeper
networks are better at representing less smooth functions, and indeed, the main
technical novelty is to fully exploit the fact that deep networks can produce
highly oscillatory functions with few activation functions.
