  We investigate the sample complexity of networks with bounds on the magnitude
of its weights. In particular, we consider the class \[
H=\left\{W_t\circ\rho\circ \ldots\circ\rho\circ W_{1} :W_1,\ldots,W_{t-1}\in
M_{d, d}, W_t\in M_{1,d}\right\} \] where the spectral norm of each $W_i$ is
bounded by $O(1)$, the Frobenius norm is bounded by $R$, and $\rho$ is the
sigmoid function $\frac{e^x}{1+e^x}$ or the smoothened ReLU function $ \ln
(1+e^x)$. We show that for any depth $t$, if the inputs are in $[-1,1]^d$, the
sample complexity of $H$ is $\tilde O\left(\frac{dR^2}{\epsilon^2}\right)$.
This bound is optimal up to log-factors, and substantially improves over the
previous state of the art of $\tilde O\left(\frac{d^2R^2}{\epsilon^2}\right)$.
  We furthermore show that this bound remains valid if instead of considering
the magnitude of the $W_i$'s, we consider the magnitude of $W_i - W_i^0$, where
$W_i^0$ are some reference matrices, with spectral norm of $O(1)$. By taking
the $W_i^0$ to be the matrices at the onset of the training process, we get
sample complexity bounds that are sub-linear in the number of parameters, in
many typical regimes of parameters.
  To establish our results we develop a new technique to analyze the sample
complexity of families $H$ of predictors. We start by defining a new notion of
a randomized approximate description of functions $f:X\to\mathbb{R}^d$. We then
show that if there is a way to approximately describe functions in a class $H$
using $d$ bits, then $d/\epsilon^2$ examples suffices to guarantee uniform
convergence. Namely, that the empirical loss of all the functions in the class
is $\epsilon$-close to the true loss. Finally, we develop a set of tools for
calculating the approximate description length of classes of functions that can
be presented as a composition of linear function classes and non-linear
functions.
