  In this paper, we study the sample complexity lower bounds for the exact
recovery of parameters and for a positive excess risk of a feed-forward,
fully-connected neural network for binary classification, using
information-theoretic tools. We prove these lower bounds by the existence of a
generative network characterized by a backwards data generating process, where
the input is generated based on the binary output, and the network is
parametrized by weight parameters for the hidden layers. The sample complexity
lower bound for the exact recovery of parameters is $\Omega(d r \log(r) + p )$
and for a positive excess risk is $\Omega(r \log(r) + p )$, where $p$ is the
dimension of the input, $r$ reflects the rank of the weight matrices and $d$ is
the number of hidden layers. To the best of our knowledge, our results are the
first information theoretic lower bounds.
