  This paper studies the landscape of empirical risk of deep neural networks by
theoretically analyzing its convergence behavior to the population risk as well
as its stationary points and properties. For an $l$-layer linear neural
network, we prove its empirical risk uniformly converges to its population risk
at the rate of $\mathcal{O}(r^{2l}\sqrt{d\log(l)}/\sqrt{n})$ with training
sample size of $n$, the total weight dimension of $d$ and the magnitude bound
$r$ of weight of each layer. We then derive the stability and generalization
bounds for the empirical risk based on this result. Besides, we establish the
uniform convergence of gradient of the empirical risk to its population
counterpart. We prove the one-to-one correspondence of the non-degenerate
stationary points between the empirical and population risks with convergence
guarantees, which describes the landscape of deep neural networks. In addition,
we analyze these properties for deep nonlinear neural networks with sigmoid
activation functions. We prove similar results for convergence behavior of
their empirical risks as well as the gradients and analyze properties of their
non-degenerate stationary points.
  To our best knowledge, this work is the first one theoretically
characterizing landscapes of deep learning algorithms. Besides, our results
provide the sample complexity of training a good deep neural network. We also
provide theoretical understanding on how the neural network depth $l$, the
layer width, the network size $d$ and parameter magnitude determine the neural
network landscapes.
