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.

这篇论文通过理论分析神经网络的收敛行为、稳定点及特性研究其经验风险的实证风险的景观，证明线性神经网络的实证风险在训练样本量为n、总权重维数为d、每层权重的度量边界为r时，具有一致收敛到其总体风险的速率。深度非线性神经网络的实证风险的收敛行为、梯度和非退化稳定点的特性也得到分析。

深度学习算法概览