A Deep Conditioning Treatment of Neural Networks

We study the role of depth in training randomly initialized overparameterized neural networks. We give a general result showing that depth improves trainability of neural networks by improving the conditioning of certain kernel matrices of the input data. This result holds for arbitrary non-linear activation functions under a certain normalization. We provide versions of the result that hold for training just the top layer of the neural network, as well as for training all layers, via the neural tangent kernel. As applications of these general results, we provide a generalization of the results of Das et al. (2019) showing that learnability of deep random neural networks with a large class of non-linear activations degrades exponentially with depth. We also show how benign overfitting can occur in deep neural networks via the results of Bartlett et al. (2019b). We also give experimental evidence that normalized versions of ReLU are a viable alternative to more complex operations like Batch Normalization in training deep neural networks.Comment: In proceedings of ALT 202

Hardness of Learning Neural Networks with Natural Weights

Neural networks are nowadays highly successful despite strong hardness results. The existing hardness results focus on the network architecture, and assume that the network's weights are arbitrary. A natural approach to settle the discrepancy is to assume that the network's weights are "well-behaved" and posses some generic properties that may allow efficient learning. This approach is supported by the intuition that the weights in real-world networks are not arbitrary, but exhibit some "random-like" properties with respect to some "natural" distributions. We prove negative results in this regard, and show that for depth-$2$ networks, and many "natural" weights distributions such as the normal and the uniform distribution, most networks are hard to learn. Namely, there is no efficient learning algorithm that is provably successful for most weights, and every input distribution. It implies that there is no generic property that holds with high probability in such random networks and allows efficient learning