A Deep Conditioning Treatment of Neural Networks
This work addresses the fundamental problem of understanding depth's role in neural network optimization for researchers, offering incremental theoretical extensions and practical insights.
The paper investigates how depth enhances the trainability of overparameterized neural networks by improving the conditioning of kernel matrices, applicable to various activation functions. It generalizes prior results on depth-related learnability degradation and demonstrates benign overfitting in deep networks, with experiments suggesting normalized ReLU can replace Batch Normalization.
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.