A Near Complete Nonasymptotic Generalization Theory For Multilayer Neural Networks: Beyond the Bias-Variance Tradeoff

We propose a first near complete (that will make explicit sense in the main text) nonasymptotic generalization theory for multilayer neural networks with arbitrary Lipschitz activations and general Lipschitz loss functions (with some very mild conditions). In particular, it doens't require the boundness of loss function, as commonly assumed in the literature. Our theory goes beyond the bias-variance tradeoff, aligned with phenomenon typically encountered in deep learning. It is therefore sharp different with other existing nonasymptotic generalization error bounds for neural networks. More explicitly, we propose an explicit generalization error upper bound for multilayer neural networks with arbitrary Lipschitz activations $\sigma$ with $\sigma(0)=0$ and broad enough Lipschitz loss functions, without requiring either the width, depth or other hyperparameters of the neural network approaching infinity, a specific neural network architect (e.g. sparsity, boundness of some norms), a particular activation function, a particular optimization algorithm or boundness of the loss function, and with taking the approximation error into consideration. General Lipschitz activation can also be accommodated into our framework. A feature of our theory is that it also considers approximation errors. Furthermore, we show the near minimax optimality of our theory for multilayer ReLU networks for regression problems. Notably, our upper bound exhibits the famous double descent phenomenon for such networks, which is the most distinguished characteristic compared with other existing results. This work emphasizes a view that many classical results should be improved to embrace the unintuitive characteristics of deep learning to get a better understanding of it.