Improve Generalization Ability of Deep Wide Residual Network with A Suitable Scaling Factor

Deep Residual Neural Networks (ResNets) have demonstrated remarkable success across a wide range of real-world applications. In this paper, we identify a suitable scaling factor (denoted by $\alpha$) on the residual branch of deep wide ResNets to achieve good generalization ability. We show that if $\alpha$ is a constant, the class of functions induced by Residual Neural Tangent Kernel (RNTK) is asymptotically not learnable, as the depth goes to infinity. We also highlight a surprising phenomenon: even if we allow $\alpha$ to decrease with increasing depth $L$, the degeneration phenomenon may still occur. However, when $\alpha$ decreases rapidly with $L$, the kernel regression with deep RNTK with early stopping can achieve the minimax rate provided that the target regression function falls in the reproducing kernel Hilbert space associated with the infinite-depth RNTK. Our simulation studies on synthetic data and real classification tasks such as MNIST, CIFAR10 and CIFAR100 support our theoretical criteria for choosing $\alpha$.

Generalization Ability of Wide Residual Networks

In this paper, we study the generalization ability of the wide residual network on $\mathbb{S}^{d-1}$ with the ReLU activation function. We first show that as the width $m\rightarrow\infty$, the residual network kernel (RNK) uniformly converges to the residual neural tangent kernel (RNTK). This uniform convergence further guarantees that the generalization error of the residual network converges to that of the kernel regression with respect to the RNTK. As direct corollaries, we then show $i)$ the wide residual network with the early stopping strategy can achieve the minimax rate provided that the target regression function falls in the reproducing kernel Hilbert space (RKHS) associated with the RNTK; $ii)$ the wide residual network can not generalize well if it is trained till overfitting the data. We finally illustrate some experiments to reconcile the contradiction between our theoretical result and the widely observed ``benign overfitting phenomenon''

Statistical Optimality of Deep Wide Neural Networks

In this paper, we consider the generalization ability of deep wide feedforward ReLU neural networks defined on a bounded domain $\mathcal X \subset \mathbb R^{d}$. We first demonstrate that the generalization ability of the neural network can be fully characterized by that of the corresponding deep neural tangent kernel (NTK) regression. We then investigate on the spectral properties of the deep NTK and show that the deep NTK is positive definite on $\mathcal{X}$ and its eigenvalue decay rate is $(d+1)/d$. Thanks to the well established theories in kernel regression, we then conclude that multilayer wide neural networks trained by gradient descent with proper early stopping achieve the minimax rate, provided that the regression function lies in the reproducing kernel Hilbert space (RKHS) associated with the corresponding NTK. Finally, we illustrate that the overfitted multilayer wide neural networks can not generalize well on $\mathbb S^{d}$.