Generalization of Overparametrized Deep Neural Network Under Noisy Observations

We study the generalization properties of the overparameterized deep neural network (DNN) with Rectified Linear Unit (ReLU) activations. Under the non-parametric regression framework, it is assumed that the ground-truth function is from a reproducing kernel Hilbert space (RKHS) induced by a neural tangent kernel (NTK) of ReLU DNN, and a dataset is given with the noises. Without a delicate adoption of early stopping, we prove that the overparametrized DNN trained by vanilla gradient descent does not recover the ground-truth function. It turns out that the estimated DNN's $L_{2}$ prediction error is bounded away from $0$. As a complement of the above result, we show that the $\ell_{2}$-regularized gradient descent enables the overparametrized DNN achieve the minimax optimal convergence rate of the $L_{2}$ prediction error, without early stopping. Notably, the rate we obtained is faster than $\mathcal{O}(n^{-1/2})$ known in the literature.