## A NON-PARAMETRIC REGRESSION VIEWPOINT : GENERALIZATION OF OVERPARAMETRIZED DEEP RELU NETWORK UNDER NOISY OBSERVATIONS

### Namjoon Suh · Hyunouk Ko · Xiaoming Huo

Keywords: [ minimax ] [ neural tangent kernel ]

[ Abstract ]
Tue 26 Apr 10:30 a.m. PDT — 12:30 p.m. PDT

Abstract: 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.

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