Overparametrized neural networks trained by gradient descent (GD) can provably overfit any training data. However, the generalization guarantee may not hold for noisy data. From a nonparametric perspective, this paper studies how well overparametrized neural networks can recover the true target function in the presence of random noises. We establish a lower bound on the $L_2$ estimation error with respect to the GD iteration, which is away from zero without a delicate choice of early stopping. In turn, through a comprehensive analysis of $\ell_2$-regularized GD trajectories, we prove that for overparametrized one-hidden-layer ReLU neural network with the $\ell_2$ regularization: (1) the output is close to that of the kernel ridge regression with the corresponding neural tangent kernel; (2) minimax {optimal} rate of $L_2$ estimation error is achieved. Numerical experiments confirm our theory and further demonstrate that the $\ell_2$ regularization approach improves the training robustness and works for a wider range of neural networks.

本文研究了超参数化神经网络在存在随机噪声的情况下恢复真实目标函数的能力，证明了采用 L2 正则化的神经网络与相应的神经切线核的核岭回归输出相似，并可以实现 L2 估计误差的最佳最小值，数值实验证实了该理论，同时表明 L2 正则化方法提高了训练的鲁棒性并适用于更广泛的神经网络。

正则化很重要：关于过度参数化神经网络的非参数视角