A fundamental question in modern machine learning is how deep neural networks can generalize. We address this question using 1) an equivalence between training infinitely wide neural networks and performing kernel regression with a deterministic kernel called the Neural Tangent Kernel (NTK) (Jacot et al. 2018), and 2) theoretical tools from statistical physics. We derive analytical expressions for learning curves for kernel regression, and use them to evaluate how the test loss of a trained neural network depends on the number of samples. Our approach allows us not only to compute the total test risk but also the decomposition of the risk due to different spectral components of the kernel. Complementary to recent results showing that during gradient descent, neural networks fit low frequency components first, we identify a new type of frequency principle: as the size of the training set size grows, kernel machines and neural networks begin to fit successively higher frequency modes of the target function. We verify our theory with simulations of kernel regression and training wide artificial neural networks.

通过高斯过程和统计物理学的理论方法，我们得到了内核回归广义性能的分析表达式，这些表达式是关于训练样本数量的函数。我们的结果适用于具有广泛神经网络的情况，这是由于训练它们和使用神经切向核(NTK)的核回归之间的等效性。通过计算核的不同谱成分对总体泛化误差的分解，我们确定了一个新的谱原理：随着训练集大小的增长，核机和神经网络逐渐适应目标功能的更高频谱模式。当数据从高维超球面上的均匀分布中采样时，点积核，包括NTK，显示出学习阶段，其中学习不同频率模式的目标函数。通过对合成数据和MNIST数据集的模拟，我们验证了我们的理论。

核回归和宽神经网络中的频谱相关的学习曲线