A series of recent works suggest that deep neural networks (DNNs), of fixed depth, are equivalent to certain Gaussian Processes (NNGP/NTK) in the highly over-parameterized regime (width or number-of-channels going to infinity). Other works suggest that this limit is relevant for real-world DNNs. These results invite further study into the generalization properties of Gaussian Processes of the NNGP and NTK type. Here we make several contributions along this line. First, we develop a formalism, based on field theory tools, for calculating learning curves perturbatively in one over the dataset size. For the case of NNGPs, this formalism naturally extends to finite width corrections. Second, in cases where one can diagonalize the covariance-function of the NNGP/NTK, we provide analytic expressions for the asymptotic learning curves of any given target function. These go beyond the standard equivalence kernel results. Last, we provide closed analytic expressions for the eigenvalues of NNGP/NTK kernels of depth 2 fully-connected ReLU networks. For datasets on the hypersphere, the eigenfunctions of such kernels, at any depth, are hyperspherical harmonics. A simple coherent picture emerges wherein fully-connected DNNs have a strong entropic bias towards functions which are low order polynomials of the input.

使用一种物理学方法对深度学习进行研究，通过重整化群、Feynman图和副本构造了一个多功能场论形式体系用于分析高度超参数化情况下的深度学习，研究表明DNNs除了高度超参数化外，仍然具有解释性和预测性——偏向于简单函数。

深度神经网络的学习曲线：高斯场理论视角