It is generally recognized that finite learning rate (LR), in contrast to
infinitesimal LR, is important for good generalization in real-life deep nets.
Most attempted explanations propose approximating finite-LR SGD with Ito
Stochastic Differential Equations (SDEs), but formal justification for this
approximation (e.g., (Li et al., 2019)) only applies to SGD with tiny LR.
Experimental verification of the approximation appears computationally
infeasible. The current paper clarifies the picture with the following
contributions: (a) An efficient simulation algorithm SVAG that provably
converges to the conventionally used Ito SDE approximation. (b) A theoretically
motivated testable necessary condition for the SDE approximation and its most
famous implication, the linear scaling rule (Goyal et al., 2017), to hold. (c)
Experiments using this simulation to demonstrate that the previously proposed
SDE approximation can meaningfully capture the training and generalization
properties of common deep nets.

通过实验和理论分析，本文揭示了有限 LR 的 SGD（随机梯度下降）可以用 Ito 随机微分方程近似，且该近似可以捕捉常见深度神经网络的训练和泛化特性，提出了线性缩放规则的必要条件。

随机微分方程模型化 SGD 的有效性

On the Validity of Modeling SGD with Stochastic Differential Equations  (SDEs)

Are neural networks biased toward simple functions? Does depth always help
learn more complex features? Is training the last layer of a network as good as
training all layers? How to set the range for learning rate tuning? These
questions seem unrelated at face value, but in this work we give all of them a
common treatment from the spectral perspective. We will study the spectra of
the *Conjugate Kernel, CK,* (also called the *Neural Network-Gaussian Process
Kernel*), and the *Neural Tangent Kernel, NTK*. Roughly, the CK and the NTK
tell us respectively "what a network looks like at initialization" and "what a
network looks like during and after training." Their spectra then encode
valuable information about the initial distribution and the training and
generalization properties of neural networks. By analyzing the eigenvalues, we
lend novel insights into the questions put forth at the beginning, and we
verify these insights by extensive experiments of neural networks. We derive
fast algorithms for computing the spectra of CK and NTK when the data is
uniformly distributed over the boolean cube, and show this spectra is the same
in high dimensions when data is drawn from isotropic Gaussian or uniformly over
the sphere. Code replicating our results is available at
github.com/thegregyang/NNspectra.

本文从谱的角度研究共轭内核（Conjugate Kernel，CK）和神经切向内核（Neural Tangent Kernel，NTK）的特性，分析它们的特征值，得出关于神经网络的初始化分布和训练、泛化特性等问题的新见解，并通过广泛的神经网络实验验证这些见解。