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? 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 believe the computational tools we develop here for analyzing the spectra of CK and NTK serve as a solid foundation for future studies of deep neural networks. We have open-sourced the code for it and for generating the plots in this paper at this http URL.
This signal propagation perspective can be refined via random matrix theory @cite_25 @cite_24 . In these works, free probability is leveraged to compute the singular value distribution of the input-output map given by the random neural network, as the input dimension and width tend to infinity together. Other works also investigate various questions of neural network training and generalization from the random matrix perspective .
@cite_24 @cite_25
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Recent work has shown that tight concentration of the entire spectrum of singular values of a deep network's input-output Jacobian around one at initialization can speed up learning by orders of magnitude. Therefore, to guide important design choices, it is important to build a full theoretical understanding of the spectra of Jacobians at initialization. To this end, we leverage powerful tools from free probability theory to provide a detailed analytic understanding of how a deep network's Jacobian spectrum depends on various hyperparameters including the nonlinearity, the weight and bias distributions, and the depth. For a variety of nonlinearities, our work reveals the emergence of new universal limiting spectral distributions that remain concentrated around one even as the depth goes to infinity. It is well known that weight initialization in deep networks can have a dramatic impact on learning speed. For example, ensuring the mean squared singular value of a network's input-output Jacobian is O(1) is essential for avoiding exponentially vanishing or exploding gradients. Moreover, in deep linear networks, ensuring that all singular values of the Jacobian are concentrated near 1 can yield a dramatic additional speed-up in learning; this is a property known as dynamical isometry. However, it is unclear how to achieve dynamical isometry in nonlinear deep networks. We address this question by employing powerful tools from free probability theory to analytically compute the entire singular value distribution of a deep network's input-output Jacobian. We explore the dependence of the singular value distribution on the depth of the network, the weight initialization, and the choice of nonlinearity. Intriguingly, we find that ReLU networks are incapable of dynamical isometry. On the other hand, sigmoidal networks can achieve isometry, but only with orthogonal weight initialization. Moreover, we demonstrate empirically that deep nonlinear networks achieving dynamical isometry learn orders of magnitude faster than networks that do not. Indeed, we show that properly-initialized deep sigmoidal networks consistently outperform deep ReLU networks. Overall, our analysis reveals that controlling the entire distribution of Jacobian singular values is an important design consideration in deep learning.