Deep Neural Networks as Gaussian Processes

It has long been known that a single-layer fully-connected neural network with an i.i.d. prior over its parameters is equivalent to a Gaussian process (GP), in the limit of infinite network width. This correspondence enables exact Bayesian inference for infinite width neural networks on regression tasks by means of evaluating the corresponding GP. Recently, kernel functions which mimic multi-layer random neural networks have been developed, but only outside of a Bayesian framework. As such, previous work has not identified that these kernels can be used as covariance functions for GPs and allow fully Bayesian prediction with a deep neural network. In this work, we derive the exact equivalence between infinitely wide deep networks and GPs. We further develop a computationally efficient pipeline to compute the covariance function for these GPs. We then use the resulting GPs to perform Bayesian inference for wide deep neural networks on MNIST and CIFAR-10. We observe that trained neural network accuracy approaches that of the corresponding GP with increasing layer width, and that the GP uncertainty is strongly correlated with trained network prediction error. We further find that test performance increases as finite-width trained networks are made wider and more similar to a GP, and thus that GP predictions typically outperform those of finite-width networks. Finally we connect the performance of these GPs to the recent theory of signal propagation in random neural networks.

The Emergence of Spectral Universality in Deep Networks

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.

Estimating the Spectral Density of Large Implicit Matrices

Many important problems are characterized by the eigenvalues of a large matrix. For example, the difficulty of many optimization problems, such as those arising from the fitting of large models in statistics and machine learning, can be investigated via the spectrum of the Hessian of the empirical loss function. Network data can be understood via the eigenstructure of a graph Laplacian matrix using spectral graph theory. Quantum simulations and other many-body problems are often characterized via the eigenvalues of the solution space, as are various dynamic systems. However, naive eigenvalue estimation is computationally expensive even when the matrix can be represented; in many of these situations the matrix is so large as to only be available implicitly via products with vectors. Even worse, one may only have noisy estimates of such matrix vector products. In this work, we combine several different techniques for randomized estimation and show that it is possible to construct unbiased estimators to answer a broad class of questions about the spectra of such implicit matrices, even in the presence of noise. We validate these methods on large-scale problems in which graph theory and random matrix theory provide ground truth.

Resurrecting the sigmoid in deep learning through dynamical isometry: theory and practice

It is well known that the initialization of weights in deep neural 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 the exponential vanishing or explosion of gradients. The stronger condition that all singular values of the Jacobian concentrate near $1$ is a property known as dynamical isometry. For deep linear networks, dynamical isometry can be achieved through orthogonal weight initialization and has been shown to dramatically speed up learning; however, it has remained unclear how to extend these results to the nonlinear setting. We address this question by employing powerful tools from free probability theory to compute analytically 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.

A Correspondence Between Random Neural Networks and Statistical Field Theory

A number of recent papers have provided evidence that practical design questions about neural networks may be tackled theoretically by studying the behavior of random networks. However, until now the tools available for analyzing random neural networks have been relatively ad-hoc. In this work, we show that the distribution of pre-activations in random neural networks can be exactly mapped onto lattice models in statistical physics. We argue that several previous investigations of stochastic networks actually studied a particular factorial approximation to the full lattice model. For random linear networks and random rectified linear networks we show that the corresponding lattice models in the wide network limit may be systematically approximated by a Gaussian distribution with covariance between the layers of the network. In each case, the approximate distribution can be diagonalized by Fourier transformation. We show that this approximation accurately describes the results of numerical simulations of wide random neural networks. Finally, we demonstrate that in each case the large scale behavior of the random networks can be approximated by an effective field theory.