Stable behaviour of infinitely wide deep neural networks

Abstract

We consider fully connected feed-forward deep neural networks (NNs) where weights and biases are independent and identically distributed as symmetric centered stable distributions. Then, we show that the infinite wide limit of the NN, under suitable scaling on the weights, is a stochastic process whose finite-dimensional distributions are multivariate stable distributions. The limiting process is referred to as the stable process, and it generalizes the class of Gaussian processes recently obtained as infinite wide limits of NNs (Matthews at al., 2018b). Parameters of the stable process can be computed via an explicit recursion over the layers of the network. Our result contributes to the theory of fully connected feed-forward deep NNs, and it paves the way to expand recent lines of research that rely on Gaussian infinite wide limits.

arXiv comment25 pages, 3 figures

References in corpus (4)

• Neural Tangent Kernel: Convergence and Generalization in Neural Networks
• Scaling Limits of Wide Neural Networks with Weight Sharing: Gaussian Process Behavior, Gradient Independence, and Neural Tangent Kernel Derivation
• On the Impact of the Activation Function on Deep Neural Networks Training
• Traditional and Heavy-Tailed Self Regularization in Neural Network Models

Cited by in corpus (7)

• Bayesian Neural Network Priors Revisited
• Hausdorff Dimension, Heavy Tails, and Generalization in Neural Networks
• Scale Mixtures of Neural Network Gaussian Processes
• $\alpha$-Stable convergence of heavy-tailed infinitely-wide neural networks
• Infinite-channel deep stable convolutional neural networks
• Dependence between Bayesian neural network units
• Avoiding Kernel Fixed Points: Computing with ELU and GELU Infinite Networks