Infinite-channel deep stable convolutional neural networks
This work is an incremental theoretical extension for researchers studying the theoretical underpinnings of neural networks and their connections to stochastic processes.
This paper explores the infinite-channel limit of deep feed-forward convolutional neural networks, demonstrating that if network parameters follow a stable distribution, the limit is a stochastic process with multivariate stable finite-dimensional distributions. This extends previous work by characterizing the limiting distribution's parameters through an explicit backward recursion across layers.
The interplay between infinite-width neural networks (NNs) and classes of Gaussian processes (GPs) is well known since the seminal work of Neal (1996). While numerous theoretical refinements have been proposed in the recent years, the interplay between NNs and GPs relies on two critical distributional assumptions on the NN's parameters: A1) finite variance; A2) independent and identical distribution (iid). In this paper, we consider the problem of removing A1 in the general context of deep feed-forward convolutional NNs. In particular, we assume iid parameters distributed according to a stable distribution and we show that the infinite-channel limit of a deep feed-forward convolutional NNs, under suitable scaling, is a stochastic process with multivariate stable finite-dimensional distributions. Such a limiting distribution is then characterized through an explicit backward recursion for its parameters over the layers. Our contribution extends results of Favaro et al. (2020) to convolutional architectures, and it paves the way to expand exciting recent lines of research that rely on classes of GP limits.