The future is log-Gaussian: ResNets and their infinite-depth-and-width limit at initializationDownload PDF

Published: 09 Nov 2021, Last Modified: 05 May 2023NeurIPS 2021 PosterReaders: Everyone
Keywords: ResNet, infinite-depth-and-width, log-Gaussian, hypoactivation, balanced ResNet
Abstract: Theoretical results show that neural networks can be approximated by Gaussian processes in the infinite-width limit. However, for fully connected networks, it has been previously shown that for any fixed network width, $n$, the Gaussian approximation gets worse as the network depth, $d$, increases. Given that modern networks are deep, this raises the question of how well modern architectures, like ResNets, are captured by the infinite-width limit. To provide a better approximation, we study ReLU ResNets in the infinite-depth-and-width limit, where \emph{both} depth and width tend to infinity as their ratio, $d/n$, remains constant. In contrast to the Gaussian infinite-width limit, we show theoretically that the network exhibits log-Gaussian behaviour at initialization in the infinite-depth-and-width limit, with parameters depending on the ratio $d/n$. Using Monte Carlo simulations, we demonstrate that even basic properties of standard ResNet architectures are poorly captured by the Gaussian limit, but remarkably well captured by our log-Gaussian limit. Moreover, our analysis reveals that ReLU ResNets at initialization are hypoactivated: fewer than half of the ReLUs are activated. Additionally, we calculate the interlayer correlations, which have the effect of exponentially increasing the variance of the network output. Based on our analysis, we introduce \emph{Balanced ResNets}, a simple architecture modification, which eliminates hypoactivation and interlayer correlations and is more amenable to theoretical analysis.
Code Of Conduct: I certify that all co-authors of this work have read and commit to adhering to the NeurIPS Statement on Ethics, Fairness, Inclusivity, and Code of Conduct.
TL;DR: Log-Gaussian properties of ResNets in infinite-depth-and-width limit match those of standard ResNets.
Supplementary Material: zip
29 Replies

Loading