The Future is Log-Gaussian: ResNets and Their Infinite-Depth-and-Width Limit at Initialization

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 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 Balanced ResNets, a simple architecture modification, which eliminates hypoactivation and interlayer correlations and is more amenable to theoretical analysis.

References in corpus (12)

• Diffusion Models Beat GANs on Image Synthesis
• Neural Tangent Kernel: Convergence and Generalization in Neural Networks
• A Convergence Theory for Deep Learning via Over-Parameterization
• Scaling Limits of Wide Neural Networks with Weight Sharing: Gaussian Process Behavior, Gradient Independence, and Neural Tangent Kernel Derivation
• The Shattered Gradients Problem: If resnets are the answer, then what is the question?
• Mean-field theory of two-layers neural networks: dimension-free bounds and kernel limit
• Tensor Programs II: Neural Tangent Kernel for Any Architecture
• Products of Many Large Random Matrices and Gradients in Deep Neural Networks
• Tensor Programs I: Wide Feedforward or Recurrent Neural Networks of Any Architecture are Gaussian Processes
• Which Neural Net Architectures Give Rise To Exploding and Vanishing Gradients?
• Beyond Signal Propagation: Is Feature Diversity Necessary in Deep Neural Network Initialization?
• Analyzing Finite Neural Networks: Can We Trust Neural Tangent Kernel Theory?

Cited by in corpus (1)

• Precise characterization of the prior predictive distribution of deep ReLU networks