Dynamical Isometry and a Mean Field Theory of CNNs: How to Train 10,000-Layer Vanilla Convolutional Neural Networks (1806.05393v2)

Abstract: In recent years, state-of-the-art methods in computer vision have utilized increasingly deep convolutional neural network architectures (CNNs), with some of the most successful models employing hundreds or even thousands of layers. A variety of pathologies such as vanishing/exploding gradients make training such deep networks challenging. While residual connections and batch normalization do enable training at these depths, it has remained unclear whether such specialized architecture designs are truly necessary to train deep CNNs. In this work, we demonstrate that it is possible to train vanilla CNNs with ten thousand layers or more simply by using an appropriate initialization scheme. We derive this initialization scheme theoretically by developing a mean field theory for signal propagation and by characterizing the conditions for dynamical isometry, the equilibration of singular values of the input-output Jacobian matrix. These conditions require that the convolution operator be an orthogonal transformation in the sense that it is norm-preserving. We present an algorithm for generating such random initial orthogonal convolution kernels and demonstrate empirically that they enable efficient training of extremely deep architectures.

• The paper proposes a mean field theory-based initialization scheme that establishes dynamical isometry across layers.
• The paper demonstrates through MNIST and CIFAR-10 experiments that 10,000-layer CNNs can be effectively trained without extra architectural elements.
• The paper highlights the use of Delta-Orthogonal kernels to maintain signal stability and prevent gradient explosion or vanishing.

Dynamical Isometry and a Mean Field Theory of CNNs: Training 10,000-Layer Neural Networks

This paper presents a theoretical and empirical paper of the conditions under which a vanilla convolutional neural network (CNN) with thousands of layers can be trained effectively without the use of architectural augmentations such as batch normalization or residual connections. By building on mean field theory and exploring matrix singular value properties, the authors propose a novel initialization scheme that enables highly effective training of extremely deep CNNs.

Theoretical Framework and Findings

The authors begin by addressing the common challenge of training deeply layered CNNs, which are prone to vanishing and exploding gradients. The use of architectural features like skip connections and batch normalization has mitigated this issue, but this research investigates whether such features are strictly necessary. A key theoretical contribution of the paper is the derivation of an initialization scheme based on mean field theory that ensures the effective propagation of signals through layers.

The research introduces the concept of dynamical isometry in CNNs, which refers to the equalization of singular values of the input-output Jacobian, critical for gradient stability. For their initialization strategy, this involves convolution operators behaving as orthogonal transformations, preserving norm through layers. The authors provide a method for generating random orthogonal convolution kernels, demonstrating that these facilitate the training of CNNs with up to 10,000 layers or more, using standard optimization algorithms.

Empirical Evidence

The paper provides comprehensive experiments using the MNIST and CIFAR-10 datasets to validate the theoretical framework. Notably, the empirical data shows that extremely deep CNNs (up to 10,000 layers) can learn efficiently without a decline in performance when initialized with the proposed Delta-Orthogonal kernels. This finding is significant given the historical difficulties associated with training such deep networks.

Numerical Results

Strong numerical results substantiate the claims, with empirical data showing training and test accuracies that contrast favorably against networks initialized with traditional Gaussian methods. For instance, on CIFAR-10, a 10,000-layer network maintains high training accuracy and achieves competitive test accuracy using the described initialization.

Implications and Speculations

The implications of this work are twofold: practically, it offers a method to train extremely deep CNNs without the complexity of additional architectural elements. Theoretically, it furthers our understanding of signal propagation in neural networks and the potential for orthogonal initializations in achieving dynamical isometry.

Looking forward, these findings open avenues for exploring the limits of neural network depth as a function of generalization capacity, unperturbed by prior training challenges. This suggests a promising frontier in deep learning robustness and model architecture diversity.

In conclusion, while this paper challenges the perceived necessity of architectural crutches in training deep networks, it advances the discourse on the fundamental principles of neural network initialization and signal stability. The results offer both a novel perspective and practical tools for future investigations in deep learning architectures.