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

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.

Going deeper with Image Transformers

“…One must adapt the architecture and the optimization procedure to train them correctly. Some approaches focus on the initialization schemes [24,27,70], others on multiple stages training [53,56], multiple loss at different depth [61], adding components in the architecture [2,75] or regularization [33]. As pointed in our paper, in that respect our LayerScale approach is more related to Rezero [2] and Skipinit [16], Fixup [75], and T-Fixup [34].…”

Dynamical Isometry and a Mean Field Theory of LSTMs and GRUs

“…We study the effect of the initialization hyperparameters on signal propagation for a very broad class of recurrent architectures, which includes as special cases many state-of-the-art RNN cells, including the GRU (Cho et al, 2014), the LSTM (Hochreiter and Schmidhuber, 1997), and the peephole LSTM (Gers et al, 2002). The analysis is based on the mean field theory of signal propagation developed in a line of prior work (Schoenholz et al, 2016;Xiao et al, 2018;Chen et al, 2018;Yang et al, 2019), as well as the concept of dynamical isometry (Saxe et al, 2013;Pennington et al, 2017; that is necessary for stable gradient backpropagation and which was shown to be crucial for training simpler RNN architectures (Chen et al, 2018). We perform a number of experiments to corroborate the results of the calculations and use them to motivate initialization schemes that outperform standard initialization approaches on a number of long sequence tasks.…”

The edge of chaos: quantum field theory and deep neural networks

“…Turning to the titular edge of chaos, we are inspired by the aforementioned works [15,16,[27][28][29][30] examining criticality in various deep network architectures. However, while many of these papers used the phrase "mean-field theory", they did not actually rely on any MFT analysis: as mentioned above, Gaussianity arises simply as a consequence of the central limit theorem (CLT).…”

“…Before turning to perturbative QFT in the next section, we note that in [29] it was reported that in the case of RNNs, the injection of time-series data 28 x destroys the ordered phase, and consequently there is no order-to-chaos phase transition. This arises due to an extra factor that appears in their analogue of (3.39) containing possible correlations in x.…”

“…It has since appeared in many fields ranging from theoretical neuroscience and neurophysiology [21][22][23], to biological and complex systems [24,25], and was also explored in an early model of bioplausible neural networks in [26]. More recently, [15,16,[27][28][29][30] demonstrated that networks initialized at criticality are trainable to far greater depths than those lying further into either phase. To see this, one identifies a correlation length that sets the scale at which correlations between local degrees of freedom -e.g., the activations of two different neurons, or the spins of two different magnetic dipoles -decay with separation or depth through the network.…”