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Title: Deep Mean Field Theory: Layerwise Variance and Width Variation as Methods to Control Gradient Explosion

Paper content:
A recent line of work has studied the statistical properties of neural networks to great success from a {\it mean field theory} perspective, making and verifying very precise predictions of neural network behavior and test time performance.
In this paper, we build upon these works to explore two methods for taming the behaviors of random residual networks (with only fully connected layers and no batchnorm).
The first method is {\it width variation (WV)}, i.e. varying the widths of layers as a function of depth.
We show that width decay reduces gradient explosion without affecting the mean forward dynamics of the random network.
The second method is {\it variance variation (VV)}, i.e. changing the initialization variances of weights and biases over depth.
We show VV, used appropriately, can reduce gradient explosion of tanh and ReLU resnets from $\exp(\Theta(\sqrt L))$ and $\exp(\Theta(L))$ respectively to constant $\Theta(1)$.
A complete phase-diagram is derived for how variance decay affects different dynamics, such as those of gradient and activation norms.
In particular, we show the existence of many phase transitions where these dynamics switch between exponential, polynomial, logarithmic, and even constant behaviors.
Using the obtained mean field theory, we are able to track surprisingly well how VV at initialization time affects training and test time performance on MNIST after a set number of epochs: the level sets of test/train set accuracies coincide with the level sets of the expectations of certain gradient norms or of metric expressivity (as defined in \cite{yang_meanfield_2017}), a measure of expansion in a random neural network.
Based on insights from past works in deep mean field theory and information geometry, we also provide a new perspective on the gradient explosion/vanishing problems: they lead to ill-conditioning of the Fisher information matrix, causing optimization troubles.
Deep mean field theory studies how random neural networks behave with increasing depth, as the width goes to infinity.
In this limit, several pieces of seminal work used statistical physics BID7 Sompolinsky et al., 1988) and Gaussian Processes (Neal, 2012) to show that neural networks exhibit remarkable regularity.
Mean field theory also has a substantial history studying Boltzmann machines BID0 and sigmoid belief networks (Saul et al., 1996) .Recently
, a number of results have revitalized the use of mean field theory in deep learning, with a focus on addressing practical design questions. In Poole
et al. (2016) , mean field theory is combined with Riemannian geometry to quantify the expressivity of random neural networks. In Schoenholz
et al. (2017) and Yang and Schoenholz (2017) , a study of the critical phenomena of mean field neural networks and residual networks 1 is leveraged to theoretically predict test time relative performance of differential initialization schemes. Additionally,
BID5 and Pennington and Bahri (2017) have used related techniques to investigate properties of the loss landscape of deep networks. Together these
results have helped a large number of experimental observations onto more rigorous footing (Montfar et al., 2014; BID9 BID3 . Finally, deep
mean field theory has proven to be a necessary underpinning for studies using random matrix theory to 1 without batchnorm and with only fully connected layers understand dynamical isometry in random neural networks (Pennington et al., 2017; Pennington and Worah, 2017) . Overall, a program
is emerging toward building a mean field theory for state-of-the-art neural architectures as used in the wild, so as to provide optimal initialization parameters quickly for any deep learning practitioner.In this paper, we contribute to this program by studying how width variation (WV), as practiced commonly, can change the behavior of quantities mentioned above, with gradient norm being of central concern. We find that WV can
dramatically reduce gradient explosion without affecting the mean dynamics of forward computation, such as the activation norms, although possibly increasing deviation from the mean in the process (Section 6).We also study a second
method, variance variation (VV), for manipulating the mean field dynamics of a random neural network (Section 7 and Appendix B). In this paper, we focus
on its application to tanh and ReLU residual networks, where we show that VV can dramatically ameliorate gradient explosion, and in the case of ReLU resnet, activation explosion 2 . Affirming the results of
Yang and Schoenholz (2017) and predicted by our theory, VV improves performances of tanh and ReLU resnets through these means.Previous works (Poole et al., 2016; Schoenholz et al., 2017; Yang and Schoenholz, 2017) have focused on how network architecture and activation functions affect the dynamics of mean field quantities, subject to the constraint that initialization variances and widths are constant across layers. In each combination of (
architecture, activation), the mean field dynamics have the same kinds of asymptotics regardless of the variances. For example, tanh feedforward
networks have exp(Θ(l)) forward and backward dynamics, while tanh residual networks have poly(l) forward and exp(Θ( √ l)) backward dynamics. Such asymptotics were considered
characteristics of the (architecture, activation) combination (Yang and Schoenholz, 2017) . We show by counterexample that this
perception is erroneous. In fact, as discussed above, WV can
control the gradient dynamics arbitrarily and VV can control forward and backward dynamics jointly, all without changing the network architecture or activation. To the best of our knowledge, this
is the first time methods for reducing gradient explosion or vanishing have been proposed that vary initialization variance and/or width across layers.With regard to ReLU resnets, we find that gradient norms and "metric expressivity" (as introduced in Yang and Schoenholz (2017) , also defined in Defn 4.2), make surprisingly good predictors, respectively in two separate phases, of how VV at initialization affects performance after a fixed amount of training time (Section 7.1). However, in one of these phases, larger
gradient explosion seems to cause better performance, with no alternative course of explanation. In this paper we have no answer for why
this occurs but hope to elucidate it for future work. With regard to tanh resnets, we find that
, just as in Yang and Schoenholz (2017) , the optimal initialization balances trainability and expressivity: Decaying the variance too little means we suffer from gradient explosion, but decaying the variance too much means we suffer from not enough metric expressivity.We want to stress that in this work, by "performance" we do not mean absolute performance but rather relative performance between different initialization schemes. For example, we do not claim to know what
initialization scheme is needed to make a particular neural network architecture solve ImageNet, but rather, conditioned on the architecture, whether one initialization is better than another in terms of test set accuracy after the same amount of training iterations.Before we begin the mean field analysis, we present a perspective on gradient explosion/vanishing problem from a combination of mean field theory and information geometry, which posits that such problem manifests in the ill-conditioning of the Fisher information matrix.
In this paper, we derived the mean field theory of width and variance variation and showed that they are powerful methods to control forward (VV) and backward (VV + WV) dynamics.
We proved that even with a fixed architecture and activation function, the mean field dynamics of a residual neural network can still be manipulated at will by these two methods.
Extraordinarily, the mean field theory we developed allowed us to accurately predict the performances of trained MNIST models relative to different initializations, but one puzzling aspect remains where test set accuracy seems to increase as gradient explosion worsens in one regime of random ReLU resnets.Open Problems.
We solved a small part, width variation, of the program to construct mean field theories of state-of-the-art neural networks used in practice.
Many open problems still remain, and the most important of them include but is not limited to
1. batchnorm,
2. convolution layers, and
3. recurrent layers.
In addition, more work is needed to mathematically justify our "physical" assumptions Axiom 1 and Axiom 2 to a "math" problem.
We hope readers will take note and contribute toward deep mean field theory.
Jeffrey Pennington, Samuel Schoenholz, and Surya Ganguli.Resurrecting the sigmoid in deep learning through dynamical isometry: theory and practice.In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Advances in Neural Information Processing Systems 30, pages 4788-4798.
Curran Associates, Inc., 2017.URL http://papers.nips.cc/paper/ 7064-resurrecting-the-sigmoid-in-deep-learning-through-dynamical-isometry-theory-a pdf.Ben Poole, Subhaneil Lahiri, Maithreyi Raghu, Jascha Sohl-Dickstein, and Surya Ganguli.
Exponential expressivity in deep neural networks through transient chaos.
In Advances In Neural Information Processing Systems, pages 3360-3368, 2016.
DISPLAYFORM0 DISPLAYFORM1 The two cases for χ/χ are resp.
for a projection and a normal residual block, assuming σπ
= 1. The V and W operators are defined in Defn C.1.

assistant: text: By setting the width or the initialization variance of each layer differently, we can actually subdue gradient explosion problems in residual networks (with fully connected layers and no batchnorm). A mathematical theory is developed that not only tells you how to do it, but also surprisingly is able to predict, after you apply such tricks, how fast your network trains to achieve a certain test set performance. This is some black magic stuff, and it's called "Deep Mean Field Theory."