Mean Field Residual Networks: On the Edge of Chaos
We study randomly initialized residual networks using mean field theory and the theory of difference equations. Classical feedforward neural networks, such as those with tanh activations, exhibit exponential behavior on the average when propagating inputs forward or gradients backward. The exponential forward dynamics causes rapid collapsing of the input space geometry, while the exponential backward dynamics causes drastic vanishing or exploding gradients. We show, in contrast, that by adding skip connections, the network will, depending on the nonlinearity, adopt subexponential forward and backward dynamics, and in many cases in fact polynomial. The exponents of these polynomials are obtained through analytic methods and proved and verified empirically to be correct. In terms of the "edge of chaos" hypothesis, these subexponential and polynomial laws allow residual networks to "hover over the boundary between stability and chaos," thus preserving the geometry of the input space and the gradient information flow. In our experiments, for each activation function we study here, we initialize residual networks with different hyperparameters and train them on MNIST. Remarkably, our initialization time theory can accurately predict test time performance of these networks, by tracking either the expected amount of gradient explosion or the expected squared distance between the images of two input vectors. Importantly, we show, theoretically as well as empirically, that common initializations such as the Xavier or the He schemes are not optimal for residual networks, because the optimal initialization variances depend on the depth. Finally, we have made mathematical contributions by deriving several new identities for the kernels of powers of ReLU functions by relating them to the zeroth Bessel function of the second kind.
This paper analytically investigates the properties of Resnets with random weights using a mean field approximation. The approach used is an extension of previous analysis of feed forward neural networks. The authors show that in contrast to feed forward networks Resnets do exhibit a sub-exponential behavior (polynomial or logarithmic) when inputs are propagated forward or gradients are propagated backwards through the layers.
The results are very interesting because they give an analytic justification and intuition of why Resnets with a large number of layers can be trained reliably. The paper also extends the mean field technique for studying neural network properties, which is of value for analyzing other architectures.  The presentation of the results is overall clear and while the paper is well written. 
Overall this is an interesting paper with convincing results.