Spherical Perspective on Learning with Batch Norm

Batch Normalization (BN) is a prominent deep learning technique. In spite of its apparent simplicity, its implications over optimization are yet to be fully understood. In this paper, we study the optimization of neural networks with BN layers from a geometric perspective. We leverage the radial invariance of groups of parameters, such as neurons for multi-layer perceptrons or filters for convolutional neural networks, and translate several popular optimization schemes on the $L_2$ unit hypersphere. This formulation and the associated geometric interpretation sheds new light on the training dynamics and the relation between different optimization schemes. In particular, we use it to derive the effective learning rate of Adam and stochastic gradient descent (SGD) with momentum, and we show that in the presence of BN layers, performing SGD alone is actually equivalent to a variant of Adam constrained to the unit hypersphere. Our analysis also leads us to introduce new variants of Adam. We empirically show, over a variety of datasets and architectures, that they improve accuracy in classification tasks. The complete source code for our experiments is available at: https://github.com/ymontmarin/adamsr

Inductive Bias of Gradient Descent for Exponentially Weight Normalized Smooth Homogeneous Neural Nets

We analyze the inductive bias of gradient descent for weight normalized smooth homogeneous neural nets, when trained on exponential or cross-entropy loss. Our analysis focuses on exponential weight normalization (EWN), which encourages weight updates along the radial direction. This paper shows that the gradient flow path with EWN is equivalent to gradient flow on standard networks with an adaptive learning rate, and hence causes the weights to be updated in a way that prefers asymptotic relative sparsity. These results can be extended to hold for gradient descent via an appropriate adaptive learning rate. The asymptotic convergence rate of the loss in this setting is given by $\Theta(\frac{1}{t(\log t)^2})$, and is independent of the depth of the network. We contrast these results with the inductive bias of standard weight normalization (SWN) and unnormalized architectures, and demonstrate their implications on synthetic data sets.Experimental results on simple data sets and architectures support our claim on sparse EWN solutions, even with SGD. This demonstrates its potential applications in learning prunable neural networks.Comment: We have modified proposition 3, removing the extra assumptions, resulting in a slightly less sharp instability result. We have also added a figure showing the norm of the weights for SWN, EWN and NWN for the MNIST training procedure (Appendix N, Figure 11). A few more references that use SWN have been added to page 3. We have also fixed a few typos and grammatical error