Theoretical Analysis of Auto Rate-Tuning by Batch Normalization
Authors: Sanjeev Arora, Zhiyuan Li, Kaifeng Lyu
Keywords: batch normalization, scale invariance, learning rate, stationary point
TL;DR: We give a theoretical analysis of the ability of batch normalization to automatically tune learning rates, in the context of finding stationary points for a deep learning objective.
Abstract: Batch Normalization (BN) has become a cornerstone of deep learning across diverse architectures, appearing to help optimization as well as generalization. While the idea makes intuitive sense, theoretical analysis of its effectiveness has been lacking. Here theoretical support is provided for one of its conjectured properties, namely, the ability to allow gradient descent to succeed with less tuning of learning rates. It is shown that even if we fix the learning rate of scale-invariant parameters (e.g., weights of each layer with BN) to a constant (say, 0.3), gradient descent still approaches a stationary point (i.e., a solution where gradient is zero) in the rate of T^{−1/2} in T iterations, asymptotically matching the best bound for gradient descent with well-tuned learning rates. A similar result with convergence rate T^{−1/4} is also shown for stochastic gradient descent.