Theoretical Analysis of Auto Rate-Tuning by Batch Normalization

TL;DR

This paper provides a theoretical analysis demonstrating that Batch Normalization enables gradient descent to succeed with less tuning of learning rates, showing convergence rates for both deterministic and stochastic methods.

Contribution

It offers the first theoretical support for BN's conjectured property of reducing the need for learning rate tuning in gradient-based optimization.

Findings

Gradient descent with fixed learning rate approaches stationarity at rate T^{-1/2}.

Stochastic gradient descent converges at rate T^{-1/4}.

Supports BN's role in simplifying hyperparameter tuning.

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.