An Exponential Learning Rate Schedule for Deep Learning

TL;DR

This paper introduces a novel exponential learning rate schedule for deep learning that, combined with Batch Normalization, enables stable training with SGD and momentum, supported by theoretical proofs and practical experiments.

Contribution

It demonstrates the first successful use of an exponential learning rate schedule with SGD and BN, providing theoretical justification and practical insights.

Findings

Exponential learning rate schedule works with BN and SGD.

The schedule is mathematically equivalent to standard training settings.

Using both weight decay and BN can hinder convergence.

Abstract

Intriguing empirical evidence exists that deep learning can work well with exoticschedules for varying the learning rate. This paper suggests that the phenomenon may be due to Batch Normalization or BN, which is ubiquitous and provides benefits in optimization and generalization across all standard architectures. The following new results are shown about BN with weight decay and momentum (in other words, the typical use case which was not considered in earlier theoretical analyses of stand-alone BN. 1. Training can be done using SGD with momentum and an exponentially increasing learning rate schedule, i.e., learning rate increases by some $(1 +\alpha)$ factor in every epoch for some $\alpha >0$. (Precise statement in the paper.) To the best of our knowledge this is the first time such a rate schedule has been successfully used, let alone for highly successful architectures. As expected, such training rapidly blows up network weights, but the net stays well-behaved due to normalization. 2. Mathematical explanation of the success of the above rate schedule: a rigorous proof that it is equivalent to the standard setting of BN + SGD + StandardRate Tuning + Weight Decay + Momentum. This equivalence holds for other normalization layers as well, Group Normalization, LayerNormalization, Instance Norm, etc. 3. A worked-out toy example illustrating the above linkage of hyper-parameters. Using either weight decay or BN alone reaches global minimum, but convergence fails when both are used.