Intriguing empirical evidence exists that deep learning can work well with exoticschedules for varying the learning rate. This paper suggests that the phenomenonmay be due to Batch Normalization or BN(Ioffe & Szegedy, 2015), which is ubiquitous and provides benefits in optimization and generalization across all standardarchitectures. The following new results are shown about BN with weight decay and momentum (in other words, the typical use case which was not considered inearlier theoretical analyses of stand-alone BN (Ioffe & Szegedy, 2015; Santurkaret al., 2018; Arora et al., 2018). 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(Wu & He, 2018), LayerNormalization(Ba et al., 2016), Instance Norm(Ulyanov et al., 2016), 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.

通过对BN的权重衰减及动量模型的应用，本文发现深度学习算法能够成功应用于具有指数增长学习速率的训练方式，证明了这种训练方式在各种标准结构中具有优秀的表现，并给出了数学解释和实例验证。

深度学习指数学习率调度