  Recent works (e.g., (Li and Arora, 2020)) suggest that the use of popular
normalization schemes (including Batch Normalization) in today's deep learning
can move it far from a traditional optimization viewpoint, e.g., use of
exponentially increasing learning rates. The current paper highlights other
ways in which behavior of normalized nets departs from traditional viewpoints,
and then initiates a formal framework for studying their mathematics via
suitable adaptation of the conventional framework namely, modeling SGD-induced
training trajectory via a suitable stochastic differential equation (SDE) with
a noise term that captures gradient noise. This yields: (a) A new ' intrinsic
learning rate' parameter that is the product of the normal learning rate and
weight decay factor. Analysis of the SDE shows how the effective speed of
learning varies and equilibrates over time under the control of intrinsic LR.
(b) A challenge -- via theory and experiments -- to popular belief that good
generalization requires large learning rates at the start of training. (c) New
experiments, backed by mathematical intuition, suggesting the number of steps
to equilibrium (in function space) scales as the inverse of the intrinsic
learning rate, as opposed to the exponential time convergence bound implied by
SDE analysis. We name it the Fast Equilibrium Conjecture and suggest it holds
the key to why Batch Normalization is effective.

Machine Learning