  Stochastic descent methods (of the gradient and mirror varieties) have become
increasingly popular in optimization. In fact, it is now widely recognized that
the success of deep learning is not only due to the special deep architecture
of the models, but also due to the behavior of the stochastic descent methods
used, which play a key role in reaching "good" solutions that generalize well
to unseen data. In an attempt to shed some light on why this is the case, we
revisit some minimax properties of stochastic gradient descent (SGD) for the
square loss of linear models---originally developed in the 1990's---and extend
them to general stochastic mirror descent (SMD) algorithms for general loss
functions and nonlinear models. In particular, we show that there is a
fundamental identity which holds for SMD (and SGD) under very general
conditions, and which implies the minimax optimality of SMD (and SGD) for
sufficiently small step size, and for a general class of loss functions and
general nonlinear models. We further show that this identity can be used to
naturally establish other properties of SMD (and SGD), namely convergence and
implicit regularization for over-parameterized linear models (in what is now
being called the "interpolating regime"), some of which have been shown in
certain cases in prior literature. We also argue how this identity can be used
in the so-called "highly over-parameterized" nonlinear setting (where the
number of parameters far exceeds the number of data points) to provide insights
into why SMD (and SGD) may have similar convergence and implicit regularization
properties for deep learning.
