Stochastic Gradient/Mirror Descent: Minimax Optimality and Implicit Regularization
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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) on the square loss of linear models---originally developed in the 1990's---and extend them to generic stochastic mirror descent (SMD) algorithms on 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 that this identity implies the minimax optimality of SMD (and SGD) with sufficiently small step size, 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), such as convergence and implicit regularization for over-parameterized linear models, which have been shown in certain cases in the literature. We also show how this identity can be used in the so-called "highly over-parameterized" nonlinear setting to provide insights into why SMD (and SGD) may have similar convergence and implicit regularization properties for deep learning.
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