Asymptotic Analysis via Stochastic Differential Equations of Gradient Descent Algorithms in Statistical and Computational Paradigms

This paper investigates asymptotic behaviors of gradient descent algorithms (particularly accelerated gradient descent and stochastic gradient descent) in the context of stochastic optimization arose in statistics and machine learning where objective functions are estimated from available data. We show that these algorithms can be modeled by continuous-time ordinary or stochastic differential equations, and their asymptotic dynamic evolutions and distributions are governed by some linear ordinary or stochastic differential equations, as the data size goes to infinity. We illustrate that our study can provide a novel unified framework for a joint computational and statistical asymptotic analysis on dynamic behaviors of these algorithms with the time (or the number of iterations in the algorithms) and large sample behaviors of the statistical decision rules (like estimators and classifiers) that the algorithms are applied to compute, where the statistical decision rules are the limits of the random sequences generated from these iterative algorithms as the number of iterations goes to infinity.