Title: Stochastic Gradient Descent in Continuous Time: A Central Limit Theorem,
Abstract: Stochastic gradient descent in continuous time (SGDCT) provides a
computationally efficient method for the statistical learning of
continuous-time models, which are widely used in science, engineering, and
finance. The SGDCT algorithm follows a (noisy) descent direction along a
continuous stream of data. The parameter updates occur in continuous time and
satisfy a stochastic differential equation. This paper analyzes the asymptotic
convergence rate of the SGDCT algorithm by proving a central limit theorem
(CLT) for strongly convex objective functions and, under slightly stronger
conditions, for non-convex objective functions as well. An L$^p$ convergence
rate is also proven for the algorithm in the strongly convex case. The
mathematical analysis lies at the intersection of stochastic analysis and
statistical learning.