  Stochastic gradient Langevin dynamics (SGLD) is a fundamental algorithm in
stochastic optimization. Recent work by Zhang et al. [2017] presents an
analysis for the hitting time of SGLD for the first and second order stationary
points. The proof in Zhang et al. [2017] is a two-stage procedure through
bounding the Cheeger's constant, which is rather complicated and leads to loose
bounds. In this paper, using intuitions from stochastic differential equations,
we provide a direct analysis for the hitting times of SGLD to the first and
second order stationary points. Our analysis is straightforward. It only relies
on basic linear algebra and probability theory tools. Our direct analysis also
leads to tighter bounds comparing to Zhang et al. [2017] and shows the explicit
dependence of the hitting time on different factors, including dimensionality,
smoothness, noise strength, and step size effects. Under suitable conditions,
we show that the hitting time of SGLD to first-order stationary points can be
dimension-independent. Moreover, we apply our analysis to study several
important online estimation problems in machine learning, including linear
regression, matrix factorization, and online PCA.
