  We study the problem of non-convex optimization using Stochastic Gradient
Langevin Dynamics (SGLD). SGLD is a natural and popular variation of stochastic
gradient descent where at each step, appropriately scaled Gaussian noise is
added. To our knowledge, the only strategy for showing global convergence of
SGLD on the loss function is to show that SGLD can sample from a stationary
distribution which assigns larger mass when the function is small (the Gibbs
measure), and then to convert these guarantees to optimization results.
  We employ a new strategy to analyze the convergence of SGLD to global minima,
based on Lyapunov potentials and optimization. We convert the same mild
conditions from previous works on SGLD into geometric properties based on
Lyapunov potentials. This adapts well to the case with a stochastic gradient
oracle, which is natural for machine learning applications where one wants to
minimize population loss but only has access to stochastic gradients via
minibatch training samples. Here we provide 1) improved rates in the setting of
previous works studying SGLD for optimization, 2) the first finite gradient
complexity guarantee for SGLD where the function is Lipschitz and the Gibbs
measure defined by the function satisfies a Poincar\'e Inequality, and 3) prove
if continuous-time Langevin Dynamics succeeds for optimization, then
discrete-time SGLD succeeds under mild regularity assumptions.
