  In this paper, we propose a new adaptive stochastic gradient Langevin
dynamics (ASGLD) algorithmic framework and its two specialized versions, namely
adaptive stochastic gradient (ASG) and adaptive gradient Langevin
dynamics(AGLD), for non-convex optimization problems. All proposed algorithms
can escape from saddle points with at most $O(\log d)$ iterations, which is
nearly dimension-free. Further, we show that ASGLD and ASG converge to a local
minimum with at most $O(\log d/\epsilon^4)$ iterations. Also, ASGLD with full
gradients or ASGLD with a slowly linearly increasing batch size converge to a
local minimum with iterations bounded by $O(\log d/\epsilon^2)$, which
outperforms existing first-order methods.
