Stochastic gradient Langevin dynamics with adaptive drifts

We propose a class of adaptive stochastic gradient Markov chain Monte Carlo (SGMCMC) algorithms, where the drift function is adaptively adjusted according to the gradient of past samples to accelerate the convergence of the algorithm in simulations of the distributions with pathological curvatures. We establish the convergence of the proposed algorithms under mild conditions. The numerical examples indicate that the proposed algorithms can significantly outperform the popular SGMCMC algorithms, such as stochastic gradient Langevin dynamics (SGLD), stochastic gradient Hamiltonian Monte Carlo (SGHMC) and preconditioned SGLD, in both simulation and optimization tasks. In particular, the proposed algorithms can converge quickly for the distributions for which the energy landscape possesses pathological curvatures.