  We study convergence rates of Hamiltonian Monte Carlo (HMC) algorithms with
leapfrog integration under mild conditions on stochastic gradient oracle for
the target distribution (SGHMC). Our method extends standard HMC by allowing
the use of general auxiliary distributions, which is achieved by a novel
procedure of Alternating Directions.
  The convergence analysis is based on the investigations of the Dirichlet
forms associated with the underlying Markov chain driving the algorithms. For
this purpose, we provide a detailed analysis on the error of the leapfrog
integrator for Hamiltonian motions with both the kinetic and potential energy
functions in general form. We characterize the explicit dependence of the
convergence rates on key parameters such as the problem dimension, functional
properties of both the target and auxiliary distributions, and the quality of
the oracle.
