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.

我们研究了具有跳跃点积分的 Hamiltonian Monte Carlo 算法在目标分布的随机梯度神谕上的收敛速率。我们的方法通过 Alternating Directions 的新颖步骤，扩展了标准 HMC 算法的使用范围，使其能够使用一般辅助分布。在此基础上，我们基于驱动算法的底层马尔可夫链的狄利克雷形式的研究进行收敛分析。为此，我们对具有一般形式的动能和势能函数的 Hamilton 运动的 leapfrog 积分器的误差进行了详细分析。我们表征了收敛速率对问题维度、目标和辅助分布的函数特性以及神谕的质量等关键参数的明确依赖关系。