On Convergence of the Alternating Directions SGHMC Algorithm
This work addresses convergence analysis for a variant of HMC, which is incremental as it extends existing methods with a novel alternating directions procedure.
The paper tackles the problem of analyzing convergence rates for Hamiltonian Monte Carlo algorithms using stochastic gradient oracles, and provides explicit convergence rates that depend on key parameters like problem dimension and oracle quality.
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.