A New Framework for Variance-Reduced Hamiltonian Monte Carlo

Abstract: We propose a new framework of variance-reduced Hamiltonian Monte Carlo (HMC) methods for sampling from an $L$-smooth and $m$-strongly log-concave distribution, based on a unified formulation of biased and unbiased variance reduction methods. We study the convergence properties for HMC with gradient estimators which satisfy the Mean-Squared-Error-Bias (MSEB) property. We show that the unbiased gradient estimators, including SAGA and SVRG, based HMC methods achieve highest gradient efficiency with small batch size under high precision regime, and require $\tilde{O}(N + \kappa^2 d^{\frac{1}{2}} \varepsilon^{-1} + N^{\frac{2}{3}} \kappa^{\frac{4}{3}} d^{\frac{1}{3}} \varepsilon^{-\frac{2}{3}} )$ gradient complexity to achieve $\epsilon$-accuracy in 2-Wasserstein distance. Moreover, our HMC methods with biased gradient estimators, such as SARAH and SARGE, require $\tilde{O}(N+\sqrt{N} \kappa^2 d^{\frac{1}{2}} \varepsilon^{-1})$ gradient complexity, which has the same dependency on condition number $\kappa$ and dimension $d$ as full gradient method, but improves the dependency of sample size $N$ for a factor of $N^\frac{1}{2}$. Experimental results on both synthetic and real-world benchmark data show that our new framework significantly outperforms the full gradient and stochastic gradient HMC approaches. The earliest version of this paper was submitted to ICML 2020 with three weak accept but was not finally accepted.