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Unadjusted Hamiltonian MCMC with Stratified Monte Carlo Time Integration

Abstract

A novel randomized time integrator is suggested for unadjusted Hamiltonian Monte Carlo (uHMC) in place of the usual Verlet integrator; namely, a stratified Monte Carlo (sMC) integrator which involves a minor modification to Verlet, and hence, is easy to implement. For target distributions of the form μ(dx)eU(x)dx\mu(dx) \propto e^{-U(x)} dx where U:RdR0U: \mathbb{R}^d \to \mathbb{R}_{\ge 0} is both KK-strongly convex and LL-gradient Lipschitz, and initial distributions ν\nu with finite second moment, coupling proofs reveal that an ε\varepsilon-accurate approximation of the target distribution μ\mu in L2L^2-Wasserstein distance W2\boldsymbol{\mathcal{W}}^2 can be achieved by the uHMC algorithm with sMC time integration using O((d/K)1/3(L/K)5/3ε2/3log(W2(μ,ν)/ε)+)O\left((d/K)^{1/3} (L/K)^{5/3} \varepsilon^{-2/3} \log( \boldsymbol{\mathcal{W}}^2(\mu, \nu) / \varepsilon)^+\right) gradient evaluations; whereas without additional assumptions the corresponding complexity of the uHMC algorithm with Verlet time integration is in general O((d/K)1/2(L/K)2ε1log(W2(μ,ν)/ε)+)O\left((d/K)^{1/2} (L/K)^2 \varepsilon^{-1} \log( \boldsymbol{\mathcal{W}}^2(\mu, \nu) / \varepsilon)^+ \right). Duration randomization, which has a similar effect as partial momentum refreshment, is also treated. In this case, without additional assumptions on the target distribution, the complexity of duration-randomized uHMC with sMC time integration improves to O(max((d/K)1/4(L/K)3/2ε1/2,(d/K)1/3(L/K)4/3ε2/3))O\left(\max\left((d/K)^{1/4} (L/K)^{3/2} \varepsilon^{-1/2},(d/K)^{1/3} (L/K)^{4/3} \varepsilon^{-2/3} \right) \right) up to logarithmic factors. The improvement due to duration randomization turns out to be analogous to that of time integrator randomization.

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