Unadjusted Hamiltonian MCMC with Stratified Monte Carlo Time Integration

• URL: http://arxiv.org/abs/2211.11003v3
• Date: Fri, 04 Oct 2024 15:01:35 GMT
• Title: Unadjusted Hamiltonian MCMC with Stratified Monte Carlo Time Integration
• Authors: Nawaf Bou-Rabee, Milo Marsden,
• Abstract summary: A randomized time integrator is suggested for unadjusted Hamiltonian Monte Carlo (uHMC) $adjustable randomized time are also provided. The complexity of the uHMC algorithm with Verlet time integration is in general $Oleft((d/K)1/2 (L/K)2 varepsilon-1 log.
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• Abstract: A randomized time integrator is suggested for unadjusted Hamiltonian Monte Carlo (uHMC) which involves a very minor modification to the usual Verlet time integrator, and hence, is easy to implement. For target distributions of the form $\mu(dx) \propto e^{-U(x)} dx$ where $U: \mathbb{R}^d \to \mathbb{R}_{\ge 0}$ is $K$-strongly convex but only $L$-gradient Lipschitz, and initial distributions $\nu$ with finite second moment, coupling proofs reveal that an $\varepsilon$-accurate approximation of the target distribution in $L^2$-Wasserstein distance $\boldsymbol{\mathcal{W}}^2$ can be achieved by the uHMC algorithm with randomized time integration using $O\left((d/K)^{1/3} (L/K)^{5/3} \varepsilon^{-2/3} \log( \boldsymbol{\mathcal{W}}^2(\mu, \nu) / \varepsilon)^+\right)$ gradient evaluations; whereas for such rough target densities the corresponding complexity of the uHMC algorithm with Verlet time integration is in general $O\left((d/K)^{1/2} (L/K)^2 \varepsilon^{-1} \log( \boldsymbol{\mathcal{W}}^2(\mu, \nu) / \varepsilon)^+ \right)$. Metropolis-adjustable randomized time integrators are also provided.

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