Unadjusted Hamiltonian MCMC with Stratified Monte Carlo Time Integration
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866916634110722048 |
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| author | Bou-Rabee, Nawaf Marsden, Milo |
| author_facet | Bou-Rabee, Nawaf Marsden, Milo |
| contents | 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 $μ(dx) \propto e^{-U(x)} dx$ where $U: \mathbb{R}^d \to \mathbb{R}_{\ge 0}$ is $K$-strongly convex but only $L$-gradient Lipschitz, and initial distributions $ν$ 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 $O\left((d/K)^{1/3} (L/K)^{5/3} \varepsilon^{-2/3} \log( \boldsymbol{\mathcal{W}}^2(μ, ν) / \varepsilon)^+\right)$ gradient evaluations; whereas for such rough target densities the corresponding complexity of the uHMC algorithm with Verlet time integration is in general $O\left((d/K)^{1/2} (L/K)^2 \varepsilon^{-1} \log( \boldsymbol{\mathcal{W}}^2(μ, ν) / \varepsilon)^+ \right)$. Metropolis-adjustable randomized time integrators are also provided. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_11003 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Unadjusted Hamiltonian MCMC with Stratified Monte Carlo Time Integration Bou-Rabee, Nawaf Marsden, Milo Probability Numerical Analysis Statistics Theory Computation Machine Learning 60J05 (Primary) 65C05, 65P10 (Secondary) 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 $μ(dx) \propto e^{-U(x)} dx$ where $U: \mathbb{R}^d \to \mathbb{R}_{\ge 0}$ is $K$-strongly convex but only $L$-gradient Lipschitz, and initial distributions $ν$ 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 $O\left((d/K)^{1/3} (L/K)^{5/3} \varepsilon^{-2/3} \log( \boldsymbol{\mathcal{W}}^2(μ, ν) / \varepsilon)^+\right)$ gradient evaluations; whereas for such rough target densities the corresponding complexity of the uHMC algorithm with Verlet time integration is in general $O\left((d/K)^{1/2} (L/K)^2 \varepsilon^{-1} \log( \boldsymbol{\mathcal{W}}^2(μ, ν) / \varepsilon)^+ \right)$. Metropolis-adjustable randomized time integrators are also provided. |
| title | Unadjusted Hamiltonian MCMC with Stratified Monte Carlo Time Integration |
| topic | Probability Numerical Analysis Statistics Theory Computation Machine Learning 60J05 (Primary) 65C05, 65P10 (Secondary) |
| url | https://arxiv.org/abs/2211.11003 |