HMC and underdamped Langevin united in the unadjusted convex smooth case
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866914804976844800 |
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| author | Gouraud, Nicolaï Bris, Pierre Le Majka, Adrien Monmarché, Pierre |
| author_facet | Gouraud, Nicolaï Bris, Pierre Le Majka, Adrien Monmarché, Pierre |
| contents | We consider a family of unadjusted generalized HMC samplers, which includes standard position HMC samplers and discretizations of the underdamped Langevin process. A detailed analysis and optimization of the parameters is conducted in the Gaussian case, which shows an improvement from $1/κ$ to $1/\sqrtκ$ for the convergence rate in terms of the condition number $κ$ by using partial velocity refreshment, with respect to classical full refreshments. A similar effect is observed empirically for two related algorithms, namely Metropolis-adjusted gHMC and kinetic piecewise-deterministic Markov processes. Then, a stochastic gradient version of the samplers is considered, for which dimension-free convergence rates are established for log-concave smooth targets over a large range of parameters, gathering in a unified framework previous results on position HMC and underdamped Langevin and extending them to HMC with inertia. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_00977 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | HMC and underdamped Langevin united in the unadjusted convex smooth case Gouraud, Nicolaï Bris, Pierre Le Majka, Adrien Monmarché, Pierre Probability Statistics Theory 65C05 We consider a family of unadjusted generalized HMC samplers, which includes standard position HMC samplers and discretizations of the underdamped Langevin process. A detailed analysis and optimization of the parameters is conducted in the Gaussian case, which shows an improvement from $1/κ$ to $1/\sqrtκ$ for the convergence rate in terms of the condition number $κ$ by using partial velocity refreshment, with respect to classical full refreshments. A similar effect is observed empirically for two related algorithms, namely Metropolis-adjusted gHMC and kinetic piecewise-deterministic Markov processes. Then, a stochastic gradient version of the samplers is considered, for which dimension-free convergence rates are established for log-concave smooth targets over a large range of parameters, gathering in a unified framework previous results on position HMC and underdamped Langevin and extending them to HMC with inertia. |
| title | HMC and underdamped Langevin united in the unadjusted convex smooth case |
| topic | Probability Statistics Theory 65C05 |
| url | https://arxiv.org/abs/2202.00977 |