Analysis of Langevin Monte Carlo from Poincaré to Log-Sobolev
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866909249788968960 |
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| author | Chewi, Sinho Erdogdu, Murat A. Li, Mufan Bill Shen, Ruoqi Zhang, Matthew |
| author_facet | Chewi, Sinho Erdogdu, Murat A. Li, Mufan Bill Shen, Ruoqi Zhang, Matthew |
| contents | Classically, the continuous-time Langevin diffusion converges exponentially fast to its stationary distribution $π$ under the sole assumption that $π$ satisfies a Poincaré inequality. Using this fact to provide guarantees for the discrete-time Langevin Monte Carlo (LMC) algorithm, however, is considerably more challenging due to the need for working with chi-squared or Rényi divergences, and prior works have largely focused on strongly log-concave targets. In this work, we provide the first convergence guarantees for LMC assuming that $π$ satisfies either a Latała--Oleszkiewicz or modified log-Sobolev inequality, which interpolates between the Poincaré and log-Sobolev settings. Unlike prior works, our results allow for weak smoothness and do not require convexity or dissipativity conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_12662 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Analysis of Langevin Monte Carlo from Poincaré to Log-Sobolev Chewi, Sinho Erdogdu, Murat A. Li, Mufan Bill Shen, Ruoqi Zhang, Matthew Statistics Theory Machine Learning Classically, the continuous-time Langevin diffusion converges exponentially fast to its stationary distribution $π$ under the sole assumption that $π$ satisfies a Poincaré inequality. Using this fact to provide guarantees for the discrete-time Langevin Monte Carlo (LMC) algorithm, however, is considerably more challenging due to the need for working with chi-squared or Rényi divergences, and prior works have largely focused on strongly log-concave targets. In this work, we provide the first convergence guarantees for LMC assuming that $π$ satisfies either a Latała--Oleszkiewicz or modified log-Sobolev inequality, which interpolates between the Poincaré and log-Sobolev settings. Unlike prior works, our results allow for weak smoothness and do not require convexity or dissipativity conditions. |
| title | Analysis of Langevin Monte Carlo from Poincaré to Log-Sobolev |
| topic | Statistics Theory Machine Learning |
| url | https://arxiv.org/abs/2112.12662 |