Ergodicity of Langevin Dynamics and its Discretizations for Non-smooth Potentials
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| Format: | Preprint |
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2024
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| author | Fruehwirth, Lorenz Habring, Andreas |
| author_facet | Fruehwirth, Lorenz Habring, Andreas |
| contents | This article is concerned with sampling from Gibbs distributions $π(x)\propto e^{-U(x)}$ using Markov chain Monte Carlo methods. In particular, we investigate Langevin dynamics in the continuous- and the discrete-time setting for such distributions with potentials $U(x)$ which are strongly-convex but possibly non-differentiable. We show that the corresponding subgradient Langevin dynamics are exponentially ergodic to the target density $π$ in the continuous setting and that certain explicit as well as semi-implicit discretizations are geometrically ergodic and approximate $π$ for vanishing discretization step size. Moreover, we prove that the discrete schemes satisfy the law of large numbers allowing to use consecutive iterates of a Markov chain in order to compute statistics of the stationary distribution posing a significant reduction of computational complexity in practice. Numerical experiments are provided confirming the theoretical findings and showcasing the practical relevance of the proposed methods in imaging applications. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_12051 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ergodicity of Langevin Dynamics and its Discretizations for Non-smooth Potentials Fruehwirth, Lorenz Habring, Andreas Numerical Analysis Optimization and Control 60J20, 68U10, 94A08 G.3; I.4.5 This article is concerned with sampling from Gibbs distributions $π(x)\propto e^{-U(x)}$ using Markov chain Monte Carlo methods. In particular, we investigate Langevin dynamics in the continuous- and the discrete-time setting for such distributions with potentials $U(x)$ which are strongly-convex but possibly non-differentiable. We show that the corresponding subgradient Langevin dynamics are exponentially ergodic to the target density $π$ in the continuous setting and that certain explicit as well as semi-implicit discretizations are geometrically ergodic and approximate $π$ for vanishing discretization step size. Moreover, we prove that the discrete schemes satisfy the law of large numbers allowing to use consecutive iterates of a Markov chain in order to compute statistics of the stationary distribution posing a significant reduction of computational complexity in practice. Numerical experiments are provided confirming the theoretical findings and showcasing the practical relevance of the proposed methods in imaging applications. |
| title | Ergodicity of Langevin Dynamics and its Discretizations for Non-smooth Potentials |
| topic | Numerical Analysis Optimization and Control 60J20, 68U10, 94A08 G.3; I.4.5 |
| url | https://arxiv.org/abs/2411.12051 |