Contraction and Convergence Rates for Discretized Kinetic Langevin Dynamics
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| Format: | Preprint |
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2023
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| author | Leimkuhler, Benedict Paulin, Daniel Whalley, Peter A. |
| author_facet | Leimkuhler, Benedict Paulin, Daniel Whalley, Peter A. |
| contents | We provide a framework to analyze the convergence of discretized kinetic Langevin dynamics for $M$-$\nabla$Lipschitz, $m$-convex potentials. Our approach gives convergence rates of $\mathcal{O}(m/M)$, with explicit stepsize restrictions, which are of the same order as the stability threshold for Gaussian targets and are valid for a large interval of the friction parameter. We apply this methodology to various integration schemes which are popular in the molecular dynamics and machine learning communities. Further, we introduce the property ``$γ$-limit convergent" (GLC) to characterize underdamped Langevin schemes that converge to overdamped dynamics in the high-friction limit and which have stepsize restrictions that are independent of the friction parameter; we show that this property is not generic by exhibiting methods from both the class and its complement. Finally, we provide asymptotic bias estimates for the BAOAB scheme, which remain accurate in the high-friction limit by comparison to a modified stochastic dynamics which preserves the invariant measure. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2302_10684 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Contraction and Convergence Rates for Discretized Kinetic Langevin Dynamics Leimkuhler, Benedict Paulin, Daniel Whalley, Peter A. Numerical Analysis Computation 65C05, 65C30, 65C40 G.1.0; G.3 We provide a framework to analyze the convergence of discretized kinetic Langevin dynamics for $M$-$\nabla$Lipschitz, $m$-convex potentials. Our approach gives convergence rates of $\mathcal{O}(m/M)$, with explicit stepsize restrictions, which are of the same order as the stability threshold for Gaussian targets and are valid for a large interval of the friction parameter. We apply this methodology to various integration schemes which are popular in the molecular dynamics and machine learning communities. Further, we introduce the property ``$γ$-limit convergent" (GLC) to characterize underdamped Langevin schemes that converge to overdamped dynamics in the high-friction limit and which have stepsize restrictions that are independent of the friction parameter; we show that this property is not generic by exhibiting methods from both the class and its complement. Finally, we provide asymptotic bias estimates for the BAOAB scheme, which remain accurate in the high-friction limit by comparison to a modified stochastic dynamics which preserves the invariant measure. |
| title | Contraction and Convergence Rates for Discretized Kinetic Langevin Dynamics |
| topic | Numerical Analysis Computation 65C05, 65C30, 65C40 G.1.0; G.3 |
| url | https://arxiv.org/abs/2302.10684 |