Contraction and Convergence Rates for Discretized Kinetic Langevin Dynamics
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We provide a framework to prove convergence rates for discretizations of kinetic Langevin dynamics for M-∇Lipschitz m-log-concave densities. Our approach provides convergence rates of 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 methods which are popular in the molecular dynamics and machine learning communities. Finally we introduce the property “γ-limit convergent" (GLC) to characterise 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.
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