Double-Loop Unadjusted Langevin Algorithm

Paul Rolland; Armin Eftekhari; Ali Kavis; Volkan Cevher

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arXiv · 2007.01147

Double-Loop Unadjusted Langevin Algorithm

Paul Rolland·Armin Eftekhari·Ali Kavis·Volkan Cevher

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Abstract

A well-known first-order method for sampling from log-concave probability distributions is the Unadjusted Langevin Algorithm (ULA). This work proposes a new annealing step-size schedule for ULA, which allows to prove new convergence guarantees for sampling from a smooth log-concave distribution, which are not covered by existing state-of-the-art convergence guarantees. To establish this result, we derive a new theoretical bound that relates the Wasserstein distance to total variation distance between any two log-concave distributions that complements the reach of Talagrand T2 inequality. Moreover, applying this new step size schedule to an existing constrained sampling algorithm, we show state-of-the-art convergence rates for sampling from a constrained log-concave distribution, as well as improved dimension dependence.

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Paul Rolland, Armin Eftekhari, Ali Kavis, Volkan Cevher. 2020-07-02. Double-Loop Unadjusted Langevin Algorithm. https://arxiv.org/abs/2007.01147

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