User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient

• URL: http://arxiv.org/abs/1710.00095v4
• Date: Fri, 23 Feb 2024 17:39:40 GMT
• Title: User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient
• Authors: Arnak S. Dalalyan and Avetik G. Karagulyan
• Abstract summary: We analyze several methods of approximate sampling based on discretizations of the Langevin diffusion. Our guarantees improve or extend the state-of-the-art results in three directions.
• Score: 8.348896353632165
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we study the problem of sampling from a given probability density function that is known to be smooth and strongly log-concave. We analyze several methods of approximate sampling based on discretizations of the (highly overdamped) Langevin diffusion and establish guarantees on its error measured in the Wasserstein-2 distance. Our guarantees improve or extend the state-of-the-art results in three directions. First, we provide an upper bound on the error of the first-order Langevin Monte Carlo (LMC) algorithm with optimized varying step-size. This result has the advantage of being horizon free (we do not need to know in advance the target precision) and to improve by a logarithmic factor the corresponding result for the constant step-size. Second, we study the case where accurate evaluations of the gradient of the log-density are unavailable, but one can have access to approximations of the aforementioned gradient. In such a situation, we consider both deterministic and stochastic approximations of the gradient and provide an upper bound on the sampling error of the first-order LMC that quantifies the impact of the gradient evaluation inaccuracies. Third, we establish upper bounds for two versions of the second-order LMC, which leverage the Hessian of the log-density. We provide nonasymptotic guarantees on the sampling error of these second-order LMCs. These guarantees reveal that the second-order LMC algorithms improve on the first-order LMC in ill-conditioned settings.

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