Title: User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient,
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 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.