Improved Bounds for Discretization of Langevin Diffusions: Near-Optimal Rates without Convexity

Mou, Wenlong; Flammarion, Nicolas; Wainwright, Martin J.

doi:10.48550/arxiv.1907.11331

Cited by 15 publications

(32 citation statements)

References 17 publications

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“…The results significantly improve those obtained e.g. in [DM17] or in [Dal17] in terms of pγ n q ně1 and seem quite consistent with more recent works (by totally different methods) like [MFWB19] (see Remark 2.4 for details).…”

Section: Xσsupporting

confidence: 91%

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Unajusted Langevin algorithm with multiplicative noise: Total variation and Wasserstein bounds

Pagès¹,

Panloup²

2020

Preprint

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In this paper, we focus on non-asymptotic bounds related to the Euler scheme of an ergodic diffusion with a possibly multiplicative diffusion term (non-constant diffusion coefficient). More precisely, the objective of this paper is to control the distance of the standard Euler scheme with decreasing step (usually called Unajusted Langevin Algorithm in the Monte-Carlo literature) to the invariant distribution of such an ergodic diffusion. In an appropriate Lyapunov setting and under uniform ellipticity assumptions on the diffusion coefficient, we establish (or improve) such bounds for Total Variation and L 1 -Wasserstein distances in both multiplicative and additive and frameworks. These bounds rely on weak error expansions using Stochastic Analysis adapted to decreasing step setting.

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Section: Xσsupporting

confidence: 91%

“…[Pel96], [MP96]) and more recently in a series of papers, still in the additive setting, motivated by applications in machine learning (in particular in Bayesian or PAC-Bayesian statistics). Among others, we refer to [DM17,Dal17,MFWB19] and to the references therein.…”

Section: ¯´1mentioning

confidence: 99%

Unajusted Langevin algorithm with multiplicative noise: Total variation and Wasserstein bounds

Pagès¹,

Panloup²

2020

Preprint

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“…The study of the long-time behavior of Euler-Maruyama schemes of (6) has been the topic of numerous papers in the last years. Among others, we can refer to [DM15, DM17, DMM19, Dal17, DK19,MFWB19] where the authors generally focus on the (Wasserstein, Total Variation,. .…”

Section: Ergodic Approximation and Gibbs Approximationmentioning

confidence: 99%

Multilevel-Langevin pathwise average for Gibbs approximation

Egea,

Panloup

2021

Preprint

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We propose and study a new multilevel method for the numerical approximation of a Gibbs distribution π on R d , based on (over-damped) Langevin diffusions. This method both inspired by [PP18] and [GMS + 20] relies on a multilevel occupation measure, i.e. on an appropriate combination of R occupation measures of (constant-step) discretized schemes of the Langevin diffusion with respective steps γr = γ02 −r , r = 0, . . . , R. For a given diffusion, we first state a result under general assumptions which guarantees an ε-approximation (in a L 2 -sense) with a cost proportional to ε −2 (i.e. proportional to a Monte-Carlo method without bias) or ε −2 | log ε| 3 under less contractive assumptions.This general result is then applied to over-damped Langevin diffusions in a strongly convex setting, with a study of the dependence in the dimension d and in the spectrum of the Hessian matrix D 2 U of the potential U : R d → R involved in the Gibbs distribution. This leads to strategies with cost in O(dε −2 log 3 (dε −2 )) and in O(dε −2 ) under an additional condition on the third derivatives of U . In particular, in our last main result, we show that, up to universal constants, an appropriate choice of the diffusion coefficient and of the parameters of the procedure leads to a cost controlled by(where λU and λ U respectively denote the supremum and the infimum of the largest and lowest eigenvalue of D 2 U ).In our numerical illustrations, we show that our theoretical bounds are confirmed in practice and finally propose an opening to some theoretical or numerical strategies in order to increase the robustness of the procedure when the largest and smallest eigenvalues of D 2 U are respectively too large or too small.

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“…One notable example is the Langevin Monte Carlo algorithm (LMC), which corresponds to Euler-Maruyama discretization of the overdamped Langevin equation. Its study dated back to at least the 90s (Roberts et al, 1996) but keeps on leading to important discoveries, for example, on non-asymptotics and dimension dependence, which are relevant to machine learning (e.g., Dalalyan (2017a,b); Cheng et al (2018a); ; Durmus and Moulines (2019); Vempala and Wibisono (2019); Dalalyan and Riou-Durand (2020); Li et al (2019); Erdogdu and Hosseinzadeh (2020); Mou et al (2019)). LMC is closely related to SGD too (e.g., Mandt et al (2017)).…”

Section: Introductionmentioning

confidence: 99%

Sqrt(d) Dimension Dependence of Langevin Monte Carlo

Zha

Tao

2021

Preprint

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This article considers the popular MCMC method of unadjusted Langevin Monte Carlo (LMC) and provides a non-asymptotic analysis of its sampling error in 2-Wasserstein distance. The proof is based on a mean-square analysis framework refined from Li et al. ( 2019), which works for a large class of sampling algorithms based on discretizations of contractive SDEs. We establish an O √ d / mixing time bound for LMC, without warm start, under the common log-smooth and log-strongly-convex conditions, plus a growth condition on the 3rd-order derivative of the potential of target measures. This bound improves the best previously known O d / result and is optimal (in terms of order) in both dimension d and accuracy tolerance for target measures satisfying the aforementioned assumptions. Our theoretical analysis is further validated by numerical experiments.Preprint. Under review.

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Improved Bounds for Discretization of Langevin Diffusions: Near-Optimal Rates without Convexity

Cited by 15 publications

References 17 publications

Unajusted Langevin algorithm with multiplicative noise: Total variation and Wasserstein bounds

Unajusted Langevin algorithm with multiplicative noise: Total variation and Wasserstein bounds

Multilevel-Langevin pathwise average for Gibbs approximation

Sqrt(d) Dimension Dependence of Langevin Monte Carlo

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