Minimax Mixing Time of the Metropolis-Adjusted Langevin Algorithm for Log-Concave Sampling

Wu, Keru; Schmidler, Scott C.; Chen, Yuansi

doi:10.48550/arxiv.2109.13055

Cited by 6 publications

(10 citation statements)

References 44 publications

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“…In other words, sampling is strictly easier in the presence of an initialization with bounded density ratio to the target (i.e., a warm start) than an initialization with bounded KL divergence. This is consistent with intuition from prior work on the complexity of the Metropolis-adjusted Langevin algorithm (see Chewi et al, 2021;Lee et al, 2021;Wu et al, 2021).…”

Section: Bump Construction and The Second Lower Boundsupporting

confidence: 89%

Fisher information lower bounds for sampling

Chewi¹,

Gerber²,

Lee³

et al. 2022

Preprint

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We prove two lower bounds for the complexity of non-log-concave sampling within the framework of , who introduced the use of Fisher information (FI) bounds as a notion of approximate first-order stationarity in sampling. Our first lower bound shows that averaged LMC is optimal for the regime of large FI by reducing the problem of finding stationary points in non-convex optimization to sampling. Our second lower bound shows that in the regime of small FI, obtaining a FI of at most ε 2 from the target distribution requires poly(1/ε) queries, which is surprising as it rules out the existence of high-accuracy algorithms (e.g., algorithms using Metropolis-Hastings filters) in this context.

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Section: Bump Construction and The Second Lower Boundsupporting

confidence: 89%

Fisher information lower bounds for sampling

Chewi¹,

Gerber²,

Lee³

et al. 2022

Preprint

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“…These algorithms can be combined with Metropolis-Hastings filters to correct the bias of the algorithm's stationary distribution; this typically improves mixing times to π from polynomial to polylogarithmic in the inverse accuracy. A notable example is the Metropolis-Adjusted Langevin Algorithm [78], whose mixing time has been the focus of much recent work [22,36] and was recently characterized in [91]. While there have been many more exciting developments, this literature is far too large to attempt to comprehensively cite here, and we refer the reader to textbooks and surveys such as [4,20,61,76,77] for further background.…”

Section: Related Workmentioning

confidence: 99%

Resolving the Mixing Time of the Langevin Algorithm to its Stationary Distribution for Log-Concave Sampling

Altschuler¹,

Talwar²

2022

Preprint

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Sampling from a high-dimensional distribution is a fundamental task in statistics, engineering, and the sciences. A canonical approach is the Langevin Algorithm, i.e., the Markov chain for the discretized Langevin Diffusion. This is the sampling analog of Gradient Descent. Despite being studied for several decades in multiple communities, tight mixing bounds for this algorithm remain unresolved even in the seemingly simple setting of log-concave distributions over a bounded domain. This paper completely characterizes the mixing time of the Langevin Algorithm to its stationary distribution in this setting (and others). This mixing result can be combined with any bound on the discretization bias in order to sample from the stationary distribution of the continuous Langevin Diffusion. In this way, we disentangle the study of the mixing and bias of the Langevin Algorithm.Our key insight is to introduce a technique from the differential privacy literature to the sampling literature. This technique, called Privacy Amplification by Iteration, uses as a potential a variant of Rényi divergence that is made geometrically aware via Optimal Transport smoothing. This gives a short, simple proof of optimal mixing bounds and has several additional appealing properties. First, our approach removes all unnecessary assumptions required by other sampling analyses. Second, our approach unifies many settings: it extends unchanged if the Langevin Algorithm uses projections, stochastic mini-batch gradients, or strongly convex potentials (whereby our mixing time improves exponentially). Third, our approach exploits convexity only through the contractivity of a gradient step-reminiscent of how convexity is used in textbook proofs of Gradient Descent. In this way, we offer a new approach towards further unifying the analyses of optimization and sampling algorithms.

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“…The behaviour of the mixing time of the Metropolis-Adjusted Langevin Algorithm (MALA) and the Unadjusted Langevin Algorithm (ULA) has been extensively studied in the last few years, with particular attention being given to its dimensional dependence. See Durmus and Moulines (2019); Li et al (2021); Chewi et al (2021); Wu et al (2021) for recent theoretical contributions to this area. In contrast to theoretical bounds, which may only be informative in the very high-dimensional regime, our empirical bounds are informative in the low-to-moderate dimensional setting as well.…”

Section: Dimensional Scaling Of Ula and Malamentioning

confidence: 99%

Bounds on Wasserstein distances between continuous distributions using independent samples

Papp¹,

Sherlock²

2022

Preprint

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The plug-in estimator of the Wasserstein distance is known to be conservative, however its usefulness is severely limited when the distributions are similar as its bias does not decay to zero with the true Wasserstein distance. We propose a linear combination of plug-in estimators for the squared 2-Wasserstein distance with a reduced bias that decays to zero with the true distance. The new estimator is provably conservative provided one distribution is appropriately overdispersed with respect the other, and is unbiased when the distributions are equal. We apply it to approximately bound from above the 2-Wasserstein distance between the target and current distribution in Markov chain Monte Carlo, running multiple identically distributed chains which start, and remain, overdispersed with respect to the target. Our bound consistently outperforms the current state-of-the-art bound, which uses coupling, improving mixing time bounds by up to an order of magnitude.

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Minimax Mixing Time of the Metropolis-Adjusted Langevin Algorithm for Log-Concave Sampling

Cited by 6 publications

References 44 publications

Fisher information lower bounds for sampling

Fisher information lower bounds for sampling

Resolving the Mixing Time of the Langevin Algorithm to its Stationary Distribution for Log-Concave Sampling

Bounds on Wasserstein distances between continuous distributions using independent samples

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