Fisher information lower bounds for sampling

Chewi, Sinho; Gerber, Patrik; Lee, Holden; Chen, Lu

doi:10.48550/arxiv.2210.02482

Cited by 2 publications

(4 citation statements)

References 16 publications

(24 reference statements)

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“…5 At the time of writing, there was no setting in which current mixing bounds for the (Unadjusted) Langevin Algorithm were known to be optimal. The concurrent work [23] also shows an optimality result. Their result concerns an entirely different setting (non-convex potentials where error is measured in Fisher information) and applies to a variant of the Langevin Algorithm (that outputs the averaged iterate) in the specific regime of very large mixing error (namely ε ≈ √ d, where d is the dimension).…”

Section: Related Workmentioning

confidence: 74%

“…An important but difficult question for bridging theory and practice is to understand to what extent fast mixing carries over to families of target distributions π under weaker assumptions than log-concavity (i.e., weaker assumptions on potentials than convexity). There have been several recent breakthrough results towards this direction under various assumptions like dissipativity or isoperimetry or under weaker notions of convergence (see the previous work section §1.3 and the very recent papers [6,23]). However, on one hand tight mixing bounds are unknown for any of these settings (see Footnote 5), and on the other hand the non-convexity assumptions made in the sampling literature are often markedly different from those in the optimization literature.…”

Section: Discussionmentioning

confidence: 99%

“…We refer the reader to the recent paper [24] (which settles this question in dimension 1 for unconstrained, smooth, strongly log-concave sampling) for a discussion of this gap (in dimensions greater than 1) and the technical challenges for overcoming it. Because of these challenges, existing sampling lower bounds are primarily algorithm-specific, e.g., for the Underdamped Langevin Algorithm [15], or the Unadjusted Langevin Algorithm [23], or the Metropolis Adjusted Langevin Algorithm [22,56,91].…”

Section: Related Workmentioning

confidence: 99%

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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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Section: Related Workmentioning

confidence: 74%

Section: Discussionmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

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

Altschuler¹,

Talwar²

2022

Preprint

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“…Balasubramanian et al (2022) show that even for non-log-concave distributions, averaged Langevin Monte Carlo converges quickly to a distribution with low relative Fisher information to the target distribution, although this does not imply that the distribution is close to the target distribution with respect to other measures such as the total variation distance. Chewi et al (2022a) prove corresponding lower bounds. Woodard et al (2009) show that the mixing time of parallel and simulated tempering for certain distributions can scale exponentially with d, but in a setting different from ours.…”

Section: Related Workmentioning

confidence: 93%

Convergence Rates for Non-Log-Concave Sampling and Log-Partition Estimation

Holzmüller¹,

Bach²

2023

Preprint

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Sampling from Gibbs distributions p(x) ∝ exp(−V (x)/ε) and computing their log-partition function are fundamental tasks in statistics, machine learning, and statistical physics. However, while efficient algorithms are known for convex potentials V , the situation is much more difficult in the non-convex case, where algorithms necessarily suffer from the curse of dimensionality in the worst case. For optimization, which can be seen as a low-temperature limit of sampling, it is known that smooth functions V allow faster convergence rates. Specifically, for m-times differentiable functions in d dimensions, the optimal rate for algorithms with n function evaluations is known to be O(n −m/d ), where the constant can potentially depend on m, d and the function to be optimized. Hence, the curse of dimensionality can be alleviated for smooth functions at least in terms of the convergence rate. Recently, it has been shown that similarly fast rates can also be achieved with polynomial runtime O(n 3.5 ), where the exponent 3.5 is independent of m or d. Hence, it is natural to ask whether similar rates for sampling and log-partition computation are possible, and whether they can be realized in polynomial time with an exponent independent of m and d. We show that the optimal rates for sampling and log-partition computation are sometimes equal and sometimes faster than for optimization. We then analyze various polynomial-time sampling algorithms, including an extension of a recent promising optimization approach, and find that they sometimes exhibit interesting behavior but no near-optimal rates. Our results also give further insights on the relation between sampling, log-partition, and optimization problems. * Work done partially while visiting INRIA. E-mail: david (dot) holzmueller (at) mathematik.uni-stuttgart.de 2 We choose X as the unit cube for convenience: It is compact, has unit volume, does not have too sharp corners, there are well-studied approximation results, and it allows to investigate algorithms for periodic functions. However, many of our results could be generalized to other domains.

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Fisher information lower bounds for sampling

Cited by 2 publications

References 16 publications

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

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

Convergence Rates for Non-Log-Concave Sampling and Log-Partition Estimation

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