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DTSTAMP:20251107T054647Z
DESCRIPTION:Title: Resolving the Mixing Time of the Langevin Algorithm to i
 ts Stationary Distribution for Log-Concave Sampling\n\nAbstract:\n\nSampli
 ng from a high-dimensional distribution is a fundamental task in statistic
 s\, engineering\, and the sciences. A canonical approach is the Langevin A
 lgorithm\, i.e.\, the Markov chain for the discretized Langevin Diffusion.
  This is the sampling analog of Gradient Descent. De- spite being studied 
 for several decades in multiple communities\, tight mixing bounds for this
  algorithm remain unresolved even in the seemingly simple setting of log-c
 oncave distributions over a bounded domain. This paper completely characte
 rizes the mixing time of the Langevin Algorithm to its stationary distribu
 tion in this setting (and others). This mixing result can be combined with
  any bound on the discretization bias in order to sample from the stationa
 ry distribution of the continuous Langevin Diffusion. In this way\, we dis
 entangle the study of the mixing and bias of the Langevin Algorithm.\n	Our 
 key insight is to introduce a technique from the differential privacy lite
 rature to the sampling literature. This technique\, called Privacy Amplifi
 cation by Iteration\, uses as a potential a variant of Renyi divergence th
 at is made geometrically aware via Optimal Transport smoothing. This gives
  a short\, simple proof of optimal mixing bounds and has several additiona
 l appealing properties. First\, our approach removes all unnecessary assum
 ptions required by other sampling analyses. Second\, our approach unifies 
 many settings: it extends unchanged if the Langevin Algorithm uses project
 ions\, stochastic mini-batch gradients\, or strongly convex potentials (wh
 ereby our mixing time improves exponentially). Third\, our approach exploi
 ts 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 opt
 imization and sampling algorithms.\n\nReferences:\n\n• https://arxiv.org/a
 bs/2210.08448\n\n \n
DTSTART:20231108T180000Z
DTEND:20231108T190000Z
LOCATION:Room 1214\, Burnside Hall\, CA\, QC\, Montreal\, H3A 0B9\, 805 rue
  Sherbrooke Ouest
SUMMARY:Alexandra Kravchuk (9IÖÆ×÷³§Ãâ·Ñ)
URL:/mathstat/channels/event/alexandra-kravchuk-mcgill
 -university-352487
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