Sampling algorithms and Markov chains László Lovász Microsoft Research One Microsoft Way, Redmond, WA 98052

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Presentation transcript:

Sampling algorithms and Markov chains László Lovász Microsoft Research One Microsoft Way, Redmond, WA

Sampling: a general algorithmic task Applications: - statistics - simulation - counting - numerical integration - optimization - …

L: a language in NP, with presentation polynomial time algorithm Find: - a certificate Given: x certificate - an optimal certificate - the number of certificates - a random certificate (uniform, or given distribution)

One general method for sampling: Markov chains (+rejection sampling, lifting,…) Construct ergodic Markov chain with states: V stationary distribution: p Want: sample from distribution p on set V Simulate (run) the chain for T steps Output the final state ???????????? mixing time

Given: poset State: compatible linear order Transition: - pick randomly label i<n ; - interchange i and i+1 if possible

Mixing time Bipartite graph?! : distribution after t steps Roughly: (enough to consider )

Conductance in sequence of independent samples: frequency of stepping from S to K\S conductance: in Markov chain:

Jerrum - Sinclair But in finer analysis? In typical sampling application: polynomial

Key lemma: Proof for l=k+1

L – Simonovits Dyer – Frieze Simple isoperimetric inequality: Improved isoperimetric inequality: Kannan-L After appropriate preprocessing,

Lifting Markov chains Diaconis – Holmes – Neal