Sampling algorithms and Markov chains Lszl Lovsz Microsoft

Sampling algorithms and Markov chains László Lovász Microsoft Research One Microsoft Way, Redmond, WA 98052 lovasz@microsoft. com

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

L: a language in NP, with presentation certificate polynomial time algorithm Given: x Find: - a 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, …) Want: sample from distribution p on set V Construct ergodic Markov chain with states: V stationary distribution: p Simulate (run) the chain for T steps Output the final state ? ? ? mixing time

5 4 2 3 3 2 1 1 Given: poset State: compatible linear order Transition: - pick randomly label i<n; - interchange i and i+1 if possible

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

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

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

Key lemma: Proof for l=k+1

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

Lifting Markov chains Diaconis – Holmes – Neal