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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2012.124
URN: urn:nbn:de:0030-drops-34374
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2012/3437/
Chung, Kai-Min ;
Lam, Henry ;
Liu, Zhenming ;
Mitzenmacher, Michael
Chernoff-Hoeffding Bounds for Markov Chains: Generalized and Simplified
Abstract
We prove the first Chernoff-Hoeffding bounds for general nonreversible finite-state Markov chains based on the standard L_1 (variation distance) mixing-time of the chain. Specifically, consider an ergodic Markov chain M and a weight function f: [n] -> [0,1] on the state space [n] of M with mean mu = E_{v <- pi}[f(v)], where pi is the stationary distribution of M. A t-step random walk (v_1,...,v_t) on M starting from the stationary distribution pi has expected total weight E[X] = mu t, where X = sum_{i=1}^t f(v_i). Let T be the L_1 mixing-time of M. We show that the probability of X deviating from its mean by a multiplicative factor of delta, i.e., Pr [ |X - mu t| >= delta mu t ], is at most exp(-Omega( delta^2 mu t / T )) for 0 <= delta <= 1, and exp(-Omega( delta mu t / T )) for delta > 1. In fact, the bounds hold even if the weight functions f_i's for i in [t] are distinct, provided that all of them have the same mean mu.
We also obtain a simplified proof for the Chernoff-Hoeffding bounds based on the spectral expansion lambda of M, which is the square root of the second largest eigenvalue (in absolute value) of M tilde{M}, where tilde{M} is the time-reversal Markov chain of M. We show that the probability Pr [ |X - mu t| >= delta mu t ] is at most exp(-Omega( delta^2 (1-lambda) mu t )) for 0 <= delta <= 1, and exp(-Omega( delta (1-lambda) mu t )) for delta > 1.
Both of our results extend to continuous-time Markov chains, and to the case where the walk starts from an arbitrary distribution x, at a price of a multiplicative factor depending on the distribution x in the concentration bounds.
BibTeX - Entry
@InProceedings{chung_et_al:LIPIcs:2012:3437,
author = {Kai-Min Chung and Henry Lam and Zhenming Liu and Michael Mitzenmacher},
title = {{Chernoff-Hoeffding Bounds for Markov Chains: Generalized and Simplified}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {124--135},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-35-4},
ISSN = {1868-8969},
year = {2012},
volume = {14},
editor = {Christoph D{\"u}rr and Thomas Wilke},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2012/3437},
URN = {urn:nbn:de:0030-drops-34374},
doi = {10.4230/LIPIcs.STACS.2012.124},
annote = {Keywords: probabilistic analysis, tail bounds, Markov chains}
}
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Keywords: |
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probabilistic analysis, tail bounds, Markov chains |
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Collection: |
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29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012) |
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Issue Date: |
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2012 |
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Date of publication: |
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24.02.2012 |
