{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article7063","name":"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.\r\n\r\nWe 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.\r\n\r\nBoth 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.","keywords":["probabilistic analysis","tail bounds","Markov chains"],"author":[{"@type":"Person","name":"Chung, Kai-Min","givenName":"Kai-Min","familyName":"Chung"},{"@type":"Person","name":"Lam, Henry","givenName":"Henry","familyName":"Lam"},{"@type":"Person","name":"Liu, Zhenming","givenName":"Zhenming","familyName":"Liu"},{"@type":"Person","name":"Mitzenmacher, Michael","givenName":"Michael","familyName":"Mitzenmacher"}],"position":13,"pageStart":124,"pageEnd":135,"dateCreated":"2012-02-24","datePublished":"2012-02-24","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Chung, Kai-Min","givenName":"Kai-Min","familyName":"Chung"},{"@type":"Person","name":"Lam, Henry","givenName":"Henry","familyName":"Lam"},{"@type":"Person","name":"Liu, Zhenming","givenName":"Zhenming","familyName":"Liu"},{"@type":"Person","name":"Mitzenmacher, Michael","givenName":"Michael","familyName":"Mitzenmacher"}],"copyrightYear":"2012","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.STACS.2012.124","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6217","volumeNumber":14,"name":"29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)","dateCreated":"2012-02-24","datePublished":"2012-02-24","editor":[{"@type":"Person","name":"D\u00fcrr, Christoph","givenName":"Christoph","familyName":"D\u00fcrr"},{"@type":"Person","name":"Wilke, Thomas","givenName":"Thomas","familyName":"Wilke"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article7063","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6217"}}}