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DOI: 10.4230/LIPIcs.STACS.2012.124

Chernoff-Hoeffding Bounds for Markov Chains: Generalized and Simplified

Authors: Kai-Min Chung, Henry Lam, Zhenming Liu, and Michael Mitzenmacher

Published in: LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

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.

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Kai-Min Chung, Henry Lam, Zhenming Liu, and Michael Mitzenmacher. Chernoff-Hoeffding Bounds for Markov Chains: Generalized and Simplified. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 124-135, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{chung_et_al:LIPIcs.STACS.2012.124,
  author =	{Chung, Kai-Min and Lam, Henry and Liu, Zhenming and Mitzenmacher, Michael},
  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 =	{D\"{u}rr, Christoph and Wilke, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.124},
  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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DOI: 10.4230/LIPIcs.STACS.2010.2449

AMS Without 4-Wise Independence on Product Domains

Authors: Vladimir Braverman, Kai-Min Chung, Zhenming Liu, Michael Mitzenmacher, and Rafail Ostrovsky

Published in: LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

Abstract

In their seminal work, Alon, Matias, and Szegedy introduced several sketching techniques, including showing that $4$-wise independence is sufficient to obtain good approximations of the second frequency moment. In this work, we show that their sketching technique can be extended to product domains $[n]^k$ by using the product of $4$-wise independent functions on $[n]$. Our work extends that of Indyk and McGregor, who showed the result for $k = 2$. Their primary motivation was the problem of identifying correlations in data streams. In their model, a stream of pairs $(i,j) \in [n]^2$ arrive, giving a joint distribution $(X,Y)$, and they find approximation algorithms for how close the joint distribution is to the product of the marginal distributions under various metrics, which naturally corresponds to how close $X$ and $Y$ are to being independent. By using our technique, we obtain a new result for the problem of approximating the $\ell_2$ distance between the joint distribution and the product of the marginal distributions for $k$-ary vectors, instead of just pairs, in a single pass. Our analysis gives a randomized algorithm that is a $(1\pm \epsilon)$ approximation (with probability $1-\delta$) that requires space logarithmic in $n$ and $m$ and proportional to $3^k$.

Cite as

Vladimir Braverman, Kai-Min Chung, Zhenming Liu, Michael Mitzenmacher, and Rafail Ostrovsky. AMS Without 4-Wise Independence on Product Domains. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 119-130, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{braverman_et_al:LIPIcs.STACS.2010.2449,
  author =	{Braverman, Vladimir and Chung, Kai-Min and Liu, Zhenming and Mitzenmacher, Michael and Ostrovsky, Rafail},
  title =	{{AMS Without 4-Wise Independence on Product Domains}},
  booktitle =	{27th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{119--130},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-16-3},
  ISSN =	{1868-8969},
  year =	{2010},
  volume =	{5},
  editor =	{Marion, Jean-Yves and Schwentick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2449},
  URN =		{urn:nbn:de:0030-drops-24496},
  doi =		{10.4230/LIPIcs.STACS.2010.2449},
  annote =	{Keywords: Data Streams, Randomized Algorithms, Streaming Algorithms, Independence, Sketches}
}

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Chernoff-Hoeffding Bounds for Markov Chains: Generalized and Simplified

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AMS Without 4-Wise Independence on Product Domains

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