TY - JOUR

T1 - Total variation discrepancy of deterministic random walks for ergodic Markov chains

AU - Shiraga, Takeharu

AU - Yamauchi, Yukiko

AU - Kijima, Shuji

AU - Yamashita, Masafumi

N1 - Funding Information:
The authors would like to thank the anonymous reviewers for their valuable comments, especially for the proof of Proposition 3.4 . This work is supported by JSPS KAKENHI Grant Numbers 15J03840 , 15K15938 , 25700002 , 15H02666 .

PY - 2017/11/7

Y1 - 2017/11/7

N2 - Motivated by a derandomization of Markov chain Monte Carlo (MCMC), this paper investigates a deterministic random walk, which is a deterministic process analogous to a random walk. There is some recent progress in the analysis of the vertex-wise discrepancy (i.e., L∞-discrepancy), while little is known about the total variation discrepancy (i.e., L1-discrepancy), which plays an important role in the analysis of an FPRAS based on MCMC. This paper investigates the L1-discrepancy between the expected number of tokens in a Markov chain and the number of tokens in its corresponding deterministic random walk. First, we give a simple but nontrivial upper bound O(mt⁎) of the L1-discrepancy for any ergodic Markov chains, where m is the number of edges of the transition diagram and t⁎ is the mixing time of the Markov chain. Then, we give a better upper bound O(mt⁎) for non-oblivious deterministic random walks, if the corresponding Markov chain is ergodic and lazy. We also present some lower bounds.

AB - Motivated by a derandomization of Markov chain Monte Carlo (MCMC), this paper investigates a deterministic random walk, which is a deterministic process analogous to a random walk. There is some recent progress in the analysis of the vertex-wise discrepancy (i.e., L∞-discrepancy), while little is known about the total variation discrepancy (i.e., L1-discrepancy), which plays an important role in the analysis of an FPRAS based on MCMC. This paper investigates the L1-discrepancy between the expected number of tokens in a Markov chain and the number of tokens in its corresponding deterministic random walk. First, we give a simple but nontrivial upper bound O(mt⁎) of the L1-discrepancy for any ergodic Markov chains, where m is the number of edges of the transition diagram and t⁎ is the mixing time of the Markov chain. Then, we give a better upper bound O(mt⁎) for non-oblivious deterministic random walks, if the corresponding Markov chain is ergodic and lazy. We also present some lower bounds.

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U2 - 10.1016/j.tcs.2016.11.017

DO - 10.1016/j.tcs.2016.11.017

M3 - Article

AN - SCOPUS:85015672779

VL - 699

SP - 63

EP - 74

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

ER -