### Abstract

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., L_{1}-discrepancy), which plays an important role in the analysis of an FPRAS based on MCMC. This paper investigates the L_{1}-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 L_{1}-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.

Original language | English |
---|---|

Pages (from-to) | 63-74 |

Number of pages | 12 |

Journal | Theoretical Computer Science |

Volume | 699 |

DOIs | |

Publication status | Published - Nov 7 2017 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Theoretical Computer Science*,

*699*, 63-74. https://doi.org/10.1016/j.tcs.2016.11.017

**Total variation discrepancy of deterministic random walks for ergodic Markov chains.** / Shiraga, Takeharu; Yamauchi, Yukiko; Kijima, Shuji; Yamashita, Masafumi.

Research output: Contribution to journal › Article

*Theoretical Computer Science*, vol. 699, pp. 63-74. https://doi.org/10.1016/j.tcs.2016.11.017

}

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

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.

UR - http://www.scopus.com/inward/record.url?scp=85015672779&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85015672779&partnerID=8YFLogxK

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 -