### Abstract

Motivated by a derandomization of Markov chain Monte Carlo (MCMC), this paper investigates deterministic random walks, which is a deterministic process analogous to a random walk. While there are some progress on the analysis of the vertex-wise discrepancy (i.e., L_{∞} discrepancy), little is known about the total variation discrepancy (i.e., Li discrepancy), which plays a significant role in the analysis of an FPRAS based on MCMC. This paper investigates upper bounds of 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(m√t∗ log t∗) for non-oblivious deterministic random walks, if the corresponding Markov chain is ergodic and lazy. We also present some lower bounds.

Original language | English |
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Title of host publication | 13th Workshop on Analytic Algorithmics and Combinatorics 2016, ANALCO 2016 |

Editors | Mark Daniel Ward, James Allen Fill |

Publisher | Society for Industrial and Applied Mathematics Publications |

Pages | 138-148 |

Number of pages | 11 |

ISBN (Electronic) | 9781510819696 |

Publication status | Published - Jan 1 2016 |

Event | 13th Workshop on Analytic Algorithmics and Combinatorics 2016, ANALCO 2016 - Arlington, United States Duration: Jan 11 2016 → … |

### Publication series

Name | 13th Workshop on Analytic Algorithmics and Combinatorics 2016, ANALCO 2016 |
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### Other

Other | 13th Workshop on Analytic Algorithmics and Combinatorics 2016, ANALCO 2016 |
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Country | United States |

City | Arlington |

Period | 1/11/16 → … |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Applied Mathematics
- Materials Chemistry
- Discrete Mathematics and Combinatorics

### Cite this

*13th Workshop on Analytic Algorithmics and Combinatorics 2016, ANALCO 2016*(pp. 138-148). (13th Workshop on Analytic Algorithmics and Combinatorics 2016, ANALCO 2016). Society for Industrial and Applied Mathematics Publications.

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

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*13th Workshop on Analytic Algorithmics and Combinatorics 2016, ANALCO 2016.*13th Workshop on Analytic Algorithmics and Combinatorics 2016, ANALCO 2016, Society for Industrial and Applied Mathematics Publications, pp. 138-148, 13th Workshop on Analytic Algorithmics and Combinatorics 2016, ANALCO 2016, Arlington, United States, 1/11/16.

}

TY - GEN

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 - 2016/1/1

Y1 - 2016/1/1

N2 - Motivated by a derandomization of Markov chain Monte Carlo (MCMC), this paper investigates deterministic random walks, which is a deterministic process analogous to a random walk. While there are some progress on the analysis of the vertex-wise discrepancy (i.e., L∞ discrepancy), little is known about the total variation discrepancy (i.e., Li discrepancy), which plays a significant role in the analysis of an FPRAS based on MCMC. This paper investigates upper bounds of 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(m√t∗ log t∗) 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 deterministic random walks, which is a deterministic process analogous to a random walk. While there are some progress on the analysis of the vertex-wise discrepancy (i.e., L∞ discrepancy), little is known about the total variation discrepancy (i.e., Li discrepancy), which plays a significant role in the analysis of an FPRAS based on MCMC. This paper investigates upper bounds of 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(m√t∗ log t∗) 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=84965173529&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84965173529&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84965173529

T3 - 13th Workshop on Analytic Algorithmics and Combinatorics 2016, ANALCO 2016

SP - 138

EP - 148

BT - 13th Workshop on Analytic Algorithmics and Combinatorics 2016, ANALCO 2016

A2 - Ward, Mark Daniel

A2 - Fill, James Allen

PB - Society for Industrial and Applied Mathematics Publications

ER -