Total variation discrepancy of deterministic random walks for ergodic Markov chains

Takeharu Shiraga, Yukiko Yamauchi, Shuji Kijima, Masafumi Yamashita

Research output: Chapter in Book/Report/Conference proceedingConference contribution

2 Citations (Scopus)

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 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.

Original languageEnglish
Title of host publication13th Workshop on Analytic Algorithmics and Combinatorics 2016, ANALCO 2016
EditorsMark Daniel Ward, James Allen Fill
PublisherSociety for Industrial and Applied Mathematics Publications
Pages138-148
Number of pages11
ISBN (Electronic)9781510819696
Publication statusPublished - Jan 1 2016
Event13th Workshop on Analytic Algorithmics and Combinatorics 2016, ANALCO 2016 - Arlington, United States
Duration: Jan 11 2016 → …

Publication series

Name13th Workshop on Analytic Algorithmics and Combinatorics 2016, ANALCO 2016

Other

Other13th Workshop on Analytic Algorithmics and Combinatorics 2016, ANALCO 2016
CountryUnited States
CityArlington
Period1/11/16 → …

Fingerprint

Total Variation
Markov processes
Discrepancy
Random walk
Markov chain
Upper bound
Markov Chain Monte Carlo
Derandomization
Mixing Time
Diagram
Lower bound
Vertex of a graph

All Science Journal Classification (ASJC) codes

  • Applied Mathematics
  • Materials Chemistry
  • Discrete Mathematics and Combinatorics

Cite this

Shiraga, T., Yamauchi, Y., Kijima, S., & Yamashita, M. (2016). Total variation discrepancy of deterministic random walks for ergodic Markov chains. In M. D. Ward, & J. A. Fill (Eds.), 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.

13th Workshop on Analytic Algorithmics and Combinatorics 2016, ANALCO 2016. ed. / Mark Daniel Ward; James Allen Fill. Society for Industrial and Applied Mathematics Publications, 2016. p. 138-148 (13th Workshop on Analytic Algorithmics and Combinatorics 2016, ANALCO 2016).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Shiraga, T, Yamauchi, Y, Kijima, S & Yamashita, M 2016, Total variation discrepancy of deterministic random walks for ergodic Markov chains. in MD Ward & JA Fill (eds), 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.
Shiraga T, Yamauchi Y, Kijima S, Yamashita M. Total variation discrepancy of deterministic random walks for ergodic Markov chains. In Ward MD, Fill JA, editors, 13th Workshop on Analytic Algorithmics and Combinatorics 2016, ANALCO 2016. Society for Industrial and Applied Mathematics Publications. 2016. p. 138-148. (13th Workshop on Analytic Algorithmics and Combinatorics 2016, ANALCO 2016).
Shiraga, Takeharu ; Yamauchi, Yukiko ; Kijima, Shuji ; Yamashita, Masafumi. / Total variation discrepancy of deterministic random walks for ergodic Markov chains. 13th Workshop on Analytic Algorithmics and Combinatorics 2016, ANALCO 2016. editor / Mark Daniel Ward ; James Allen Fill. Society for Industrial and Applied Mathematics Publications, 2016. pp. 138-148 (13th Workshop on Analytic Algorithmics and Combinatorics 2016, ANALCO 2016).
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