Total variation discrepancy of deterministic random walks for ergodic Markov chains

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., $L_{\infty }$-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 $\mathrm{O}(m{t}^{⁎})$ 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 $\mathrm{O}(m\sqrt{t^{⁎}})$ for non-oblivious deterministic random walks, if the corresponding Markov chain is ergodic and lazy. We also present some lower bounds.