Total Variation Discrepancy of Deterministic Random Walks for Ergodic Markov Chains

TL;DR

This paper analyzes the total variation discrepancy of deterministic random walks in ergodic Markov chains, providing upper bounds that relate discrepancy to chain properties and improving understanding of derandomized MCMC methods.

Contribution

It introduces new upper bounds on the total variation discrepancy for deterministic random walks in ergodic Markov chains, advancing derandomization techniques in MCMC.

Findings

Upper bound of O(m t*) for total variation discrepancy.

Improved upper bound of O(m√(t* log t*)) for lazy, non-oblivious walks.

Presented lower bounds for discrepancy measures.

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 several progresses on the analysis of the vertex-wise discrepancy (i.e., $L_\infty$ discrepancy), little is known about the {\em total variation discrepancy} (i.e., $L_1$ 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 ${\rm O}(mt^*)$ 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 ${\rm O}(m\sqrt{t^*\log t^*})$ for non-oblivious deterministic random walks, if the corresponding Markov chain is ergodic and lazy. We also present some lower bounds.