Distances for Markov chains from sample streams

TL;DR

This paper introduces a stochastic optimization approach to estimate bisimulation metrics for Markov chains using only sample trajectories, removing the need for explicit transition models and enabling practical applications.

Contribution

It presents a new LP formulation and a stochastic primal-dual algorithm for estimating bisimulation metrics from samples, with theoretical guarantees and empirical validation.

Findings

The method accurately estimates bisimulation metrics from sample data.

The approach has provable sample complexity bounds.

Empirical results demonstrate effectiveness on real-world data.

Abstract

Bisimulation metrics are powerful tools for measuring similarities between stochastic processes, and specifically Markov chains. Recent advances have uncovered that bisimulation metrics are, in fact, optimal-transport distances, which has enabled the development of fast algorithms for computing such metrics with provable accuracy and runtime guarantees. However, these recent methods, as well as all previously known methods, assume full knowledge of the transition dynamics. This is often an impractical assumption in most real-world scenarios, where typically only sample trajectories are available. In this work, we propose a stochastic optimization method that addresses this limitation and estimates bisimulation metrics based on sample access, without requiring explicit transition models. Our approach is derived from a new linear programming (LP) formulation of bisimulation metrics, which we solve using a stochastic primal-dual optimization method. We provide theoretical guarantees on the sample complexity of the algorithm and validate its effectiveness through a series of empirical evaluations.