Distances for Markov chains from sample streams

Sergio Calo, Anders Jonsson, Gergely Neu, Ludovic Schwartz, Javier Segovia-Aguas

Keywords: Optimal Transport, Markov Chains, bisimulation, primal dual, stochastic methods

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.

Primary Area: Theory (e.g., control theory, learning theory, algorithmic game theory)

Submission Number: 20782