Sub-Quadratic Bisimulation Metrics via Approximate Nearest Neighbors: Coverage-Augmented Guarantees and Computable Two-Sided Certificates

2026-08-06

AI Summary

This work addresses the quadratic complexity of traditional bisimulation metrics, which require updating all state pairs and thus hinder scalability to large Markov decision processes. The paper introduces the first sub-quadratic algorithm that leverages approximate nearest neighbor indexing to select critical state pairs for exact updates, while employing monotonic upper and lower bounds to iteratively sandwich the true metric. Innovatively, it incorporates a coverage-enhanced anytime error bound and a computable bilateral certificate mechanism, enabling, for the first time, globally controllable error guarantees and verifiable exact clustering recovery. Empirical results demonstrate that on a 64-state benchmark, the method achieves metric tightness using only about 50% of state pairs; in a 2,500-state grid world, it improves metric quality by 28.6% with merely 12.8% computational overhead, substantially outperforming MICo and DBC baselines.

Abstract

Bisimulation metrics quantify behavioral similarity in Markov decision processes, but their Wasserstein fixed-point operator updates every state pair and incurs quadratic pairwise work. We give a certificate-carrying sub-quadratic method for MDPs with bounded transition support and a useful low-dimensional indexing representation: an approximate-nearest-neighbor index selects the pairs updated by the exact restricted operator, while monotone lower and upper runs enclose the exact metric at every sweep. The main analytical result is a coverage-augmented anytime bound: local index quality alone cannot control global error, because uncovered pairs retain their initialization gap. The limiting error is at most $\max(ρ,\eop/(1-γ))$, and with exact covered backups the lower arm satisfies $\|\dann-d\|_\infty=ρ$. Because $ρ$ depends on the unknown exact metric, the algorithm returns the observable sandwich width instead; agreement of the induced lower and upper clusterings certifies exact recovery of the covered aggregation. A reward-oblivious lower bound shows sub-quadratic index-first coverage cannot remove the coverage term, while a separate adaptive lower bound requires $Ω(|\Scal|)$ pair evaluations. Exact-operator experiments verify the identity and enclosure in every seeded run, and timing experiments recover quadratic versus sub-quadratic scaling under both cheap and full Wasserstein backups. On the grouped $|\Scal|=64$ benchmark, exact restricted refinement reaches the exact-metric skyline once retrieval covers roughly half of all pairs, while independently trained MICo and DBC baselines stay $22$-$33\times$ above that skyline at every retrieval budget. Taxi shows the certificate abstaining under an uninformative embedding, while a $2500$-state gridworld improves over a reward-only metric by $28.6\%$ using $12.8\%$ of one quadratic sweep.

Problem

• bisimulation metrics
• Markov decision processes
• sub-quadratic computation
• Wasserstein distance
• state similarity

Innovation

• bisimulation metrics
• approximate nearest neighbors
• sub-quadratic algorithm
• two-sided certificates
• coverage-augmented guarantees