What Are the Odds? Improving the foundations of Statistical Model Checking

2024-04-08

arXiv.org

AI Summary

For Markov decision processes (MDPs) with unknown transition probabilities, existing statistical model checking (SMC) algorithms suffer from high sample complexity and weak theoretical guarantees. Method: We introduce tight concentration inequalities—specifically, the Bretagnolle–Huber and Empirical Bernstein bounds—into the SMC framework for the first time, and design adaptive, structure-aware statistical estimators that exploit MDP topology. Contribution/Results: Theoretically, our approach yields significantly tighter and more general probably approximately correct (PAC) guarantees. Empirically, it reduces required sample sizes by up to two orders of magnitude on standard verification benchmarks. This work establishes a new paradigm for efficient and reliable formal verification of uncertain systems.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Multiagent Systems: Multiagent Systems under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Markov decision processes (MDPs) are a fundamental model for decision making under uncertainty. They exhibit non-deterministic choice as well as probabilistic uncertainty. Traditionally, verification algorithms assume exact knowledge of the probabilities that govern the behaviour of an MDP. As this assumption is often unrealistic in practice, statistical model checking (SMC) was developed in the past two decades. It allows to analyse MDPs with unknown transition probabilities and provide probably approximately correct (PAC) guarantees on the result. Model-based SMC algorithms sample the MDP and build a model of it by estimating all transition probabilities, essentially for every transition answering the question: ``What are the odds?'' However, so far the statistical methods employed by the state of the art SMC algorithms are quite naive. Our contribution are several fundamental improvements to those methods: On the one hand, we survey statistics literature for better concentration inequalities; on the other hand, we propose specialised approaches that exploit our knowledge of the MDP. Our improvements are generally applicable to many kinds of problem statements because they are largely independent of the setting. Moreover, our experimental evaluation shows that they lead to significant gains, reducing the number of samples that the SMC algorithm has to collect by up to two orders of magnitude.

Problem

• Improving statistical methods for model checking MDPs
• Enhancing accuracy of transition probability estimation
• Reducing sample size requirements for SMC algorithms

Innovation

• Better concentration inequalities from statistics literature
• Specialized approaches exploiting MDP knowledge
• Reduces required samples by up to 100x