Value Iteration with Guessing for Markov Chains and Markov Decision Processes (2505.06769v1)

Abstract: Two standard models for probabilistic systems are Markov chains (MCs) and Markov decision processes (MDPs). Classic objectives for such probabilistic models for control and planning problems are reachability and stochastic shortest path. The widely studied algorithmic approach for these problems is the Value Iteration (VI) algorithm which iteratively applies local updates called BeLLMan updates. There are many practical approaches for VI in the literature but they all require exponentially many BeLLMan updates for MCs in the worst case. A preprocessing step is an algorithm that is discrete, graph-theoretical, and requires linear space. An important open question is whether, after a polynomial-time preprocessing, VI can be achieved with sub-exponentially many BeLLMan updates. In this work, we present a new approach for VI based on guessing values. Our theoretical contributions are twofold. First, for MCs, we present an almost-linear-time preprocessing algorithm after which, along with guessing values, VI requires only subexponentially many BeLLMan updates. Second, we present an improved analysis of the speed of convergence of VI for MDPs. Finally, we present a practical algorithm for MDPs based on our new approach. Experimental results show that our approach provides a considerable improvement over existing VI-based approaches on several benchmark examples from the literature.

Analysis of "Value Iteration with Guessing for Markov Chains and Markov Decision Processes"

The paper, "Value Iteration with Guessing for Markov Chains and Markov Decision Processes," presents novel algorithmic advancements in solving control and planning problems using Value Iteration (VI). The principal focus is on Markov chains (MCs) and Markov decision processes (MDPs), which are foundational models for probabilistic systems. Two objectives are central to the paper: reachability objectives and stochastic shortest path (SSP) objectives.

Problem Scope and Motivation

Value Iteration is a classical approach featuring prominently in probabilistic system analysis due to its simplicity and space efficiency. Yet, a significant limitation remains: in worst-case scenarios, it requires exponentially many BeLLMan updates relative to the number of states. This paper addresses an open challenge in the field: whether, following polynomial-time preprocessing, VI can be executed with sub-exponential iterations in terms of BeLLMan updates.

Theoretical Contributions

Sub-exponential VI for MCs: The paper introduces a preprocessing algorithm for Markov chains that operates in almost-linear time and requires a reduced number of BeLLMan updates—specifically, sub-exponentially many updates compared to traditional methods. This preprocessing achieves an almost-linear complexity because it effectively organizes states into levels facilitating efficient guessing in value iteration. Crucially, the number of levels is reduced by selectively marking states to be guessed, hence allowing the significant reduction in update requirements.

VI Convergence for MDPs: The paper advances an analysis of the convergence speed of VI for Markov decision processes. The authors present an improved understanding of how the levels derived from optimal strategies can enhance VI execution, highlighting potential faster convergence rates than previously acknowledged in literature.

Practical Implementation

The authors' theoretical advancements have been translated into practical algorithms, with Guessing VI as the centerpiece. Implementations are benchmarked using a comprehensive probabilistic model checker tool, STORM. Experimental results from the Quantitative Verification Benchmark Set demonstrate consistent improvements over existing VI approaches across several examples.

Experimental Results

The experiments cover 474 distinct cases where Guessing VI showed an average performance improvement, indicating its practical viability. Notably, in 86 instances (highlighted as Group 4), Guessing VI offered significant speed advantages, underscoring the algorithm's efficiency in real-world computational settings.

Implications and Future Work

The implications of these advancements are twofold: practically, they offer immediate improvements in probabilistic verification tools by reducing computational burdens; theoretically, they suggest new avenues for enhancing VI-based approaches across even broader classes of models and objectives. The transitioning of the proof methods for MCs to MDPs present compelling questions for future research, especially in achieving polynomial preprocessing while maintaining sub-exponential update efficiencies in MDPs.

Conclusion

Overall, the paper delivers a substantive contribution to the domain by overcoming significant limitations of traditional VI, influencing both algorithmic efficiency and practical application in probabilistic modeling frameworks. Moving forward, extending the scope of Guessing VI to more complex models, such as multi-agent stochastic games, remains an intriguing opportunity, promising further computational benefits within the field of AI and probabilistic system verification.