Solving Long-run Average Reward Robust Markov Decision Processes via Reduction to Turn-based Stochastic Games

Core Concepts

The core message of this work is that the problem of solving long-run average reward polytopic robust Markov decision processes (RMDPs) can be reduced in linear time to the problem of solving long-run average reward turn-based stochastic games (TBSGs). This reduction enables the transfer of results from the TBSG literature to the RMDP setting, leading to novel insights on the computational complexity and efficient algorithms for solving long-run average reward polytopic RMDPs.

Abstract

The authors consider the problem of solving long-run average reward robust Markov decision processes (RMDPs) with polytopic uncertainty sets. They present a novel perspective on this problem by showing that it can be reduced in linear time to the problem of solving long-run average reward turn-based stochastic games (TBSGs). The key highlights and insights are: Reduction to TBSGs: The authors formally define a linear-time reduction from polytopic RMDPs to TBSGs. This reduction allows them to leverage results from the TBSG literature to obtain new insights on RMDPs. Computational Complexity: Using the reduction, the authors show that the threshold decision problem for long-run average reward polytopic RMDPs is in NP ∩ CONP. They also show that these RMDPs admit a randomized algorithm with sub-exponential expected runtime. Efficient Algorithms: The authors propose Robust Polytopic Policy Iteration (RPPI), a novel policy iteration algorithm for solving long-run average reward polytopic RMDPs. RPPI does not impose any structural restrictions on the RMDP, unlike prior value iteration-based algorithms. Experimental Evaluation: The authors implement RPPI and experimentally compare it against state-of-the-art value iteration-based methods. The results demonstrate significant computational runtime gains provided by the policy iteration-based RPPI, especially on non-unichain polytopic RMDPs to which existing methods are not applicable.

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Key Insights Distilled From

Solving Long-run Average Reward Robust MDPs via Stochastic Games

by Kris... at arxiv.org 05-01-2024

https://arxiv.org/pdf/2312.13912.pdf

Deeper Inquiries

What are the potential limitations or challenges in extending the reduction from polytopic RMDPs to RMDPs with non-polytopic uncertainty sets

Extending the reduction from polytopic RMDPs to RMDPs with non-polytopic uncertainty sets may face several limitations and challenges. One major challenge is the complexity of representing and manipulating non-polytopic uncertainty sets. Polytopic uncertainty sets have a simple geometric structure that allows for efficient computation and optimization. Non-polytopic uncertainty sets, on the other hand, can be more complex and may not have a straightforward representation in terms of vertices or convex hulls. This complexity can make it challenging to apply the same reduction techniques used for polytopic RMDPs. Another limitation is the computational complexity of dealing with non-polytopic uncertainty sets. Algorithms designed for polytopic uncertainty sets may not be directly applicable to non-polytopic sets, as the computational complexity of solving RMDPs can increase significantly with the complexity of the uncertainty sets. This could lead to challenges in developing efficient algorithms and may require novel approaches to handle the increased complexity. Furthermore, the theoretical foundations and results established for polytopic RMDPs may not directly translate to RMDPs with non-polytopic uncertainty sets. The properties and characteristics of non-polytopic uncertainty sets may introduce new challenges and considerations that need to be addressed in the algorithm design and analysis.

How could the insights from the reduction to TBSGs be leveraged to design efficient algorithms for solving discounted-sum reward polytopic RMDPs

The insights from the reduction to TBSGs can be leveraged to design efficient algorithms for solving discounted-sum reward polytopic RMDPs by utilizing the principles of policy iteration and Blackwell optimality. Policy iteration has been shown to be sound and efficient for solving TBSGs, and the reduction to TBSGs allows for the application of policy iteration techniques to RMDPs. One approach could be to adapt the policy iteration algorithm used for long-run average TBSGs to handle discounted-sum objectives. By incorporating the principles of discounted-sum reward optimization and leveraging the reduction to TBSGs, it is possible to design an algorithm that iteratively refines policies to converge to optimal solutions for discounted-sum reward polytopic RMDPs. Additionally, the reduction to TBSGs provides a framework for analyzing the relationship between discounted-sum reward RMDPs and TBSGs, allowing for the transfer of algorithmic techniques and insights between the two domains. By leveraging the connection established through the reduction, researchers can develop efficient algorithms that exploit the similarities and differences between the two problem domains to improve computational efficiency and solution quality.

Given the connection between RMDPs and TBSGs, are there any interesting applications or connections to other areas of computer science that could be explored

The connection between RMDPs and TBSGs opens up interesting applications and connections to other areas of computer science. One potential application is in the field of reinforcement learning, where RMDPs are commonly used to model decision-making processes under uncertainty. By leveraging insights from TBSGs, researchers can develop more efficient and robust reinforcement learning algorithms that can handle complex uncertainty structures and optimize long-run average rewards. Furthermore, the connection to TBSGs can be explored in the context of game theory and decision-making under uncertainty. TBSGs provide a formal framework for analyzing strategic interactions in the presence of both stochastic and adversarial elements, making them relevant to various decision-making scenarios in economics, operations research, and multi-agent systems. The insights from the reduction to TBSGs can also be applied to algorithm design in other areas of computer science, such as optimization, machine learning, and algorithmic game theory. By understanding the connections between RMDPs and TBSGs, researchers can develop novel algorithms and techniques that leverage the principles of policy iteration, Blackwell optimality, and stochastic games to solve complex decision-making problems efficiently and effectively.