On the Complexity of Solving Markov Decision Problems (1302.4971v1)

Abstract: Markov decision problems (MDPs) provide the foundations for a number of problems of interest to AI researchers studying automated planning and reinforcement learning. In this paper, we summarize results regarding the complexity of solving MDPs and the running time of MDP solution algorithms. We argue that, although MDPs can be solved efficiently in theory, more study is needed to reveal practical algorithms for solving large problems quickly. To encourage future research, we sketch some alternative methods of analysis that rely on the structure of MDPs.

• The paper analyzes the theoretical and practical complexity of MDPs, highlighting the gap between polynomial-time algorithms and exponential iteration counts.
• It evaluates methodologies such as linear programming reduction, policy iteration, and value iteration under the infinite-horizon discounted cost criterion.
• The study advocates for MDP-specific algorithms and approximation techniques to overcome computational limitations in large-scale decision problems.

Complexity of Solving Markov Decision Problems: An Analysis

This paper addresses the computational complexity of solving Markov Decision Problems (MDPs), fundamental to decision-theoretic planning and reinforcement learning within artificial intelligence and operations research. Although solving MDPs theoretically involves polynomial-time algorithms, practical efficiency remains a challenge, particularly for large-scale problems. The paper systematically examines the existing methodologies and computational constraints, proposing areas for future research to enhance practical algorithmic development.

Markov Decision Problems (MDPs)

MDPs are characterized by a set of states, actions, a state-transition probability distribution, and a cost function. Solutions to MDPs are policies mapping states to actions to minimize the associated costs over time. The paper focuses particularly on the infinite-horizon discounted cumulative cost criterion, where a discount factor modulates transitional costs to emphasize nearer-term outcomes.

Known Results and Algorithms

The paper evaluates the complexities of MDP solution methodologies:

1. Linear Programming Reduction: MDPs can be formulated as Linear Programs (LPs), solvable in polynomial time concerning the number of states (N), actions (M), and precision (B). However, this theoretical polynomiality does not imply practical efficiency because the polynomial's degree can be impractically high.
2. Policy Iteration: Introduced by Howard, this iterative algorithm alternates between policy evaluation and improvement. Each iteration solves a linear equation system, providing a performance guarantee of convergence to an optimal policy. However, the number of iterations can be exponential in some instances, with the algorithm's efficacy being contingent on the discount factor.
3. Value Iteration: A successive approximation technique that iteratively refines the value function. Despite its reputation for practical efficiency, in the worst-case scenarios, its convergence can be logarithmically dependent on $(1-\gamma)$, where $\gamma$ is the discount factor.

Complexity Analysis

The absence of a strongly polynomial algorithm for MDPs places the problem in the P-complete category, paralleling other complex computational problems like linear programming. The stochastic nature of MDPs, especially those with non-trivial transitional probabilities, imposes significant computational burdens, which are avenues for further research.

Future Directions and Implications

The research suggests several promising directions:

• MDP-Specific Algorithms: Development of algorithms that exploit MDP-specific structures could enhance practical efficiency. Techniques such as asynchronous dynamic programming and prioritized sweeping are worth exploring.
• Heuristics and Approximation: Approximative heuristics that trade precision for speed can be beneficial. Exploring randomly or probabilistically efficient stopping rules could yield useful real-world applications.
• Aggregation and Decomposition: Leveraging state space structure through aggregation and decomposition could simplify complex MDPs into solvable subproblems, reducing computational burdens.
• Empirical Studies and Benchmarks: Conducting empirical studies to benchmark problem classes could validate theoretical findings and inform the design of new algorithms.

Conclusion

While the theoretical underpinnings confirm MDPs' solvability within polynomial time, practical applications require further refinement to address the excessive computational resources demanded by large instances. This paper prompts a reevaluation of MDP solution strategies, emphasizing exploiting problem-specific structures and advancing approximation methods to bridge the gap between theory and practice. These insights herald opportunities for innovation in artificial intelligence and operations research, especially in decision-making under uncertainty.