How to Find the Exact Pareto Front for Multi-Objective MDPs?

2024-10-21

arXiv.org

AI Summary

Computing the exact Pareto frontier in multi-objective Markov decision processes (MO-MDPs) is computationally challenging due to exponential policy enumeration or reliance on continuous preference sampling. Method: This paper proposes a geometry-driven algorithm that exploits the structural property that the exact Pareto frontier lies precisely on the boundary of a convex polytope in value space—whose vertices correspond to deterministic policies, and whose adjacent vertices differ in action choice at exactly one state-action pair. The algorithm reduces global optimization to local edge traversal over this polytope, integrating dynamic programming with geometric analysis and using only a single scalarized MDP solution as its primitive operation. Contribution/Results: It is the first method to construct the exact Pareto frontier in polynomial time under known models, outperforming existing approximation schemes and exponential enumeration approaches. It overcomes fundamental limitations of prior methods—namely, dependence on dense preference sampling or restrictive assumptions about deterministic policies.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Planning with Markov Models (MDPs, POMDPs)
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• 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
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Multi-Objective Markov Decision Processes (MO-MDPs) are receiving increasing attention, as real-world decision-making problems often involve conflicting objectives that cannot be addressed by a single-objective MDP. The Pareto front identifies the set of policies that cannot be dominated, providing a foundation for finding Pareto optimal solutions that can efficiently adapt to various preferences. However, finding the Pareto front is a highly challenging problem. Most existing methods either (i) rely on traversing the continuous preference space, which is impractical and results in approximations that are difficult to evaluate against the true Pareto front, or (ii) focus solely on deterministic Pareto optimal policies, from which there are no known techniques to characterize the full Pareto front. Moreover, finding the structure of the Pareto front itself remains unclear even in the context of dynamic programming, where the MDP is fully known in advance. In this work, we address the challenge of efficiently discovering the Pareto front. By investigating the geometric structure of the Pareto front in MO-MDPs, we uncover a key property: the Pareto front is on the boundary of a convex polytope whose vertices all correspond to deterministic policies, and neighboring vertices of the Pareto front differ by only one state-action pair of the deterministic policy, almost surely. This insight transforms the global comparison across all policies into a localized search among deterministic policies that differ by only one state-action pair, drastically reducing the complexity of searching for the exact Pareto front. We develop an efficient algorithm that identifies the vertices of the Pareto front by solving a single-objective MDP only once and then traversing the edges of the Pareto front, making it more efficient than existing methods.

Problem

• Finding exact Pareto front
• Efficient algorithm development
• Geometric structure analysis

Innovation

• Convex polytope boundary analysis
• Single-state-action policy variation
• Efficient Pareto front algorithm