Optimizing Minimax Regret in Uncertain MDPs with Sm...

Optimizing Minimax Regret in Uncertain MDPs with Sm... | AI Research

Key Takeaways

Optimizing Minimax Regret in Uncertain MDPs with Small Sets of Policies addresses the challenge of decision-making under model uncertainty.
Sequential decision-making in real-world applications often involves uncertainty about the environment's model.
Uncertain Markov decision processes (UMDPs) represent the possible environments as a set of MDPs with shared states and actions but potentially different transition probabilities and rewards.
We consider settings in which model uncertainty is resolved shortly before execution, allowing the most suitable policy to be selected from a limited set prepared in advance.
We introduce $k$-adaptable policy synthesis, which optimizes such a set of $k$ policies under a minimax-regret objective.

Paper AbstractExpand

Sequential decision-making in real-world applications often involves uncertainty about the environment's model. Uncertain Markov decision processes (UMDPs) represent the possible environments as a set of MDPs with shared states and actions but potentially different transition probabilities and rewards. Optimizing a single policy across all possible MDPs may sacrifice performance, while preparing an individually optimized policy for every MDP may violate operational, regulatory, or interpretability constraints on the number of policies that can be prepared and deployed. We consider settings in which model uncertainty is resolved shortly before execution, allowing the most suitable policy to be selected from a limited set prepared in advance. We introduce $k$-adaptable policy synthesis, which optimizes such a set of $k$ policies under a minimax-regret objective. We prove that the problem is NP-hard and develop KAPS, an exact nested branch-and-bound algorithm with problem-specific bounds and heuristics. KAPS jointly optimizes which MDPs share a policy and the policies themselves. Experiments across various UMDP benchmarks show that the largest reduction in regret consistently occurs when increasing from one to two policies. In the single-policy setting, KAPS is competitive with existing methods in solution quality and proves optimality substantially more often.

Optimizing Minimax Regret in Uncertain MDPs with Small Sets of Policies addresses the challenge of decision-making under model uncertainty. When the exact environment model is unknown, researchers Sterre Lutz, Daniël Vos, Matthijs T.J. Spaan, and Anna Lukina propose a method to prepare a small, pre-defined set of policies that can be selected once the environment's specific parameters are revealed.

The Challenge of Model Uncertainty

In many real-world scenarios, agents operate within Uncertain Markov Decision Processes (UMDPs), where transition probabilities and rewards may vary. While an agent could theoretically optimize a unique policy for every possible environment, doing so is often impractical due to operational, regulatory, or interpretability constraints. Conversely, relying on a single policy for all scenarios can lead to poor performance. The authors address this by introducing $k$-adaptable policy synthesis, which allows for a limited set of $k$ policies to be prepared in advance.

The KAPS Algorithm

To solve this, the researchers developed KAPS, an exact nested branch-and-bound algorithm. KAPS is designed to jointly optimize two components: which specific MDPs within the uncertainty set should share a policy, and the optimal policies themselves. The objective is to minimize the "minimax regret," which seeks to minimize the maximum possible loss an agent might incur across the set of potential environments. The authors prove that this optimization problem is NP-hard, necessitating the use of problem-specific bounds and heuristics within the branch-and-bound framework.

Performance and Results

Experiments conducted across various UMDP benchmarks indicate that the most significant improvements in performance occur when moving from a single policy to two policies. In the single-policy setting, the authors report that KAPS is competitive with existing methods regarding solution quality. Furthermore, the researchers note that KAPS proves optimality more frequently than existing approaches in these single-policy tests.

Franklin Analysis

The research suggests that for organizations constrained by the number of policies they can deploy, even a very small increase in the number of available policies—specifically moving from one to two—can yield substantial gains in mitigating regret. The reliance on an exact algorithm like KAPS provides a rigorous way to handle these constraints, though the NP-hard nature of the problem implies that computational complexity remains a factor to consider as the number of policies or the size of the uncertainty set grows.