Contextual Markov Decision Processes (1502.02259v1)

Abstract: We consider a planning problem where the dynamics and rewards of the environment depend on a hidden static parameter referred to as the context. The objective is to learn a strategy that maximizes the accumulated reward across all contexts. The new model, called Contextual Markov Decision Process (CMDP), can model a customer's behavior when interacting with a website (the learner). The customer's behavior depends on gender, age, location, device, etc. Based on that behavior, the website objective is to determine customer characteristics, and to optimize the interaction between them. Our work focuses on one basic scenario--finite horizon with a small known number of possible contexts. We suggest a family of algorithms with provable guarantees that learn the underlying models and the latent contexts, and optimize the CMDPs. Bounds are obtained for specific naive implementations, and extensions of the framework are discussed, laying the ground for future research.

• The paper proposes a novel CMDP model that extends traditional MDPs by incorporating static latent contexts affecting both transitions and rewards.
• Researchers introduce a modular CECE framework, sequentially clustering, exploring, classifying, and exploiting to optimize decision-making in CMDPs.
• Empirical and theoretical analyses provide quantifiable regret bounds and highlight the trade-off in trajectory length for effective clustering and policy performance.

Contextual Markov Decision Processes: An Overview

The paper "Contextual Markov Decision Processes" introduces a novel framework designed to address scenarios where both dynamics and rewards in a Markovian environment depend on static external parameters or contexts. This setup is pertinent in various domains where decision-making strategies need to adapt to different contextual information, which remains consistent within each episode of decision-making.

Problem and Model Definition

The authors propose the Contextual Markov Decision Process (CMDP) as a model to account for contextual variations influencing observed behaviors. In traditional MDPs, the observed trajectories are primarily attributed to stationary transition dynamics, which can be estimated using maximum likelihood strategies. However, CMDPs extend this concept to incorporate latent contextual information within the trajectory, allowing for models that better capture the nuances of scenarios like personalized content recommendations or targeted advertising.

The CMDP is defined by a set of contexts, each associated with a distinct MDP. The context influences the transition probabilities and rewards in the environment, but remains hidden from the decision-maker. The finite horizon episodic setting is explored, with a specific focus on scenarios where the number of contexts is small and known.

Algorithmic Framework

To tackle the CMDP challenge, the paper introduces a CECE framework, which systematically addresses the CMDP problem through a modular process:

1. Clustering (Cluster): This initial step involves grouping observed trajectories to estimate transition probabilities for each context.
2. Exploration (Explore): A subsequent exploration phase is carried out to collect additional samples conducive to distinguishing among contexts.
3. Classification (Classify): The partially observed trajectory is classified to a model or context based on previously learned clusters.
4. Exploitation (Exploit): The final step involves selecting actions based on the identified context to maximize rewards.

Theoretical Guarantees

The CMDP setting confronts significant challenges, particularly around context exploration and classification. The authors provide an analytical foundation for their approach through regret analysis, which measures the performance gap between the ideal contextual learning strategy and the cumulative reward achieved by their method. The paper includes quantifiable bounds on regret, dependent on factors such as trajectory length and the accuracy of context estimation. The assumptions underpinning this analysis include sufficient separability between contexts and conditions on trajectory length to assure efficient clustering.

Empirical Investigations

Experiments conducted elucidate the impact of trajectory length and number of episodes on the clustering accuracy and resultant policy performance. For instance, a phase transition in clustering effectiveness suggests a critical trajectory length beyond which model estimation becomes reliable. These experimental outcomes highlight the practical considerations in CMDP applications, such as the trade-off between exploration duration and exploitation potential.

Discussion and Implications

The CMDP framework holds significant implications for domains where static contextual parameters critically influence decision-making. Unlike more general models such as POMDPs, CMDPs offer a computationally feasible solution by leveraging the static nature of contexts. Future work might explore enhancements in each module of the CECE framework, including more advanced clustering techniques or optimized exploration strategies. Moreover, extensions to scenarios with infinite contexts or concurrent RL setups are suggested, posing promising avenues for subsequent research.

Conclusion

This paper lays a robust foundation for CMDPs, demonstrating their utility in scenarios where context-driven decision-making is pivotal. While initial findings and algorithms provide valuable insights, the domain remains rich with potential advancements and open questions—particularly regarding scalability, efficiency, and applicability across broader, possibly dynamic contexts.