Source-linked AI summary
Contextual Markov Decision Processes
Assaf Hallak, Dotan Di Castro, Shie Mannor
TL;DR
The paper addresses planning when unknown, static context affects environment dynamics and rewards, especially for identifying users and tailoring website interactions. It defines CMDPs as context-indexed MDPs and develops algorithms for finite-horizon episodic settings with latent contexts. The framework provides provable guarantees for learning model parameters and contexts while optimizing actions, but the presented solutions are preliminary and have scope limitations.
Problem
The paper studies how to learn an optimal control strategy when both latent context and model parameters are unknown, including user identification from website interactions without prior profile information.
Method
The paper defines CMDPs as sets of MDPs sharing state and action spaces and presents algorithms for finite-horizon episodic problems with a finite number of latent contexts.
Results
The proposed framework provides provable guarantees for learning model parameters and latent contexts while optimizing online actions, with regret analysis for a specific implementation.
Takeaways & Limitations
CMDPs offer a flexible and computationally tractable framework for modeling multiple Markovian sources with sequential decision making and static context.
Takeaways & Limitations
The study focuses on finite-horizon problems with a small known number of contexts, and its suggested solutions are described as preliminary.
Abstract
from arXiv · showhide
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.
1. Introduction
The paper introduces CMDPs for sequential decision problems whose dynamics depend on static, possibly hidden context, motivated by identifying users and adapting website interactions. It positions the framework against related models and provides preliminary algorithms with guarantees for finite contextual settings.
- Motivation: Static exogenous variables can affect transition dynamics, so encoding them into the state creates disconnected chains and weakens generalization.The paper proposes separate transition matrices for each context as a more compact representation.
- Motivation: Latent context learning can identify website users from interaction trajectories when cookies or prior profile information are unavailable.The motivating setting includes a cold-start problem involving user age, gender, and other profile characteristics.
- Motivation: Once user groups are identified, contextual policies can optimize content and advertisements, with user identification linked to higher click-through rates in online advertising.The broader objective is online learning when both context and model parameters are unknown.
- Contributions: The main contribution is a general algorithm with provable guarantees for finite-horizon episodic CMDPs, alongside regret analysis for a specific implementation.The paper also discusses infinitely many contexts and concurrent reinforcement learning as extensions, while describing the solutions as preliminary.
- Related models: CMDPs differ from contextual HMMs, multi-model RL, representation learning, robust MDPs, and POMDPs in how context affects dynamics, model composition, state spaces, uncertainty, and control.POMDPs can encompass CMDPs but are described as more general and complex, usually assuming a context distribution.
2. Contextual Markov Decision Processes
A CMDP is a collection of MDPs sharing state and action spaces, with a context selecting the transition, reward, and initial-state model. The paper focuses on finite latent contexts and describes episodic interaction in which contexts generate trajectories.
- MDP setup: An MDP specifies states, actions, transition probabilities, rewards, and an initial-state distribution.The learner acts through a policy over a finite-horizon interaction.
- MDP setup: When MDP parameters are unknown, selecting a reward-maximizing policy is treated as adaptive control or reinforcement learning.When parameters are known, the corresponding problem is called planning.
- CMDP definition: A CMDP maps each context to an MDP while keeping the state and action spaces common across models.Thus, a CMDP is a set of models sharing the same state and action space.
- Context settings: For finitely many observable contexts, the learner can independently learn one model for each context without further assumptions.The paper then focuses on latent contexts rather than model aggregation as the number of clusters grows with trajectories.
- Episodic interaction: In the general episodic setup, each episode begins with an environment-selected context, followed by an initial state and a trajectory generated by the selected MDP.The context may be chosen randomly, adversarially, or by another process, and trajectory length is a stopping time.
3. Problem Definition and Solution
The finite-sources episodic CMDP problem seeks low-regret control when each trajectory comes from one of several hidden contexts. CECE clusters trajectories, explores to identify contexts, classifies them, and exploits learned models, with guarantees under stated assumptions and several scope limitations.
- Problem Definition: Regret compares the agent with an all-knowing policy that selects the optimal policy for each trajectory’s correct context.Because context identification causes loss in every new trajectory, the paper’s regret is necessarily linear in H under this comparator.
- Solution: CECE partitions trajectories into minibatches and repeatedly clusters observed trajectories, explores each new trajectory, classifies its context, and exploits the identified model.The framework comprises Cluster, Explore, Classify, and Exploit stages.
- Guarantees: The framework’s guarantees require approximated models for each context, reliable context identification, and additional assumptions summarized in Theorem 1.The theorem states CECE’s performance in the L’th minibatch when the assumptions hold.
- Exploration: Exploration seeks a policy that identifies the context quickly because regret grows linearly with the number of exploration steps.Choosing actions solely for immediate model distinction can be suboptimal because actions have future state-space consequences.
- Limitations and Extensions: The framework is preliminary and becomes more difficult with infinitely many contexts, short trajectories, outliers, or models that cannot be reliably distinguished.Increasing context counts raise clustering and model-identification sample requirements; short trajectories may prevent model approximation altogether.
- Specific Instance: The specific clustering realization exhaustively searches trajectory partitions, making it highly inefficient despite accommodating the required modeling assumption.Polynomial-time alternatives exist, but their guarantees and assumptions would need adjustment for this setting.
4. Experiments
The experiments evaluate CECE’s clustering and full CMDP learning behavior under varying trajectory lengths, episode counts, and numbers of contexts. Clustering exhibits a threshold effect, while full-algorithm reward generally improves with more data and declines as more models are introduced.
- Experiment 1: The first experiment uses K = 5 equal-probability contexts, |A| = 2 actions, |S| = 100 states, and uniformly sampled actions across H trajectories.Transition matrices are drawn from a uniform distribution, and clustering uses K-means on vectorized empirical transition matrices averaged over 100 trials.
- Experiment 1: Clustering fails below T = 4000 but succeeds almost certainly at T = 8000, indicating a phase transition with trajectory length.Between these regimes, performance passes through a short adjustment period.
- Experiment 1: If trajectories are too short, increasing the number of episodes does not rescue clustering; sufficiently long trajectories benefit from additional episodes.The study varies T across 2000, 5000, and 8000 while measuring clustering as H changes.
- Experiment 2: Average reward increases with episode count and trajectory length because clustering and model learning improve as more data become available.The full CECE experiment uses |S| = 100, |A| = 4, K = 20, H = 100, and T = 2000.
- Experiment 2: Average reward decreases as more models are introduced, since clustering and learning each model become harder.The full-algorithm results are presented in Figure 2 and averaged over 20 experiments.
- Experiment 2: A fixed identification fraction leaves a persistent gap between optimal and achieved values because identification consumes trajectory steps.The identification portion is set using η = 0.3 in the full CECE experiment.
5. Conclusions and Future Work
The paper introduces a flexible CMDP framework for sequential decision making with observable or latent static contexts, and analyzes basic finite-context algorithms. It identifies computational and theoretical limitations while outlining extensions and future research directions.
- Conclusions: The framework models multiple Markovian sources with sequential decision making and supports both observable and latent static contexts.The proposed setup remains computationally tractable under certain conditions.
- Conclusions: Under certain conditions, the algorithms address learning model parameters and optimizing actions online within a reinforcement-learning framework.The analyzed algorithms assume a finite number of contexts.
- Future Work: The framework is presented as a first step, with modular upgrades proposed for clustering, trajectory classification, reward-oriented context classification, and exploration during exploitation.The proposed improvements target efficiency, classification speed, regret, and protection against misclassification.
- Future Work: The paper discusses alternative CMDP approaches, including belief-based exploration-classification-exploitation, POMDP, and robust MDP formulations.POMDPs are described as more general and complex, while robust MDPs address uncertainty when data are insufficient.
- Future Work: The theoretical trade-off among learning, exploration, optimization, and control remains an open question, especially for concurrent reinforcement learning and many or infinitely many contexts.The exact theoretical setup for these extensions requires more precise definitions and assumptions.
- Future Work: Computational efficiency and sample complexity were not addressed, and large state or action spaces can make straightforward model-based implementations fail as sample requirements grow.The paper leaves finite-sample error and regret rates for future analysis.
B. Useful Lemmas
This section collects lemmas used in the proofs, covering concentration for empirical distributions and performance preservation under approximate MDP models.
- Useful Lemmas: The proofs use a concentration lemma for independent identically distributed samples from a finite probability distribution.The lemma is attributed to Weissman et al. (2003).
- Useful Lemmas: A second lemma relates an O(ϵ)-approximation of an MDP to the behavior of any policy and its corresponding optimal policy.The lemma is attributed to Kearns and Singh (2002).
C. Proof of Lemma 1
The proof establishes that trajectory-level and cluster-level transition estimates concentrate around their true models, then uses cluster separation to control matching, model error, and classification.
- Lemma 1: Under Assumption 4, the realizations of Algorithms 1–4 satisfy Assumptions 1–3 with the stated probability guarantee.Lemma 1 summarizes the conditions established by the proof.
- Setup: The proof defines estimated transition matrices for individual trajectories and clusters, together with the true and algorithmically found trajectory clusterings.These quantities form the basis for comparing estimated and true models.
- Model Estimation: With at least αT samples from each state-action pair, an individual trajectory model is within ϵ of its true transition model with probability at least 1 −SAeS−αT ϵ2/2.This follows from Lemma 2 and a union bound.
- Model Estimation: With at least βH trajectories from each model, the corresponding cluster estimate is within ϵ of the true model with probability at least 1 −SAeS−αT βHϵ2/2.The bound applies the available trajectory count to the cluster-level estimate.
- Bound Simplification: For large H, the proof approximates the relevant accumulated quantity using the boundedness of each summand and the maximal L1 distance between distributions.The resulting probability expression is retained in the proof’s guarantee.
- Cluster Matching: If every pair of true clusters is separated by D > O(ϵ + δ), matched clusters agree on O(1 −δ) of trajectories and each model error is of order O(Kδ).Choosing ϵ of order D leads to the stated sample-size condition and resulting approximation guarantee.
- Classification: The proof bounds classification error using a union bound over the complementary event, yielding a probability of correct classification of at least 1 −δ.The argument separately bounds the two terms in the classification comparison.