Sample Complexity Characterization for Linear Contextual MDPs

Junze Deng, Yuan Cheng, Shaofeng Zou, Yingbin Liang
Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, PMLR 238:1693-1701, 2024.

Abstract

Contextual Markov decision processes (CMDPs) describe a class of reinforcement learning problems in which the transition kernels and reward functions can change over time with different MDPs indexed by a context variable. While CMDPs serve as an important framework to model many real-world applications with time-varying environments, they are largely unexplored from theoretical perspective. In this paper, we study CMDPs under two linear function approximation models: Model I with context-varying representations and common linear weights for all contexts; and Model II with common representations for all contexts and context-varying linear weights. For both models, we propose novel model-based algorithms and show that they enjoy guaranteed $\epsilon$-suboptimality gap with desired polynomial sample complexity. In particular, instantiating our result for the first model to the tabular CMDP improves the existing result by removing the reachability assumption. Our result for the second model is the first-known result for such a type of function approximation models. Comparison between our results for the two models further indicates that having context-varying features leads to much better sample efficiency than having common representations for all contexts under linear CMDPs.

Cite this Paper

BibTeX
@InProceedings{pmlr-v238-deng24a, title = { Sample Complexity Characterization for Linear Contextual {MDP}s }, author = {Deng, Junze and Cheng, Yuan and Zou, Shaofeng and Liang, Yingbin}, booktitle = {Proceedings of The 27th International Conference on Artificial Intelligence and Statistics}, pages = {1693--1701}, year = {2024}, editor = {Dasgupta, Sanjoy and Mandt, Stephan and Li, Yingzhen}, volume = {238}, series = {Proceedings of Machine Learning Research}, month = {02--04 May}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v238/deng24a/deng24a.pdf}, url = {https://proceedings.mlr.press/v238/deng24a.html}, abstract = { Contextual Markov decision processes (CMDPs) describe a class of reinforcement learning problems in which the transition kernels and reward functions can change over time with different MDPs indexed by a context variable. While CMDPs serve as an important framework to model many real-world applications with time-varying environments, they are largely unexplored from theoretical perspective. In this paper, we study CMDPs under two linear function approximation models: Model I with context-varying representations and common linear weights for all contexts; and Model II with common representations for all contexts and context-varying linear weights. For both models, we propose novel model-based algorithms and show that they enjoy guaranteed $\epsilon$-suboptimality gap with desired polynomial sample complexity. In particular, instantiating our result for the first model to the tabular CMDP improves the existing result by removing the reachability assumption. Our result for the second model is the first-known result for such a type of function approximation models. Comparison between our results for the two models further indicates that having context-varying features leads to much better sample efficiency than having common representations for all contexts under linear CMDPs. } }
Endnote
%0 Conference Paper %T Sample Complexity Characterization for Linear Contextual MDPs %A Junze Deng %A Yuan Cheng %A Shaofeng Zou %A Yingbin Liang %B Proceedings of The 27th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2024 %E Sanjoy Dasgupta %E Stephan Mandt %E Yingzhen Li %F pmlr-v238-deng24a %I PMLR %P 1693--1701 %U https://proceedings.mlr.press/v238/deng24a.html %V 238 %X Contextual Markov decision processes (CMDPs) describe a class of reinforcement learning problems in which the transition kernels and reward functions can change over time with different MDPs indexed by a context variable. While CMDPs serve as an important framework to model many real-world applications with time-varying environments, they are largely unexplored from theoretical perspective. In this paper, we study CMDPs under two linear function approximation models: Model I with context-varying representations and common linear weights for all contexts; and Model II with common representations for all contexts and context-varying linear weights. For both models, we propose novel model-based algorithms and show that they enjoy guaranteed $\epsilon$-suboptimality gap with desired polynomial sample complexity. In particular, instantiating our result for the first model to the tabular CMDP improves the existing result by removing the reachability assumption. Our result for the second model is the first-known result for such a type of function approximation models. Comparison between our results for the two models further indicates that having context-varying features leads to much better sample efficiency than having common representations for all contexts under linear CMDPs.
APA
Deng, J., Cheng, Y., Zou, S. & Liang, Y.. (2024). Sample Complexity Characterization for Linear Contextual MDPs . Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 238:1693-1701 Available from https://proceedings.mlr.press/v238/deng24a.html.

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