Polynomial-Time Approximability of Constrained Reinforcement Learning

Jeremy Mcmahan
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:43417-43439, 2025.

Abstract

We study the computational complexity of approximating general constrained Markov decision processes. Our primary contribution is the design of a polynomial time $(0,\epsilon)$-additive bicriteria approximation algorithm for finding optimal constrained policies across a broad class of recursively computable constraints, including almost-sure, chance, expectation, and their anytime variants. Matching lower bounds imply our approximation guarantees are optimal so long as $P \neq NP$. The generality of our approach results in answers to several long-standing open complexity questions in the constrained reinforcement learning literature. Specifically, we are the first to prove polynomial-time approximability for the following settings: policies under chance constraints, deterministic policies under multiple expectation constraints, policies under non-homogeneous constraints (i.e., constraints of different types), and policies under constraints for continuous-state processes.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-mcmahan25b, title = {Polynomial-Time Approximability of Constrained Reinforcement Learning}, author = {Mcmahan, Jeremy}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {43417--43439}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/mcmahan25b/mcmahan25b.pdf}, url = {https://proceedings.mlr.press/v267/mcmahan25b.html}, abstract = {We study the computational complexity of approximating general constrained Markov decision processes. Our primary contribution is the design of a polynomial time $(0,\epsilon)$-additive bicriteria approximation algorithm for finding optimal constrained policies across a broad class of recursively computable constraints, including almost-sure, chance, expectation, and their anytime variants. Matching lower bounds imply our approximation guarantees are optimal so long as $P \neq NP$. The generality of our approach results in answers to several long-standing open complexity questions in the constrained reinforcement learning literature. Specifically, we are the first to prove polynomial-time approximability for the following settings: policies under chance constraints, deterministic policies under multiple expectation constraints, policies under non-homogeneous constraints (i.e., constraints of different types), and policies under constraints for continuous-state processes.} }
Endnote
%0 Conference Paper %T Polynomial-Time Approximability of Constrained Reinforcement Learning %A Jeremy Mcmahan %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-mcmahan25b %I PMLR %P 43417--43439 %U https://proceedings.mlr.press/v267/mcmahan25b.html %V 267 %X We study the computational complexity of approximating general constrained Markov decision processes. Our primary contribution is the design of a polynomial time $(0,\epsilon)$-additive bicriteria approximation algorithm for finding optimal constrained policies across a broad class of recursively computable constraints, including almost-sure, chance, expectation, and their anytime variants. Matching lower bounds imply our approximation guarantees are optimal so long as $P \neq NP$. The generality of our approach results in answers to several long-standing open complexity questions in the constrained reinforcement learning literature. Specifically, we are the first to prove polynomial-time approximability for the following settings: policies under chance constraints, deterministic policies under multiple expectation constraints, policies under non-homogeneous constraints (i.e., constraints of different types), and policies under constraints for continuous-state processes.
APA
Mcmahan, J.. (2025). Polynomial-Time Approximability of Constrained Reinforcement Learning. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:43417-43439 Available from https://proceedings.mlr.press/v267/mcmahan25b.html.

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