A Unifying Framework for Linearly Solvable Control

Krishnamurthy Dvijotham, Emanuel Todorov
Proceedings of the 27th Conference on Uncertainty in Artificial Intelligence, PMLR R9:217-224, 2011.

Abstract

Recent work has led to the development of an elegant theory of Linearly Solvable Markov Decision Processes (LMDPs) and related Path-Integral Control Problems. Traditionally, MDPs have been formulated using stochastic policies and a control cost based on the KL divergence. In this paper, we extend this framework to a more general class of divergences: the Renyi divergences. These are a more general class of divergences parameterized by a continuous parameter that include the KL divergence as a special case. The resulting control problems can be interpreted as solving a risk-sensitive version of the LMDP problem. For a > 0, we get risk-averse behavior (the degree of risk-aversion increases with a) and for a < 0, we get risk-seeking behavior. We recover LMDPs in the limit as a -> 0. This work generalizes the recently developed risk-sensitive path-integral control formalism which can be seen as the continuous-time limit of results obtained in this paper. To the best of our knowledge, this is a general theory of linearly solvable control and includes all previous work as a special case. We also present an alternative interpretation of these results as solving a 2-player (cooperative or competitive) Markov Game. From the linearity follow a number of nice properties including compositionality of control laws and a path-integral representation of the value function. We demonstrate the usefulness of the framework on control problems with noise where different values of lead to qualitatively different control behaviors.

Cite this Paper

BibTeX
@InProceedings{pmlr-vR9-dvijotham11a, title = {A Unifying Framework for Linearly Solvable Control}, author = {Dvijotham, Krishnamurthy and Todorov, Emanuel}, booktitle = {Proceedings of the 27th Conference on Uncertainty in Artificial Intelligence}, pages = {217--224}, year = {2011}, editor = {Cozman, Fabio and Pfeffer, Avi}, volume = {R9}, series = {Proceedings of Machine Learning Research}, month = {14--17 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/r9/main/assets/dvijotham11a/dvijotham11a.pdf}, url = {https://proceedings.mlr.press/r9/dvijotham11a.html}, abstract = {Recent work has led to the development of an elegant theory of Linearly Solvable Markov Decision Processes (LMDPs) and related Path-Integral Control Problems. Traditionally, MDPs have been formulated using stochastic policies and a control cost based on the KL divergence. In this paper, we extend this framework to a more general class of divergences: the Renyi divergences. These are a more general class of divergences parameterized by a continuous parameter that include the KL divergence as a special case. The resulting control problems can be interpreted as solving a risk-sensitive version of the LMDP problem. For a > 0, we get risk-averse behavior (the degree of risk-aversion increases with a) and for a < 0, we get risk-seeking behavior. We recover LMDPs in the limit as a -> 0. This work generalizes the recently developed risk-sensitive path-integral control formalism which can be seen as the continuous-time limit of results obtained in this paper. To the best of our knowledge, this is a general theory of linearly solvable control and includes all previous work as a special case. We also present an alternative interpretation of these results as solving a 2-player (cooperative or competitive) Markov Game. From the linearity follow a number of nice properties including compositionality of control laws and a path-integral representation of the value function. We demonstrate the usefulness of the framework on control problems with noise where different values of lead to qualitatively different control behaviors.}, note = {Reissued by PMLR on 04 October 2026.} }
Endnote
%0 Conference Paper %T A Unifying Framework for Linearly Solvable Control %A Krishnamurthy Dvijotham %A Emanuel Todorov %B Proceedings of the 27th Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2011 %E Fabio Cozman %E Avi Pfeffer %F pmlr-vR9-dvijotham11a %I PMLR %P 217--224 %U https://proceedings.mlr.press/r9/dvijotham11a.html %V R9 %X Recent work has led to the development of an elegant theory of Linearly Solvable Markov Decision Processes (LMDPs) and related Path-Integral Control Problems. Traditionally, MDPs have been formulated using stochastic policies and a control cost based on the KL divergence. In this paper, we extend this framework to a more general class of divergences: the Renyi divergences. These are a more general class of divergences parameterized by a continuous parameter that include the KL divergence as a special case. The resulting control problems can be interpreted as solving a risk-sensitive version of the LMDP problem. For a > 0, we get risk-averse behavior (the degree of risk-aversion increases with a) and for a < 0, we get risk-seeking behavior. We recover LMDPs in the limit as a -> 0. This work generalizes the recently developed risk-sensitive path-integral control formalism which can be seen as the continuous-time limit of results obtained in this paper. To the best of our knowledge, this is a general theory of linearly solvable control and includes all previous work as a special case. We also present an alternative interpretation of these results as solving a 2-player (cooperative or competitive) Markov Game. From the linearity follow a number of nice properties including compositionality of control laws and a path-integral representation of the value function. We demonstrate the usefulness of the framework on control problems with noise where different values of lead to qualitatively different control behaviors. %Z Reissued by PMLR on 04 October 2026.
APA
Dvijotham, K. & Todorov, E.. (2011). A Unifying Framework for Linearly Solvable Control. Proceedings of the 27th Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research R9:217-224 Available from https://proceedings.mlr.press/r9/dvijotham11a.html. Reissued by PMLR on 04 October 2026.

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