Suboptimal coverings for continuous spaces of control tasks

James A. Preiss, Gaurav S. Sukhatme
Proceedings of the 3rd Conference on Learning for Dynamics and Control, PMLR 144:547-558, 2021.

Abstract

We propose the α-suboptimal covering number to characterize multi-task control problems where the set of dynamical systems and/or cost functions is infinite, analogous to the cardinality of finite task sets. This notion may help quantify the function class expressiveness needed to represent a good multi-task policy, which is important for learning-based control methods that use parameterized function approximation. We study suboptimal covering numbers for linear dynamical systems with quadratic cost (LQR problems) and construct a class of multi-task LQR problems amenable to analysis. For the scalar case, we show logarithmic dependence on the "breadth" of the space. For the matrix case, we present experiments 1) measuring the efficiency of a particular constructive cover, and 2) visualizing the behavior of two candidate systems for the lower bound.

Cite this Paper

BibTeX
@InProceedings{pmlr-v144-preiss21a, title = {Suboptimal coverings for continuous spaces of control tasks}, author = {Preiss, James A. and Sukhatme, Gaurav S.}, booktitle = {Proceedings of the 3rd Conference on Learning for Dynamics and Control}, pages = {547--558}, year = {2021}, editor = {Jadbabaie, Ali and Lygeros, John and Pappas, George J. and A. Parrilo, Pablo and Recht, Benjamin and Tomlin, Claire J. and Zeilinger, Melanie N.}, volume = {144}, series = {Proceedings of Machine Learning Research}, month = {07 -- 08 June}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v144/preiss21a/preiss21a.pdf}, url = {https://proceedings.mlr.press/v144/preiss21a.html}, abstract = {We propose the α-suboptimal covering number to characterize multi-task control problems where the set of dynamical systems and/or cost functions is infinite, analogous to the cardinality of finite task sets. This notion may help quantify the function class expressiveness needed to represent a good multi-task policy, which is important for learning-based control methods that use parameterized function approximation. We study suboptimal covering numbers for linear dynamical systems with quadratic cost (LQR problems) and construct a class of multi-task LQR problems amenable to analysis. For the scalar case, we show logarithmic dependence on the "breadth" of the space. For the matrix case, we present experiments 1) measuring the efficiency of a particular constructive cover, and 2) visualizing the behavior of two candidate systems for the lower bound.} }
Endnote
%0 Conference Paper %T Suboptimal coverings for continuous spaces of control tasks %A James A. Preiss %A Gaurav S. Sukhatme %B Proceedings of the 3rd Conference on Learning for Dynamics and Control %C Proceedings of Machine Learning Research %D 2021 %E Ali Jadbabaie %E John Lygeros %E George J. Pappas %E Pablo A. Parrilo %E Benjamin Recht %E Claire J. Tomlin %E Melanie N. Zeilinger %F pmlr-v144-preiss21a %I PMLR %P 547--558 %U https://proceedings.mlr.press/v144/preiss21a.html %V 144 %X We propose the α-suboptimal covering number to characterize multi-task control problems where the set of dynamical systems and/or cost functions is infinite, analogous to the cardinality of finite task sets. This notion may help quantify the function class expressiveness needed to represent a good multi-task policy, which is important for learning-based control methods that use parameterized function approximation. We study suboptimal covering numbers for linear dynamical systems with quadratic cost (LQR problems) and construct a class of multi-task LQR problems amenable to analysis. For the scalar case, we show logarithmic dependence on the "breadth" of the space. For the matrix case, we present experiments 1) measuring the efficiency of a particular constructive cover, and 2) visualizing the behavior of two candidate systems for the lower bound.
APA
Preiss, J.A. & Sukhatme, G.S.. (2021). Suboptimal coverings for continuous spaces of control tasks. Proceedings of the 3rd Conference on Learning for Dynamics and Control, in Proceedings of Machine Learning Research 144:547-558 Available from https://proceedings.mlr.press/v144/preiss21a.html.

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