Towards a Dimension-Free Understanding of Adaptive Linear Control

Juan C Perdomo, Max Simchowitz, Alekh Agarwal, Peter Bartlett
Proceedings of Thirty Fourth Conference on Learning Theory, PMLR 134:3681-3770, 2021.

Abstract

We study the problem of adaptive control of the linear quadratic regulator for systems in very high, or even infinite dimension. We demonstrate that while sublinear regret requires finite dimensional inputs, the ambient state dimension of the system need not be bounded in order to perform online control. We provide the first regret bounds for LQR which hold for infinite dimensional systems, replacing dependence on ambient dimension with more natural notions of problem complexity. Our guarantees arise from a novel perturbation bound for certainty equivalence which scales with the prediction error in estimating the system parameters, without requiring consistent parameter recovery in more stringent measures like the operator norm. When specialized to finite dimensional settings, our bounds recover near optimal dimension and time horizon dependence.

Cite this Paper

BibTeX
@InProceedings{pmlr-v134-perdomo21a, title = {Towards a Dimension-Free Understanding of Adaptive Linear Control}, author = {Perdomo, Juan C and Simchowitz, Max and Agarwal, Alekh and Bartlett, Peter}, booktitle = {Proceedings of Thirty Fourth Conference on Learning Theory}, pages = {3681--3770}, year = {2021}, editor = {Belkin, Mikhail and Kpotufe, Samory}, volume = {134}, series = {Proceedings of Machine Learning Research}, month = {15--19 Aug}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v134/perdomo21a/perdomo21a.pdf}, url = {https://proceedings.mlr.press/v134/perdomo21a.html}, abstract = {We study the problem of adaptive control of the linear quadratic regulator for systems in very high, or even infinite dimension. We demonstrate that while sublinear regret requires finite dimensional inputs, the ambient state dimension of the system need not be bounded in order to perform online control. We provide the first regret bounds for LQR which hold for infinite dimensional systems, replacing dependence on ambient dimension with more natural notions of problem complexity. Our guarantees arise from a novel perturbation bound for certainty equivalence which scales with the prediction error in estimating the system parameters, without requiring consistent parameter recovery in more stringent measures like the operator norm. When specialized to finite dimensional settings, our bounds recover near optimal dimension and time horizon dependence.} }
Endnote
%0 Conference Paper %T Towards a Dimension-Free Understanding of Adaptive Linear Control %A Juan C Perdomo %A Max Simchowitz %A Alekh Agarwal %A Peter Bartlett %B Proceedings of Thirty Fourth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2021 %E Mikhail Belkin %E Samory Kpotufe %F pmlr-v134-perdomo21a %I PMLR %P 3681--3770 %U https://proceedings.mlr.press/v134/perdomo21a.html %V 134 %X We study the problem of adaptive control of the linear quadratic regulator for systems in very high, or even infinite dimension. We demonstrate that while sublinear regret requires finite dimensional inputs, the ambient state dimension of the system need not be bounded in order to perform online control. We provide the first regret bounds for LQR which hold for infinite dimensional systems, replacing dependence on ambient dimension with more natural notions of problem complexity. Our guarantees arise from a novel perturbation bound for certainty equivalence which scales with the prediction error in estimating the system parameters, without requiring consistent parameter recovery in more stringent measures like the operator norm. When specialized to finite dimensional settings, our bounds recover near optimal dimension and time horizon dependence.
APA
Perdomo, J.C., Simchowitz, M., Agarwal, A. & Bartlett, P.. (2021). Towards a Dimension-Free Understanding of Adaptive Linear Control. Proceedings of Thirty Fourth Conference on Learning Theory, in Proceedings of Machine Learning Research 134:3681-3770 Available from https://proceedings.mlr.press/v134/perdomo21a.html.

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