Learning Linear Dynamical Systems with Semi-Parametric Least Squares

Max Simchowitz, Ross Boczar, Benjamin Recht
Proceedings of the Thirty-Second Conference on Learning Theory, PMLR 99:2714-2802, 2019.

Abstract

We analyze a simple prefiltered variation of the least squares estimator for the problem of estimation with biased, \emph{semi-parametric} noise, an error model studied more broadly in causal statistics and active learning. We prove an oracle inequality which demonstrates that this procedure provably mitigates the variance introduced by long-term dependencies. % We then demonstrate that prefiltered least squares yields, to our knowledge, the first algorithm that provably estimates the parameters of partially-observed linear systems that attains rates which do not not incur a worst-case dependence on the rate at which these dependencies decay. % The algorithm is provably consistent even for systems which satisfy the weaker \emph{marginal stability} condition obeyed by many classical models based on Newtonian mechanics. In this context, our semi-parametric framework yields guarantees for both stochastic and worst-case noise.

Cite this Paper

BibTeX
@InProceedings{pmlr-v99-simchowitz19a, title = {Learning Linear Dynamical Systems with Semi-Parametric Least Squares}, author = {Simchowitz, Max and Boczar, Ross and Recht, Benjamin}, booktitle = {Proceedings of the Thirty-Second Conference on Learning Theory}, pages = {2714--2802}, year = {2019}, editor = {Beygelzimer, Alina and Hsu, Daniel}, volume = {99}, series = {Proceedings of Machine Learning Research}, month = {25--28 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v99/simchowitz19a/simchowitz19a.pdf}, url = {https://proceedings.mlr.press/v99/simchowitz19a.html}, abstract = {We analyze a simple prefiltered variation of the least squares estimator for the problem of estimation with biased, \emph{semi-parametric} noise, an error model studied more broadly in causal statistics and active learning. We prove an oracle inequality which demonstrates that this procedure provably mitigates the variance introduced by long-term dependencies. % We then demonstrate that prefiltered least squares yields, to our knowledge, the first algorithm that provably estimates the parameters of partially-observed linear systems that attains rates which do not not incur a worst-case dependence on the rate at which these dependencies decay. % The algorithm is provably consistent even for systems which satisfy the weaker \emph{marginal stability} condition obeyed by many classical models based on Newtonian mechanics. In this context, our semi-parametric framework yields guarantees for both stochastic and worst-case noise.} }
Endnote
%0 Conference Paper %T Learning Linear Dynamical Systems with Semi-Parametric Least Squares %A Max Simchowitz %A Ross Boczar %A Benjamin Recht %B Proceedings of the Thirty-Second Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2019 %E Alina Beygelzimer %E Daniel Hsu %F pmlr-v99-simchowitz19a %I PMLR %P 2714--2802 %U https://proceedings.mlr.press/v99/simchowitz19a.html %V 99 %X We analyze a simple prefiltered variation of the least squares estimator for the problem of estimation with biased, \emph{semi-parametric} noise, an error model studied more broadly in causal statistics and active learning. We prove an oracle inequality which demonstrates that this procedure provably mitigates the variance introduced by long-term dependencies. % We then demonstrate that prefiltered least squares yields, to our knowledge, the first algorithm that provably estimates the parameters of partially-observed linear systems that attains rates which do not not incur a worst-case dependence on the rate at which these dependencies decay. % The algorithm is provably consistent even for systems which satisfy the weaker \emph{marginal stability} condition obeyed by many classical models based on Newtonian mechanics. In this context, our semi-parametric framework yields guarantees for both stochastic and worst-case noise.
APA
Simchowitz, M., Boczar, R. & Recht, B.. (2019). Learning Linear Dynamical Systems with Semi-Parametric Least Squares. Proceedings of the Thirty-Second Conference on Learning Theory, in Proceedings of Machine Learning Research 99:2714-2802 Available from https://proceedings.mlr.press/v99/simchowitz19a.html.

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