Spectral Filtering for General Linear Dynamical Systems

Hazan, Elad; Lee, Holden; Singh, Karan; Zhang, Cyril; Zhang, Yi

doi:10.48550/arxiv.1802.03981

Cited by 12 publications

(16 citation statements)

References 11 publications

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“…[25] proved a non-asymptotic guarantee for the identification of partially observed linear dynamics of unknown order. [11] and [12] proposed spectral filtering methods that predict with sublinear regret the next output of unknown partially observed linear systems. When actuation is available Wagenmaker and Jamieson [34] showed that active learning can be used for a faster identification of linear dynamics.…”

Section: Related Workmentioning

confidence: 99%

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Time varying regression with hidden linear dynamics

Jadbabaie¹,

Mania²,

Shah³

et al. 2021

Preprint

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We revisit a model for time-varying linear regression that assumes the unknown parameters evolve according to a linear dynamical system. Counterintuitively, we show that when the underlying dynamics are stable the parameters of this model can be estimated from data by combining just two ordinary least squares estimates. We offer a finite sample guarantee on the estimation error of our method and discuss certain advantages it has over Expectation-Maximization (EM), which is the main approach proposed by prior work.

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Section: Related Workmentioning

confidence: 99%

“…where ξ t is the noise term shown in (12). Note that the covariance of vec(x t x ⊤ t+1 ) is the identity matrix, whose minimum eigenvalue is one.…”

Section: A Proof Of Bound 6a Of Theoremmentioning

confidence: 99%

Time varying regression with hidden linear dynamics

Jadbabaie¹,

Mania²,

Shah³

et al. 2021

Preprint

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“…As has been suggested in the previous section, we consider the problem of predicting ŷk+1 in the Huber-like extension of LDS (2), under several assumptions, starting with the identifiability of Hazan et al [18]: Assumption 1. The outputs are generated by the stochastic difference equation (2), assuming:…”

Section: The Problemmentioning

confidence: 99%

“…Throughout both Algorithms 1 and 2, we predict the next output ŷt of the system from inputs X t until time t and outputs Y t−1 until t−1 in an online fashion. There, leading methods [19,18,28,17,30] consider an overparametrisation, where the vector Xt is composed of the inputs to the system at all time-levels up to the current one, convolved with the eigenvectors of a certain Hankel matrix, as well as the outputs at the previous time level, and inputs at the current and previous time levels. Notice that the Hankel matrix is constant and its eigenvectors can be precomputed.…”

Section: The Algorithmsmentioning

confidence: 99%

“…In improper learning of such an LDS (which we refer to as an identification problem), one wishes to estimate ŷk such that ŷk are close to the best estimates y * k of y k possible at time k. When there is no hidden state, the identification problem is convex and a variety of methods work well. When there is a hidden state, the problem is non-convex and only rather recently spectral filtering [19,18] has been used to obtain identification procedures with regret bounded by Õ(log 7 √ k) at time k, where Õ(•) hides terms that depend polynomially on the dimension of the system and norms of the inputs and outputs and the noise.…”

Section: Introductionmentioning

confidence: 99%

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Robust Spectral Filtering and Anomaly Detection

Marecek,

Tchrakian

2018

Preprint

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We consider a setting, where the output of a linear dynamical system (LDS) is, with an unknown but fixed probability, replaced by noise. There, we present a robust method for the prediction of the outputs of the LDS and identification of the samples of noise, and prove guarantees on its statistical performance. One application lies in anomaly detection: the samples of noise, unlikely to have been generated by our estimate of the unknown dynamics, can be flagged to operators of the system for further study.

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On the Sample Complexity of the Linear Quadratic Regulator

et al. 2019

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This paper addresses the optimal control problem known as the Linear Quadratic Regulator in the case when the dynamics are unknown. We propose a multi-stage procedure, called Coarse-ID control, that estimates a model from a few experimental trials, estimates the error in that model with respect to the truth, and then designs a controller using both the model and uncertainty estimate. Our technique uses contemporary tools from random matrix theory to bound the error in the estimation procedure. We also employ a recently developed approach to control synthesis called System Level Synthesis that enables robust control design by solving a quasiconvex optimization problem. We provide end-to-end bounds on the relative error in control cost that are optimal in the number of parameters and that highlight salient properties of the system to be controlled such as closed-loop sensitivity and optimal control magnitude. We show experimentally that the Coarse-ID approach enables efficient computation of a stabilizing controller in regimes where simple control schemes that do not take the model uncertainty into account fail to stabilize the true system.

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Spectral Filtering for General Linear Dynamical Systems

Cited by 12 publications

References 11 publications

Time varying regression with hidden linear dynamics

Time varying regression with hidden linear dynamics

Robust Spectral Filtering and Anomaly Detection

On the Sample Complexity of the Linear Quadratic Regulator

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