Sample Complexity Lower Bounds for Linear System Identification

Jedra, Yassir; Proutière, Alexandre

doi:10.48550/arxiv.1903.10343

Cited by 3 publications

(7 citation statements)

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“…Proof. The proof of this result is essentially identical to the proof of Theorem 1 in [23] and we omit it here.…”

Section: G Lower Boundmentioning

confidence: 86%

“…We base our analysis off the lower bound presented in [23]. A slight modification of their analysis to our situation yields the following result.…”

Section: G Lower Boundmentioning

confidence: 97%

“…The sample complexity τ δ is the infimum of all times τ satisfying the above definition. This condition was introduced in [23] and allows us to avoid trivial algorithms that simply return Ât = A * for all time. Also define:…”

Section: Asymptotic Optimality Of Algorithmmentioning

confidence: 99%

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Active Learning for Identification of Linear Dynamical Systems

Wagenmaker¹,

Jamieson²

2020

Preprint

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We propose an algorithm to actively estimate the parameters of a linear dynamical system. Given complete control over the system's input, our algorithm adaptively chooses the inputs to accelerate estimation. We show a finite time bound quantifying the estimation rate our algorithm attains and prove matching upper and lower bounds which guarantee its asymptotic optimality, up to constants. In addition, we show that this optimal rate is unattainable when using Gaussian noise to excite the system, even with optimally tuned covariance, and analyze several examples where our algorithm provably improves over rates obtained by playing noise. Our analysis critically relies on a novel result quantifying the error in estimating the parameters of a dynamical system when arbitrary periodic inputs are being played. We conclude with numerical examples that illustrate the effectiveness of our algorithm in practice.

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“…Proof. The proof of this result is essentially identical to the proof of Theorem 1 in [23] and we omit it here.…”

Section: G Lower Boundmentioning

confidence: 86%

“…We base our analysis off the lower bound presented in [23]. A slight modification of their analysis to our situation yields the following result.…”

Section: G Lower Boundmentioning

confidence: 97%

See 1 more Smart Citation

Active Learning for Identification of Linear Dynamical Systems

Wagenmaker¹,

Jamieson²

2020

Preprint

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“…trajectories. There is also a substantial body of literature on (sub-)optimal finite-sample concentration bounds for linear systems identified via least squares estimation (Simchowitz et al 2018;Jedra and Proutiere 2019;Sarkar and Rakhlin 2019;Jedra and Proutiere 2020;Sarkar, Rakhlin, and Dahleh 2020). These approaches offer fast learning rates but cannot guarantee stability of the identified systems for finite sample sizes.…”

Section: Related Workmentioning

confidence: 99%

Efficient Learning of a Linear Dynamical System with Stability Guarantees

Jongeneel¹,

Sutter²,

Kühn³

2021

Preprint

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We propose a principled method for projecting an arbitrary square matrix to the nonconvex set of asymptotically stable matrices. Leveraging ideas from large deviations theory, we show that this projection is optimal in an information-theoretic sense and that it simply amounts to shifting the initial matrix by an optimal linear quadratic feedback gain, which can be computed exactly and highly efficiently by solving a standard linear quadratic regulator problem. The proposed approach allows us to learn the system matrix of a stable linear dynamical system from a single trajectory of correlated state observations. The resulting estimator is guaranteed to be stable and offers explicit statistical bounds on the estimation error.

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“…Recently, results in statistical learning theory, self-normalizing processes [3] and high dimensional probability [4], have motivated control, signal processing, and machine learning communities to explore finite-time properties of linear system estimates by least squares. Among topics of interest, we can find sample complexity bounds [5], 1 − δ probability bounds on parameter errors [6], confidence bounds [7,Chap. 20], Cramér-Rao lower bounds [8], and bounds on quadratic criterion cost deviation [9,10].…”

Section: Introductionmentioning

confidence: 99%

Finite Sample Deviation and Variance Bounds for First Order Autoregressive Processes

González

Rojas

2020

ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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In this paper, we study non-asymptotic deviation bounds of the least squares estimator in Gaussian AR(n) processes. By relying on martingale concentration inequalities and a tail-bound for χ 2 distributed variables, we provide a concentration bound for the sample covariance matrix of the process output. With this, we present a problem-dependent finite-time bound on the deviation probability of any fixed linear combination of the estimated parameters of the AR(n) process. We discuss extensions and limitations of our approach.

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Sample Complexity Lower Bounds for Linear System Identification

Cited by 3 publications

References 0 publications

Active Learning for Identification of Linear Dynamical Systems

Active Learning for Identification of Linear Dynamical Systems

Efficient Learning of a Linear Dynamical System with Stability Guarantees

Finite Sample Deviation and Variance Bounds for First Order Autoregressive Processes

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