Improved rates for prediction and identification of partially observed linear dynamical systems

Lee, Holden

doi:10.48550/arxiv.2011.10006

Cited by 3 publications

(6 citation statements)

References 12 publications

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“…With the advances in high-dimensional statistics [6], there has been a recent shift from asymptotic analysis with infinite data to statistical analysis of system identification with finite samples. Over the past two years there have been significant advances in understanding finite sample system identification for both fully-observed systems [7][8][9][10][11][12][13][14] as well as partially-observed systems [15][16][17][18][19][20][21][22][23][24]. A tutorial can be found in [25].…”

Section: Introductionmentioning

confidence: 99%

Linear Systems can be Hard to Learn

Tsiamis¹,

Pappas²

2021

Preprint

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In this paper, we investigate when system identification is statistically easy or hard, in the finite sample regime. Statistically easy to learn linear system classes have sample complexity that is polynomial with the system dimension. Most prior research in the finite sample regime falls in this category, focusing on systems that are directly excited by process noise. Statistically hard to learn linear system classes have worst-case sample complexity that is at least exponential with the system dimension, regardless of the identification algorithm. Using tools from minimax theory, we show that classes of linear systems can be hard to learn. Such classes include, for example, under-actuated or under-excited systems with weak coupling among the states. Having classified some systems as easy or hard to learn, a natural question arises as to what system properties fundamentally affect the hardness of system identifiability. Towards this direction, we characterize how the controllability index of linear systems affects the sample complexity of identification. More specifically, we show that the sample complexity of robustly controllable linear systems is upper bounded by an exponential function of the controllability index. This implies that identification is easy for classes of linear systems with small controllability index and potentially hard if the controllability index is large. Our analysis is based on recent statistical tools for finite sample analysis of system identification as well as a novel lower bound that relates controllability index with the least singular value of the controllability Gramian.

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Section: Introductionmentioning

confidence: 99%

Linear Systems can be Hard to Learn

Tsiamis¹,

Pappas²

2021

Preprint

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“…Theorem 6 shows that the stochastic Ho-Kalman Algorithm has the same error bounds as its deterministic counterpart, which says the estimation errors for system matrix decrease as fast as O( 1N 1/4 ). Our analysis framework can be easily extended to achieve the optimal error bounds O( 1 √ N ) mentioned in [33,8,18,16].…”

Section: Resultsmentioning

confidence: 99%

“…This result can be improved to O( 1 √ N ) from [8,18,16]. Note that the computational complexity of the Ho-Kalman Algorithm in Alg.…”

Section: System Realization Via Noisy Markov Parameter Gmentioning

confidence: 98%

“…The process of obtaining estimates of the system matrices from the Markov parameters is referred to as system realization which is the main focus of this paper. Using this framework, the authors of [8,6,[15][16][17][18] have derived non-asymptotic estimation error bounds for the system parameters which decay at a rate O( 1 √ N ). Note that these papers make different assumptions about the stability, system order, and the number of required trajectories to excite the unknown system.…”

Section: Introductionmentioning

confidence: 99%

“…From the view of computational complexity, the system identification methods proposed in [8,6,[15][16][17][18] are not scalable since the size of the Hankel matrix increases quadratically with the length of output signal T and cubically with the system state dimension n. The result is the singular value decomposition used in the Ho-Kalman Algorithm cannot be computed. This quadratic/cubic dependence on the problem size greatly limits its application in large scale system identification problems.…”

Section: Introductionmentioning

confidence: 99%

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Large-Scale System Identification Using a Randomized SVD

Wang¹,

Anderson²

2021

Preprint

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Learning a dynamical system from input/output data is a fundamental task in the control design pipeline. In the partially observed setting there are two components to identification: parameter estimation to learn the Markov parameters, and system realization to obtain a state space model. In both sub-problems it is implicitly assumed that standard numerical algorithms such as the singular value decomposition (SVD) can be easily and reliably computed. When trying to fit a high-dimensional model to data, for example in the cyberphysical system setting, even computing an SVD is intractable. In this work we show that an approximate matrix factorization obtained using randomized methods can replace the standard SVD in the realization algorithm while maintaining the non-asymptotic (in dataset size) performance and robustness guarantees of classical methods. Numerical examples illustrate that for large system models, this is the only method capable of producing a model.

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Learning to Control Linear Systems can be Hard

Tsiamis¹,

Ziemann²,

Morari³

et al. 2022

Preprint

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In this paper, we study the statistical difficulty of learning to control linear systems. We focus on two standard benchmarks, the sample complexity of stabilization, and the regret of the online learning of the Linear Quadratic Regulator (LQR). Prior results state that the statistical difficulty for both benchmarks scales polynomially with the system state dimension up to systemtheoretic quantities. However, this does not reveal the whole picture. By utilizing minimax lower bounds for both benchmarks, we prove that there exist non-trivial classes of systems for which learning complexity scales dramatically, i.e. exponentially, with the system dimension. This situation arises in the case of underactuated systems, i.e. systems with fewer inputs than states. Such systems are structurally difficult to control and their system theoretic quantities can scale exponentially with the system dimension dominating learning complexity. Under some additional structural assumptions (bounding systems away from uncontrollability), we provide qualitatively matching upper bounds. We prove that learning complexity can be at most exponential with the controllability index of the system, that is the degree of underactuation.

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Improved rates for prediction and identification of partially observed linear dynamical systems

Cited by 3 publications

References 12 publications

Linear Systems can be Hard to Learn

Linear Systems can be Hard to Learn

Large-Scale System Identification Using a Randomized SVD

Learning to Control Linear Systems can be Hard

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