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Finite Sample Analysis of Stochastic System Identification

arXiv 1903.09122 ↗First posted 21 Mar 2019cs.LG cs.SY eess.SYcited by 67

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Abstract (arXiv)

In this paper, we analyze the finite sample complexity of stochastic system identification using modern tools from machine learning and statistics. An unknown discrete-time linear system evolves over time under Gaussian noise without external inputs. The objective is to recover the system parameters as well as the Kalman filter gain, given a single trajectory of output measurements over a finite horizon of length N. Based on a subspace identification algorithm and a finite number of N output samples, we provide non-asymptotic high-probability upper bounds for the system parameter estimation errors. Our analysis uses recent results from random matrix theory, self-normalized martingales and SVD robustness, in order to show that with high probability the estimation errors decrease with a rate of 1/√N. Our non-asymptotic bounds not only agree with classical asymptotic results, but are also valid even when the system is marginally stable.

Key facts. Cited by 67 papers in the map (data through 7 Oct 2026). First posted on arXiv 21 Mar 2019.

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Citing papers per year: 2021: 12; 2022: 19; 2023: 12; 2024: 12; 2025: 7; 2026: 5.

67 papers.

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  1. 2022Statistical Learning Theory for Control: A Finite Sample Perspectivecited by 52
  2. 2022Single Trajectory Nonparametric Learning of Nonlinear Dynamicscited by 26
  3. 2023A Tutorial on the Non-Asymptotic Theory of System Identificationcited by 26
  4. 2023A New Approach to Learning Linear Dynamical Systemscited by 23
  5. 2021Linear Systems can be Hard to Learncited by 20
  6. 2021Streaming Linear System Identification with Reverse Experience Replaycited by 17
  7. 2021Task-Optimal Exploration in Linear Dynamical Systemscited by 15
  8. 2022Finite Sample Identification of Bilinear Dynamical Systemscited by 12
  9. 2022Probabilities Are Not Enough: Formal Controller Synthesis for Stochastic Dynamical Models with Epistemic Uncertaintycited by 11
  10. 2022Learning to Control Linear Systems can be Hardcited by 10
  11. 2023Tensor Decompositions Meet Control Theory: Learning General Mixtures of Linear Dynamical Systemscited by 10
  12. 2021Learning from Censored and Dependent Data: The case of Linear Dynamicscited by 9
38 more
  1. 2023Smoothed Online Learning for Prediction in Piecewise Affine Systemscited by 9
  2. 2022Learning the Dynamics of Autonomous Linear Systems From Multiple Trajectoriescited by 8
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  4. 2022Efficient learning of hidden state LTI state space models of unknown ordercited by 7
  5. 2021Regret Analysis of Distributed Online LQR Control for Unknown LTI Systemscited by 6
  6. 2021Deep inference of latent dynamics with spatio-temporal super-resolution using selective backpropagation through timecited by 6
  7. 2024Learning Linear Dynamics from Bilinear Observationscited by 6
  8. 2023Oracle-Efficient Smoothed Online Learning for Piecewise Continuous Decision Makingcited by 5
  9. 2021Finite-time System Identification and Adaptive Control in Autoregressive Exogenous Systemscited by 4
  10. 2021Time varying regression with hidden linear dynamicscited by 4
  11. 2022Fairness in Forecasting of Observations of Linear Dynamical Systemscited by 4
  12. 2023Non-Asymptotic Pointwise and Worst-Case Bounds for Classical Spectrum Estimatorscited by 4
  13. 2023PAC-Bayes Generalisation Bounds for Dynamical Systems Including Stable RNNscited by 4
  14. 2024Outlier-Robust Linear System Identification Under Heavy-tailed Noisecited by 4
  15. 2025Finite Sample Identification of Partially Observed Bilinear Dynamical Systemscited by 4
  16. 2021Online Stochastic Gradient Descent Learns Linear Dynamical Systems from A Single Trajectorycited by 3
  17. 2022Stochastic Multi-armed Bandits with Non-stationary Rewards Generated by a Linear Dynamical Systemcited by 3
  18. 2022Distributed Online System Identification for LTI Systems Using Reverse Experience Replaycited by 3
  19. 2022Dynamical Linear Banditscited by 3
  20. 2023PAC-Bayesian bounds for learning LTI-ss systems with input from empirical losscited by 3
  21. 2024Stochastic Online Optimization for Cyber-Physical and Robotic Systemscited by 3
  22. 2024Optimal Transport-Assisted Risk-Sensitive Q-Learningcited by 3
  23. 2021Large-Scale System Identification Using a Randomized SVDcited by 2
  24. 2024Length independent generalization bounds for deep SSM architectures via Rademacher contraction and stability constraintscited by 2
  25. 2024State space models, emergence, and ergodicity: How many parameters are needed for stable predictions?cited by 2
  26. 2025Nonconvex Linear System Identification with Minimal State Representationcited by 2
  27. 2025Model-free Online Learning for the Kalman Filter: Forgetting Factor and Logarithmic Regretcited by 2
  28. 2026Spectral Filtering for Complex Linear Dynamical Systemscited by 2
  29. 2022Online greedy identification of linear dynamical systemscited by 1
  30. 2022Robust Data-Driven Output Feedback Control via Bootstrapped Multiplicative Noisecited by 1
  31. 2022A note on the smallest eigenvalue of the empirical covariance of causal Gaussian processescited by 1
  32. 2022Likelihood-based generalization of Markov parameter estimation and multiple shooting objectives in system identificationcited by 1
  33. 2022PAC-Bayesian-Like Error Bound for a Class of Linear Time-Invariant Stochastic State-Space Modelscited by 1
  34. 2023Regret Analysis of Distributed Online Control for LTI Systems with Adversarial Disturbancescited by 1
  35. 2023Regret Analysis of Learning-Based Linear Quadratic Gaussian Control with Additive Explorationcited by 1
  36. 2024Finite Sample Frequency Domain Identificationcited by 1
  37. 2024Rate-Optimal Non-Asymptotics for the Quadratic Prediction Error Methodcited by 1
  38. 2024Exact Recovery Guarantees for Parameterized Nonlinear System Identification Problem under Sparse Disturbances or Semi-Oblivious Attackscited by 1

50 most cited of 67 All 67 (JSON)