# Learning Linear Dynamical Systems with Semi-Parametric Least Squares

arXiv: https://arxiv.org/abs/1902.00768 · HTML: https://mlsanity.com/paper/1902.00768/

## Key facts

- Cited by 58 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Feb 2019.

## Built on by (most cited first, 30 of 58)

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- [Finite Sample Identification of Bilinear Dynamical Systems](https://mlsanity.com/paper/2208.13915/) (2022)
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- [Learning from Censored and Dependent Data: The case of Linear Dynamics](https://mlsanity.com/paper/2104.05087/) (2021)
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- [Learning Dynamical Systems by Leveraging Data from Similar Systems](https://mlsanity.com/paper/2302.04344/) (2023)
- [Regret Analysis of Distributed Online LQR Control for Unknown LTI Systems](https://mlsanity.com/paper/2105.07310/) (2021)
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- [Learning Linear Dynamics from Bilinear Observations](https://mlsanity.com/paper/2409.16499/) (2024)
- [Finite Sample Identification of Partially Observed Bilinear Dynamical Systems](https://mlsanity.com/paper/2501.07652/) (2025)
- [Universal Learning of Nonlinear Dynamics](https://mlsanity.com/paper/2508.11990/) (2025)
- [Robust Online Control with Model Misspecification](https://mlsanity.com/paper/2107.07732/) (2021)
- [Finite-time System Identification and Adaptive Control in Autoregressive Exogenous Systems](https://mlsanity.com/paper/2108.11959/) (2021)
- [Time varying regression with hidden linear dynamics](https://mlsanity.com/paper/2112.14862/) (2021)

## Abstract (arXiv)

We analyze a simple prefiltered variation of the least squares estimator for the problem of estimation with biased, 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 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.