# Rate-Optimal Non-Asymptotics for the Quadratic Prediction Error Method

arXiv: https://arxiv.org/abs/2404.07937 · HTML: https://mlsanity.com/paper/2404.07937/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 43 references that are in the map.
- First posted on arXiv 11 Apr 2024.

## Builds on (most cited first, 14 of 14)

- [Learning Without Mixing: Towards A Sharp Analysis of Linear System Identification](https://mlsanity.com/paper/1802.08334/) (2018)
- [Active Learning for Nonlinear System Identification with Guarantees](https://mlsanity.com/paper/2006.10277/) (2020)
- [Finite Sample Analysis of Stochastic System Identification](https://mlsanity.com/paper/1903.09122/) (2019)
- [Finite Time Identification in Unstable Linear Systems](https://mlsanity.com/paper/1710.01852/) (2017)
- [Statistical Learning Theory for Control: A Finite Sample Perspective](https://mlsanity.com/paper/2209.05423/) (2022)
- [Learning nonlinear dynamical systems from a single trajectory](https://mlsanity.com/paper/2004.14681/) (2020)
- [Non-asymptotic and Accurate Learning of Nonlinear Dynamical Systems](https://mlsanity.com/paper/2002.08538/) (2020)
- [Learning with Square Loss: Localization through Offset Rademacher Complexity](https://mlsanity.com/paper/1502.06134/) (2015)
- [Learning with little mixing](https://mlsanity.com/paper/2206.08269/) (2022)
- [Single Trajectory Nonparametric Learning of Nonlinear Dynamics](https://mlsanity.com/paper/2202.08311/) (2022)
- [A Tutorial on the Non-Asymptotic Theory of System Identification](https://mlsanity.com/paper/2309.03873/) (2023)
- [Near-optimal Offline and Streaming Algorithms for Learning Non-Linear Dynamical Systems](https://mlsanity.com/paper/2105.11558/) (2021)
- [On Empirical Risk Minimization with Dependent and Heavy-Tailed Data](https://mlsanity.com/paper/2109.02224/) (2021)
- [Sharp Rates in Dependent Learning Theory: Avoiding Sample Size Deflation for the Square Loss](https://mlsanity.com/paper/2402.05928/) (2024)

## Built on by (most cited first, 3 of 3)

- [Policy Gradient Bounds in Multitask LQR](https://mlsanity.com/paper/2509.19266/) (2025)
- [Multitask LQG Control: Performance and Generalization Bounds](https://mlsanity.com/paper/2604.16730/) (2026)
- [Adaptive prediction theory combining offline and online learning](https://mlsanity.com/paper/2512.00342/) (2025)

## Abstract (arXiv)

We study the quadratic prediction error method -- i.e., nonlinear least squares -- for a class of time-varying parametric predictor models satisfying a certain identifiability condition. While this method is known to asymptotically achieve the optimal rate for a wide range of problems, there have been no non-asymptotic results matching these optimal rates outside of a select few, typically linear, model classes. By leveraging modern tools from learning with dependent data, we provide the first rate-optimal non-asymptotic analysis of this method for our more general setting of nonlinearly parametrized model classes. Moreover, we show that our results can be applied to a particular class of identifiable AutoRegressive Moving Average (ARMA) models, resulting in the first optimal non-asymptotic rates for identification of ARMA models.