# On-Line Learning of Linear Dynamical Systems: Exponential Forgetting in Kalman Filters

arXiv: https://arxiv.org/abs/1809.05870 · HTML: https://mlsanity.com/paper/1809.05870/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Sep 2018.

## Built on by (most cited first, 16 of 16)

- [Statistical Learning Theory for Control: A Finite Sample Perspective](https://mlsanity.com/paper/2209.05423/) (2022)
- [Linear Systems can be Hard to Learn](https://mlsanity.com/paper/2104.01120/) (2021)
- [Streaming Linear System Identification with Reverse Experience Replay](https://mlsanity.com/paper/2103.05896/) (2021)
- [Can Transformers Learn Optimal Filtering for Unknown Systems?](https://mlsanity.com/paper/2308.08536/) (2023)
- [Kalman Filter for Online Classification of Non-Stationary Data](https://mlsanity.com/paper/2306.08448/) (2023)
- [Online Linear Regression in Dynamic Environments via Discounting](https://mlsanity.com/paper/2405.19175/) (2024)
- [Flash STU: Fast Spectral Transform Units](https://mlsanity.com/paper/2409.10489/) (2024)
- [Universal Sequence Preconditioning](https://mlsanity.com/paper/2502.06545/) (2025)
- [Predictive Linear Online Tracking for Unknown Targets](https://mlsanity.com/paper/2402.10036/) (2024)
- [Universal Learning of Nonlinear Dynamics](https://mlsanity.com/paper/2508.11990/) (2025)
- [Fairness in Forecasting of Observations of Linear Dynamical Systems](https://mlsanity.com/paper/2209.05274/) (2022)
- [Model-free Online Learning for the Kalman Filter: Forgetting Factor and Logarithmic Regret](https://mlsanity.com/paper/2505.08982/) (2025)
- [The Complexity of Sequential Prediction in Dynamical Systems](https://mlsanity.com/paper/2402.06614/) (2024)
- [Learning Linear Models Using Distributed Iterative Hessian Sketching](https://mlsanity.com/paper/2112.04101/) (2021)
- [Optimisation Strategies for Ensuring Fairness in Machine Learning: With and Without Demographics](https://mlsanity.com/paper/2411.09056/) (2024)
- [Online Nonstochastic Prediction: Logarithmic Regret via Predictive Online Least Squares](https://mlsanity.com/paper/2605.04364/) (2026)

## Abstract (arXiv)

Kalman filter is a key tool for time-series forecasting and analysis. We show that the dependence of a prediction of Kalman filter on the past is decaying exponentially, whenever the process noise is non-degenerate. Therefore, Kalman filter may be approximated by regression on a few recent observations. Surprisingly, we also show that having some process noise is essential for the exponential decay. With no process noise, it may happen that the forecast depends on all of the past uniformly, which makes forecasting more difficult. Based on this insight, we devise an on-line algorithm for improper learning of a linear dynamical system (LDS), which considers only a few most recent observations. We use our decay results to provide the first regret bounds w.r.t. to Kalman filters within learning an LDS. That is, we compare the results of our algorithm to the best, in hindsight, Kalman filter for a given signal. Also, the algorithm is practical: its per-update run-time is linear in the regression depth.