# Model-free Online Learning for the Kalman Filter: Forgetting Factor and Logarithmic Regret

arXiv: https://arxiv.org/abs/2505.08982 · HTML: https://mlsanity.com/paper/2505.08982/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 42 references that are in the map.
- First posted on arXiv 13 May 2025.

## Builds on (most cited first, 19 of 19)

- [Gradient Descent Learns Linear Dynamical Systems](https://mlsanity.com/paper/1609.05191/) (2016)
- [Finite Sample Analysis of Stochastic System Identification](https://mlsanity.com/paper/1903.09122/) (2019)
- [Learning Linear Dynamical Systems via Spectral Filtering](https://mlsanity.com/paper/1711.00946/) (2017)
- [Logistic Regression: The Importance of Being Improper](https://mlsanity.com/paper/1803.09349/) (2018)
- [Non-asymptotic Identification of Linear Dynamical Systems Using Multiple Trajectories](https://mlsanity.com/paper/2009.00739/) (2020)
- [Long Short-Term Memory Kalman Filters:Recurrent Neural Estimators for Pose Regularization](https://mlsanity.com/paper/1708.01885/) (2017)
- [Online Learning for Time Series Prediction](https://mlsanity.com/paper/1302.6927/) (2013)
- [Online Learning of the Kalman Filter with Logarithmic Regret](https://mlsanity.com/paper/2002.05141/) (2020)
- [No-Regret Prediction in Marginally Stable Systems](https://mlsanity.com/paper/2002.02064/) (2020)
- [On-Line Learning of Linear Dynamical Systems: Exponential Forgetting in Kalman Filters](https://mlsanity.com/paper/1809.05870/) (2018)
- [Improved rates for prediction and identification of partially observed linear dynamical systems](https://mlsanity.com/paper/2011.10006/) (2020)
- [Discounted Adaptive Online Learning: Towards Better Regularization](https://mlsanity.com/paper/2402.02720/) (2024)
- [SLIP: Learning to Predict in Unknown Dynamical Systems with Long-Term Memory](https://mlsanity.com/paper/2010.05899/) (2020)
- [Online Linear Regression in Dynamic Environments via Discounting](https://mlsanity.com/paper/2405.19175/) (2024)
- [A Linear Dynamical System Model for Text](https://mlsanity.com/paper/1502.04081/) (2015)
- [Predictive Linear Online Tracking for Unknown Targets](https://mlsanity.com/paper/2402.10036/) (2024)
- [Willems' Fundamental Lemma for Nonlinear Systems with Koopman Linear Embedding](https://mlsanity.com/paper/2409.16389/) (2024)
- [Generalized Forgetting Recursive Least Squares: Stability and Robustness Guarantees](https://mlsanity.com/paper/2308.04259/) (2023)
- [Regret Analysis with Almost Sure Convergence for OBF-ARX Filter](https://mlsanity.com/paper/2409.05390/) (2024)

## Built on by (most cited first, 4 of 4)

- [Logarithmic Regret and Polynomial Scaling in Online Multi-step-ahead Prediction](https://mlsanity.com/paper/2511.12467/) (2025)
- [Gradient Dominance in the Linear Quadratic Regulator: A Unified Analysis for Continuous-Time and Discrete-Time Systems](https://mlsanity.com/paper/2602.22577/) (2026)
- [Online Learning of Kalman Filtering: From Output to State Estimation](https://mlsanity.com/paper/2603.27159/) (2026)
- [Online Nonstochastic Prediction: Logarithmic Regret via Predictive Online Least Squares](https://mlsanity.com/paper/2605.04364/) (2026)

## Abstract (arXiv)

We consider the problem of online prediction for an unknown, non-explosive linear stochastic system. With a known system model, the optimal predictor is the celebrated Kalman filter. In the case of unknown systems, existing approaches based on recursive least squares and its variants may suffer from degraded performance due to the highly imbalanced nature of the regression model. This imbalance can easily lead to overfitting and thus degrade prediction accuracy. We tackle this problem by injecting an inductive bias into the regression model via exponential forgetting. While exponential forgetting is a common wisdom in online learning, it is typically used for re-weighting data. In contrast, our approach focuses on balancing the regression model. This achieves a better trade-off between regression and regularization errors, and simultaneously reduces the accumulation error. With new proof techniques, we also provide a sharper logarithmic regret bound of O(log3 N), where N is the number of observations.