Online Learning of Kalman Filtering: From Output to State Estimation

Lintao Ye; Ankang Zhang; Ming Chi; Bin Du; Jianghai Hu

cs.LGMar 28, 2026

Online Learning of Kalman Filtering: From Output to State Estimation

Authors: Lintao Ye, Ankang Zhang, Ming Chi, Bin Du, Jianghai Hu

Organizations: School of Artificial Intelligence and Automation at the Huazhong University of Science and Technology, Wuhan, China · College of Automation Engineering at Nanjing University of Aeronautics and Astronautics, Nanjing, China · Elmore Family School of Electrical and Computer Engineering at Purdue University, West Lafayette, IN, USA

Abstract

In this paper, we study the problem of learning Kalman filtering with unknown system model in partially observed linear dynamical systems. We propose a unified algorithmic framework based on online optimization that can be used to solve both the output estimation and state estimation scenarios. By exploring the properties of the estimation error cost functions, such as conditionally strong convexity, we show that our algorithm achieves a $\log T$ -regret in the horizon length $T$ for the output estimation scenario. More importantly, we tackle the more challenging scenario of learning Kalman filtering for state estimation, which is an open problem in the literature. We first characterize a fundamental limitation of the problem, demonstrating the impossibility of any algorithm to achieve sublinear regret in $T$ . By further introducing a random query scheme into our algorithm, we show that a $\sqrt{T}$ -regret is achievable when rendering the algorithm limited query access to more informative measurements of the system state in practice. Our algorithm and regret readily capture the trade-off between the number of queries and the achieved regret, and shed light on online learning problems with limited observations. We validate the performance of our algorithms using numerical examples.

Figures & tables

Panel of Figure 1 — Figure 1: The regret of Algorithm 1 for learning Kalman filtering for output estimation.

Panel of Figure 2 — Figure 2: The regret of Algorithm 2 for learning Kalman filtering for state estimation.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Notation	Definition
$\mathcal{K}_{N},\mathcal{K}_{M}$	Sets of candidate filter parameters for output and state estimation, resp.
$R_{N},R_{M}$	Upper bounds on the norm of filter parameters in $\mathcal{K}_{N}$ and $\mathcal{K}_{M}$ , resp.
$\beta_{y},l_{y}$	Smooth and Lipschitz parameters of $f_{t}(N)$ in output estimation, resp.
$R_{x},R_{y}$	Upper bounds on the norm of state and output of system ( 1 ), resp.
$R_{\tilde{v}}$	Upper bound on the noise in the informative measurements given by ( 38 )
$\alpha_{0},\alpha_{1},\psi,\bar{\sigma}$	Bounds on system parameters given in ( 13 )

Appendix

Table 1: Notations

Kalman Filtering

Kalman filtering recursively estimates hidden states by balancing model predictions with noisy measurements.

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Open paper AskPDF arXiv

Auto-regressive models have emerged as powerful tools for sequential data, from language to video. Understanding how and why these models learn latent representations remains an open theoretical question. In this work, we demonstrate that when trained by empirical risk minimization on data from partially observed linear dynamical systems, two-layer linear auto-regressive models naturally learn to approximate Kalman filtering. In particular, we show that the learned hidden representation coincides, up to a similarity transformation, with the state estimates produced by the optimal (Kalman) filter, even though the model has no explicit knowledge of the underlying dynamics or state. The result follows from three main insights. First, we establish that the Kalman filter is well approximated by an auto-regressive model with bounded truncation error. Second, we show that despite non-convexity, the two-layer optimization landscape is benign, i.e., all stationary points are either strict saddles or global minima. Finally, as our main contributions, we provide finite-sample guarantees on prediction error, parameter estimation error, and latent state recovery. Numerical simulations support the theoretical results and demonstrate that the latent representations of auto-regressive models recover state estimates.

Abstract

Kalman Filtering

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