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

2025-05-13

AI Summary

This paper addresses the online Kalman prediction problem for unknown, non-explosive linear stochastic systems, where conventional recursive least squares (RLS)-based methods suffer from overfitting and performance degradation due to ill-conditioned regression matrices. We propose a model-free online prediction framework that—uniquely—employs an exponential forgetting mechanism not merely for data weighting, but to dynamically balance regression model structure. Integrated with refined recursive updates and a novel error decomposition analysis, we rigorously establish a regret upper bound of $O(log^3 N)$, markedly improving upon the $O(sqrt{N})$ bound of RLS-type methods. The theoretical analysis guarantees stable learning under minimal assumptions. Empirical evaluations demonstrate over 37% reduction in prediction error. Our work provides both tight theoretical guarantees and an efficient implementation for high-accuracy, real-time state estimation in the absence of prior system knowledge.

Technology Category

Machine Learning: Online Learning & BanditsIntelligent Robots: State EstimationReasoning under Uncertainty: Stochastic Optimization

Application Category

Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingGraph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSystems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

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 $O(log^3 N)$, where $N$ is the number of observations.

Problem

• Online prediction for unknown linear stochastic systems
• Addressing imbalance in regression to prevent overfitting
• Achieving better error trade-off with exponential forgetting

Innovation

• Uses exponential forgetting for model balance
• Balances regression and regularization errors
• Achieves logarithmic regret bound O(log^3 N)