用指数遗忘提升未知系统在线预测精度，降低误差累积。
Model-free Online Learning for the Kalman Filter: Forgetting Factor and Logarithmic Regret
- 引入指数遗忘机制平衡回归模型，避免过拟合。
- 实现 $O(\log^3 N)$ 的更优对数后悔界，性能随观测数提升。
- 适合在线学习中系统模型未知的场景，如自适应控制与信号处理。
研究未知非爆炸性线性随机系统的在线预测问题。已知系统模型时，最优预测器为著名的卡尔曼滤波器。当系统未知时，基于递归最小二乘及其变体的方法可能因回归模型严重失衡而性能下降，易导致过拟合并降低预测准确率。本文通过指数遗忘注入归纳偏置，使回归模型保持平衡。不同于通常用于重加权数据的指数遗忘，本方法聚焦于调节回归与正则化误差间的权衡，同时减少累积误差。借助新的证明技巧，我们给出了更紧的对数后悔界 $O(\log^3 N)$，其中 $N$ 为观测次数。
原文摘要 · Abstract (English)
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.
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