带输入和次高斯噪声的部分观测随机线性时不变状态空间系统的PAC-Bayes界
PAC-Bayesian Bounds for Learning Partially Observed Stochastic Linear Time-Invariant State-Space Systems with Inputs and Sub-Gaussian Noise
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中文总结 AI 辅助
本文为带输入和次高斯噪声的部分观测LTI状态空间系统推导PAC-Bayes误差界，连接期望预测误差与训练数据误差，并推广至参数估计误差，为系统辨识算法提供有限样本界，且为RNN的PAC-Bayes界奠定基础。
中文摘要 AI 辅助
本文推导了带输入和次高斯噪声的状态空间形式的部分观测线性时不变（LTI）随机动力系统的可能近似正确（PAC）-Bayes误差界。此类界在机器学习中广泛存在，对于刻画从有限数据点学习到的模型的预测能力非常有用。本文推导的界将预测误差的期望与模型在学习所用数据上产生的预测误差联系起来。此外，我们表明该界还可用于推导参数估计误差的界。这进而使我们能够为一大类系统辨识算法提供预测误差和参数估计误差的有限样本误差界。此外，由于LTI系统是循环神经网络（RNN）的子类，这些误差界可能是迈向RNN的PAC-Bayes界的第一步。
英文摘要
In this paper we derive a Probably Approximately Correct (PAC)-Bayesian error bound for partially observed linear time-invariant (LTI) stochastic dynamical systems in state-space form with inputs and sub-Gaussian noise. Such bounds are widespread in machine learning, and they are useful for characterizing the predictive power of models learned from finitely many data points. The bound derived in this paper relates the expectation of prediction errors with the prediction error generated by the model on the data used for learning. In addition, we show that it can also be used to derive bounds for the parameter estimation error. In turn, this allows us to provide finite-sample error bounds for the prediction error and parameter estimation error for a wide class of system identification algorithms. Furthermore, as LTI systems are a sub-class of recurrent neural networks (RNNs), these error bounds could be a first step towards PAC-Bayesian bounds for RNNs.
发表机构
- CRIStAL, Centrale Lille, Université de Lille（里尔中央理工学院，里尔大学，CRIStAL）
- Department of Electronic Systems, Aalborg University（奥尔堡大学电子系统系）
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