In this paper we derive a Probably Approxilmately Correct(PAC)-Bayesian error bound for linear time-invariant (LTI) stochastic dynamical systems with inputs. Such bounds are widespread in machine learning, and they are useful for characterizing the predictive power of models learned from finitely many data points. In particular, with the bound derived in this paper relates future average prediction errors with the prediction error generated by the model on the data used for learning. In turn, this allows us to provide finite-sample error bounds for a wide class of learning/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.

翻译：本文针对具有输入的线性时不变随机动态系统，推导了概率近似正确（PAC）-贝叶斯误差界。这类误差界在机器学习中广泛存在，对于刻画基于有限数据点学习所得模型的预测能力具有重要意义。特别地，本文推导的误差界将未来平均预测误差与模型在学习所用数据上产生的预测误差联系起来。进而，这使我们能够为广泛的学习/系统辨识算法提供有限样本误差界。此外，由于LTI系统是循环神经网络（RNN）的子类，这些误差界可视为向RNN的PAC-贝叶斯界迈出的第一步。