无混合学习：线性系统辨识的精确分析
Learning Without Mixing: Towards A Sharp Analysis of Linear System Identification
摘要：
我们证明了普通的最小二乘（OLS）估计达到近极小极大最优性能的线性动力系统从一个单一的观察轨迹的识别。我们的上限依赖于一个广义的门德尔松的小球方法依赖的数据，避免使用标准的混合时间参数。我们的下限表明，这些上限匹配对数因子。特别是，我们捕捉到正确的信号噪声行为的问题，表明更不稳定的线性系统更容易估计。这种行为是定性不同的参数依赖于混合时间的计算，表明不稳定的系统更难以估计。我们推广我们的技术，以提供更一般的一类线性响应时间序列的界限。
英文摘要：
We prove that the ordinary least-squares (OLS) estimator attains nearly minimax optimal performance for the identification of linear dynamical systems from a single observed trajectory. Our upper bound relies on a generalization of Mendelson's small-ball method to dependent data, eschewing the use of standard mixing-time arguments. Our lower bounds reveal that these upper bounds match up to logarithmic factors. In particular, we capture the correct signal-to-noise behavior of the problem, showing that more unstable linear systems are easier to estimate. This behavior is qualitatively different from arguments which rely on mixing-time calculations that suggest that unstable systems are more difficult to estimate. We generalize our technique to provide bounds for a more general class of linear response time-series.
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