提出更精确的线性系统辨识误差界，突破现有方法对复杂度的高估。
CLT-Optimal Parameter Error Bounds for Linear System Identification
- 通过二阶分解重构参数误差，引入矩阵鞅项捕捉中心极限定理尺度。
- 在稳定系统与多轨迹设置下，误差界逼近实例最优率，仅差常数因子。
- 适用于系统辨识理论研究者，尤其关注有限样本精度的学者。
过去十年中，针对从观测行为中恢复未知系统参数的有限样本、非渐近误差界取得了显著进展。然而，令人惊讶的是，我们发现当前最先进的边界并未准确刻画系统辨识的统计复杂度，即使在最基础的离散时间线性动态系统（LDS）通过普通最小二乘法（OLS）估计的情形下也是如此。具体而言，利用渐近正态性，我们识别出一类问题实例，其中现有边界在谱范数和Frobenius范数下将平方参数误差高估了系统状态维度的倍数。基于此偏差，我们通过一种新颖的二阶参数误差分解，显著改进了OLS误差界：关键在于低阶项为一个矩阵值鞅，其正确捕捉了中心极限定理（CLT）的缩放特性。由此分析，我们得到了两类情形下的有限样本界：(i) 稳定系统；(ii) 多轨迹设置，其在Frobenius范数下与实例最优率仅差常数因子，在谱范数下差多项式对数量级的状态维度因子。
原文摘要 · Abstract (English)
There has been remarkable progress over the past decade in establishing finite-sample, non-asymptotic bounds on recovering unknown system parameters from observed system behavior. Surprisingly, however, we show that the current state-of-the-art bounds do not accurately capture the statistical complexity of system identification, even in the most fundamental setting of estimating a discrete-time linear dynamical system (LDS) via ordinary least-squares regression (OLS). Specifically, we utilize asymptotic normality to identify classes of problem instances for which current bounds overstate the squared parameter error, in both spectral and Frobenius norm, by a factor of the state-dimension of the system. Informed by this discrepancy, we then sharpen the OLS parameter error bounds via a novel second-order decomposition of the parameter error, where crucially the lower-order term is a matrix-valued martingale that we show correctly captures the CLT scaling. From our analysis we obtain finite-sample bounds for both (i) stable systems and (ii) the many-trajectories setting that match the instance-specific optimal rates up to constant factors in Frobenius norm, and polylogarithmic state-dimension factors in spectral norm.
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