提出长时序线性系统识别新方法，突破慢混合限制。
Long-Context Linear System Identification
- 基于固定上下文窗口的线性依赖建模，实现高效学习。
- 样本复杂度逼近独立同分布最优率，且不依赖系统混合速度。
- 适用于低秩结构与上下文误设场景，适合动态系统建模研究者。
本文研究长时序线性系统识别问题，其中系统状态 $x_t$ 在时间 $t$ 上线性依赖于过去 $p$ 个时刻的状态。我们建立了样本复杂度上界，对一大类系统而言，该上界在对数因子内达到 i.i.d. 参数化率的最优水平，扩展了以往仅考虑一阶依赖的研究。结果揭示了一种‘无需混合即能学习’的现象，表明长时序自回归模型的学习不受扩展上下文窗可能带来的慢混合性质阻碍。此外，我们将结果推广至 (i) 共享低秩表示情形，其中秩正则化估计器改善了速率对维度的依赖；(ii) 严格稳定系统中上下文长度误设的情形，发现更短的上下文反而带来统计优势。
原文摘要 · Abstract (English)
This paper addresses the problem of long-context linear system identification, where the state $x_t$ of a dynamical system at time $t$ depends linearly on previous states $x_s$ over a fixed context window of length $p$. We establish a sample complexity bound that matches the i.i.d. parametric rate up to logarithmic factors for a broad class of systems, extending previous works that considered only first-order dependencies. Our findings reveal a learning-without-mixing phenomenon, indicating that learning long-context linear autoregressive models is not hindered by slow mixing properties potentially associated with extended context windows. Additionally, we extend these results to (i) shared low-rank representations, where rank-regularized estimators improve the dependence of the rates on the dimensionality, and (ii) misspecified context lengths in strictly stable systems, where shorter contexts offer statistical advantages.
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