从单条有限轨迹学习遍历系统的动态，突破传统独立同分布假设限制。
Learning Ergodic Dynamical Systems from a Finite Trajectory
- 用非线性最小二乘法估计最优一步预测函数，考虑轨迹相关性。
- 在不变测度下给出高概率保证，揭示非独立数据对学习的影响。
- 方法可扩展至高阶系统与柯尔普曼算子，适合研究复杂动力系统者。
我们研究从一个遍历随机动力系统的单一有限轨迹中学习的问题。具体而言，关注定义时间齐次马尔可夫过程的离散时间自治随机系统。首先，通过非线性最小二乘法估计最优一步预测函数，并在过程的不变测度下建立高概率保证。这些结果明确揭示了轨迹数据的非独立、非同分布特性如何改变经典统计学习分析。随后，将框架扩展至高阶系统和有限状态空间。最后，证明相同的最小二乘与集中性论证自然适用于学习柯尔普曼算子。该方法结合了统计学习理论与马尔可夫链的定量遍历理论，特别依赖于希尔伯特空间值加性泛函在均匀几何遍历马尔可夫链下的集中不等式。
原文摘要 · Abstract (English)
We consider the problem of learning from a single finite trajectory of an ergodic stochastic dynamical system. More precisely, we study discrete-time autonomous stochastic systems defining time-homogeneous Markov processes. We first focus on estimating the optimal one-step prediction function by nonlinear least squares, and derive high-probability guarantees measured with respect to the invariant measure of the process. These results make explicit how the non-independent and non-identically distributed nature of trajectory data modifies the classical statistical learning analysis. We then extend the framework to higher-order systems and finite-state spaces. Finally, we show that the same least squares and concentration arguments naturally extend to learning Koopman operators. Our approach combines tools from statistical learning theory and quantitative ergodic theory for Markov chains. It relies, in particular, on a concentration inequality for Hilbert-space-valued additive functionals of uniformly geometrically ergodic Markov chains.
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