Single Trajectory Nonparametric Learning of Nonlinear Dynamics
Single Trajectory Nonparametric Learning of Nonlinear Dynamics
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发表时间：
2022-02
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通讯作者：
Ingvar M. Ziemann;H. Sandberg;N. Matni
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作者：
Ingvar M. Ziemann;H. Sandberg;N. Matni
给定一个动力系统的单轨迹，我们分析了非参数最小二乘估计（LSE）的性能。更确切地说，我们给出了LSE和真实回归函数之间的非渐近期望的l^2 $-距离界，其中期望是在一个新的、反事实的轨迹上评估的。我们利用最近开发的信息理论方法，建立最优的LSE的非参数假设类的上确界范数度量熵和亚高斯参数。接下来，我们使用动力系统理论的概念将此亚高斯参数与底层过程的稳定性联系起来。当结合起来，这些发展导致率最优的误差界，规模为$T^{-1/（2+q）}$的适当稳定的过程和假设类的度量熵增长的顺序$\delta^{-q}$。这里，$T$是观察到的轨迹的长度，$\delta \in \mathbb{R}_+$是填充粒度，$q\in（0，2）$是复杂度项。最后，我们专门研究我们的结果，一些场景的实际利益，如Lipschitz动力学，广义线性模型，并在某些类的再生核希尔伯特空间（RKHS）的功能描述的动态。
Given a single trajectory of a dynamical system, we analyze the performance of the nonparametric least squares estimator (LSE). More precisely, we give nonasymptotic expected $l^2$-distance bounds between the LSE and the true regression function, where expectation is evaluated on a fresh, counterfactual, trajectory. We leverage recently developed information-theoretic methods to establish the optimality of the LSE for nonparametric hypotheses classes in terms of supremum norm metric entropy and a subgaussian parameter. Next, we relate this subgaussian parameter to the stability of the underlying process using notions from dynamical systems theory. When combined, these developments lead to rate-optimal error bounds that scale as $T^{-1/(2+q)}$ for suitably stable processes and hypothesis classes with metric entropy growth of order $\delta^{-q}$. Here, $T$ is the length of the observed trajectory, $\delta \in \mathbb{R}_+$ is the packing granularity and $q\in (0,2)$ is a complexity term. Finally, we specialize our results to a number of scenarios of practical interest, such as Lipschitz dynamics, generalized linear models, and dynamics described by functions in certain classes of Reproducing Kernel Hilbert Spaces (RKHS).