从单条轨迹学习可切换非线性动态系统，给出首个非渐近理论保证。
Learning switched non-linear dynamical systems from a single trajectory
- 基于函数类度量熵推导预测风险上界
- 收敛速率依赖有效样本量 $Tp_i$，与模式出现概率相关
- 适用于多模式动态系统建模，理论严谨且有仿真验证
我们研究了从单一轨迹中学习具有随时间切换的非线性动态系统的经验风险最小化问题。在稳定性假设下，且模式切换满足独立同分布（i.i.d.）于 $K$ 个模式的情形，我们得到了以函数类度量熵表示的预测风险非渐近界。针对 Hölder 和线性函数类进行了具体实例化，获得显式收敛速率，其依赖于有效样本量 $Tp_i$，其中 $T$ 为轨迹长度，$p_i$ 为模式 $i$ 的观测概率。数值模拟支持了理论结果。据我们所知，这是首个针对从单条轨迹学习切换非线性动态系统的非渐近保证。
原文摘要 · Abstract (English)
We study empirical risk minimization for learning non-linear dynamical systems whose transition dynamics may switch over time. Under stability assumptions, and i.i.d switching over a set of $K$ modes, we derive non-asymptotic bounds on the prediction risk expressed in terms of the metric entropy of the underlying function class. We instantiate our general result for Hölder and linear function classes, obtaining explicit convergence rates that depend on the effective sample size $Tp_i$, where $T$ is the trajectory length and $p_i$ is the probability of observing mode $i$. Numerical simulations support our theoretical findings. To the best of our knowledge, these results are the first non-asymptotic guarantees for learning switched nonlinear dynamical systems from a single trajectory.
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