Empirical risk minimization and complexity of dynamical models
Empirical risk minimization and complexity of dynamical models
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DOI：
10.1214/19-aos1876
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发表时间：
2016-11
期刊：
影响因子：
--
通讯作者：
K. Mcgoff;A. Nobel
中科院分区：
文献类型：
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作者：
K. Mcgoff;A. Nobel
一个动力学模型由紧致状态空间$\mathcal {X}$的连续自映射$T：\mathcal {X} \to \mathcal{X}$和连续观测函数$f：\mathcal{X} \to \mathbb{R}$组成。本文考虑了用经验风险最小化方法拟合一个参数化的动力学模型族到一个实值随机过程。最小风险参数的极限行为进行了研究，在一般设置。建立了最小风险估计和遍历观测的一般收敛定理。然后，我们研究的条件下，经验风险最小化可以有效地分离的信号从噪声中的加性观测噪声模型。在后者的结果的关键，必要条件是，家庭的动力学模型具有有限的复杂性，这是量化的概念，通过熵的家庭的无限序列。建立了平稳过程的熵和极限平均宽度之间的密切联系。
A dynamical model consists of a continuous self-map $T: \mathcal{X} \to \mathcal{X}$ of a compact state space $\mathcal{X}$ and a continuous observation function $f: \mathcal{X} \to \mathbb{R}$. This paper considers the fitting of a parametrized family of dynamical models to an observed real-valued stochastic process using empirical risk minimization. The limiting behavior of the minimum risk parameters is studied in a general setting. We establish a general convergence theorem for minimum risk estimators and ergodic observations. We then study conditions under which empirical risk minimization can effectively separate the signal from the noise in an additive observational noise model. The key, necessary condition in the latter results is that the family of dynamical models has limited complexity, which is quantified through a notion of entropy for families of infinite sequences. Close connections between entropy and limiting average mean widths for stationary processes are established.