We study the quadratic prediction error method -- i.e., nonlinear least
squares -- for a class of time-varying parametric predictor models satisfying a
certain identifiability condition. While this method is known to asymptotically
achieve the optimal rate for a wide range of problems, there have been no
non-asymptotic results matching these optimal rates outside of a select few,
typically linear, model classes. By leveraging modern tools from learning with
dependent data, we provide the first rate-optimal non-asymptotic analysis of
this method for our more general setting of nonlinearly parametrized model
classes. Moreover, we show that our results can be applied to a particular
class of identifiable AutoRegressive Moving Average (ARMA) models, resulting in
the first optimal non-asymptotic rates for identification of ARMA models.

我们研究了二次预测误差方法，即非线性最小二乘法，针对一类满足一定可辨识条件的时变参数预测模型。通过利用依赖数据学习的现代工具，我们提供了这种方法在非线性参数化模型类的更一般设置下的首个速率最优的非渐近分析。此外，我们展示了我们的结果可应用于一类可辨识的自回归滑动平均（ARMA）模型，从而导致了 ARMA 模型鉴别的首个最优非渐近速率。

二次预测误差方法的最优速率非渐近特性

Rate-Optimal Non-Asymptotics for the Quadratic Prediction Error Method

Score-based methods for learning Bayesain networks(BN) aim to maximizing the
global score functions. However, if local variables have direct and indirect
dependence simultaneously, the global optimization on score functions misses
edges between variables with indirect dependent relationship, of which scores
are smaller than those with direct dependent relationship. In this paper, we
present an identifiability condition based on a determined subset of parents to
identify the underlying DAG. By the identifiability condition, we develop a
two-phase algorithm namely optimal-tuning (OT) algorithm to locally amend the
global optimization. In the optimal phase, an optimization problem based on
first-order Hilbert-Schmidt independence criterion (HSIC) gives an estimated
skeleton as the initial determined parents subset. In the tuning phase, the
skeleton is locally tuned by deletion, addition and DAG-formalization
strategies using the theoretically proved incremental properties of high-order
HSIC. Numerical experiments for different synthetic datasets and real-world
datasets show that the OT algorithm outperforms existing methods. Especially in
Sigmoid Mix model with the size of the graph being ${\rm\bf d=40}$, the
structure intervention distance (SID) of the OT algorithm is 329.7 smaller than
the one obtained by CAM, which indicates that the graph estimated by the OT
algorithm misses fewer edges compared with CAM.

通过确定一组父节点的可识别性条件，本文提出了一种基于最优调整算法的学习贝叶斯网络的方法，使用 Hilbert-Schmidt 独立性准则进行优化问题，证明了高阶 HSIC 的增量属性，实验证明该算法在不同合成数据集和实际数据集上优于现有方法，在 Sigmoid 混合模型中，算法的结构干预距离（SID）比 CAM 算法小 329.7，表明该算法对图的估计缺少的边更少。