标题： 自适应最小最大估计 over 稀疏$\ell_q$-Hulls 显示英文标题

标题： Adaptive Minimax Estimation over Sparse $\ell_q$-Hulls

摘要： 给定一个$M_n$的初始估计值的字典，我们旨在构建线性聚合估计量，在线性系数的稀疏$q$-范数（$0 \leq q \leq 1$）约束下，针对所有线性组合中的最佳性能。 Besides identifying the optimal rates of aggregation for these $\ell_q$-aggregation problems, our multi-directional (or universal)aggregation strategies by model mixing or model selection achieve the optimal rates simultaneously over the full range of $0\leq q \leq 1$ for general $M_n$ and upper bound $t_n$ of the $q$-norm. Both random and fixed designs, with known or unknown error variance, are handled, and the $\ell_q$-aggregations examined in this work cover major types of aggregation problems previously studied in the literature. 在$\ell_q$约束的真实系数下，最小最大率自适应回归的后果 ($0 \leq q \leq 1$) 也提供了。 我们的结果表明，$\ell_q$-聚合（$0 \leq q \leq 1$）的最小最大率基本上由一个有效模型大小决定，这是一个稀疏性指标，它以一种易于解释的方式依赖于$q$、$t_n$、$M_n$以及样本大小$n$，这是基于处理大量模型的经典模型选择理论。 此外，在固定设计的情况下，模型选择方法不仅在期望意义上能够达到最优收敛速度，而且偏差概率具有指数衰减。 相反，模型混合方法在Oracle不等式中可以在目标风险前具有常数1，但并不能在偏差概率上提供最优性。 显示英文摘要

摘要： Given a dictionary of $M_n$ initial estimates of the unknown true regression function, we aim to construct linearly aggregated estimators that target the best performance among all the linear combinations under a sparse $q$-norm ($0 \leq q \leq 1$) constraint on the linear coefficients. Besides identifying the optimal rates of aggregation for these $\ell_q$-aggregation problems, our multi-directional (or universal)aggregation strategies by model mixing or model selection achieve the optimal rates simultaneously over the full range of $0\leq q \leq 1$ for general $M_n$ and upper bound $t_n$ of the $q$-norm. Both random and fixed designs, with known or unknown error variance, are handled, and the $\ell_q$-aggregations examined in this work cover major types of aggregation problems previously studied in the literature. Consequences on minimax-rate adaptive regression under $\ell_q$-constrained true coefficients ($0 \leq q \leq 1$) are also provided. Our results show that the minimax rate of $\ell_q$-aggregation ($0 \leq q \leq 1$) is basically determined by an effective model size, which is a sparsity index that depends on $q$, $t_n$, $M_n$, and the sample size $n$ in an easily interpretable way based on a classical model selection theory that deals with a large number of models. In addition, in the fixed design case, the model selection approach is seen to yield optimal rates of convergence not only in expectation but also with exponential decay of deviation probability. In contrast, the model mixing approach can have leading constant one in front of the target risk in the oracle inequality while not offering optimality in deviation probability.