标题： 自适应 minimax 估计在稀疏$\ell_q$-凸壳上的研究 显示英文标题

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

摘要： 给定一个包含$M_n$个未知真实回归函数的初始估计值的字典，我们的目标是构建线性聚合估计器，这些估计器针对所有线性组合中最佳性能，其中对线性系数施加稀疏的$q$-范数（$0 \leq q \leq 1$）约束。 除了确定这些$\ell_q$- 聚合问题的最优聚合速率外，我们的多方向（或通用）聚合策略通过模型混合或模型选择同时实现了$0\leq q \leq 1$范围内一般$M_n$的最优速率和$t_n$的上界$q$- 范数。 无论是随机设计还是固定设计，已知或未知误差方差的情况都被处理了，并且本文研究的$\ell_q$- 聚合涵盖了之前文献中研究过的大多数类型的聚合问题。 关于在真实系数受$\ell_q$-约束的自适应回归下的最小最大速率后果 ($0 \leq q \leq 1$) 也已给出。 我们的结果显示，$\ell_q$-聚合（$0 \leq q \leq 1$）的极小极大速率基本上由一个有效的模型大小决定，这是一个依赖于$q$、$t_n$、$M_n$和样本量$n$的稀疏性指标，并且可以通过经典的模型选择理论以易于解释的方式确定，该理论处理大量模型。 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而且在偏离概率的指数衰减方面都得到了收敛的最优速率。 相比之下，模型混合方法可以在oracle不等式的目标风险前得到常数为一的领先项，但不能提供偏离概率的最优性。 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.