In the problem of aggregation, the aim is to combine a given class of base predictors to achieve predictions nearly as accurate as the best one. In this flexible framework, no assumption is made on the structure of the class or the nature of the target. Aggregation has been studied in both sequential and statistical contexts. Despite some important differences between the two problems, the classical results in both cases feature the same global complexity measure. In this paper, we revisit and tighten classical results in the theory of aggregation in the statistical setting by replacing the global complexity with a smaller, local one. Some of our proofs build on the PAC-Bayes localization technique introduced by Catoni. Among other results, we prove localized versions of the classical bound for the exponential weights estimator due to Leung and Barron and deviation-optimal bounds for the Q-aggregation estimator. These bounds improve over the results of Dai, Rigollet and Zhang for fixed design regression and the results of Lecu\'e and Rigollet for random design regression.

翻译：在聚合问题中，目标是结合给定的一类基预测器，以实现几乎与最佳预测器同样准确的预测。在这一灵活框架下，不对基预测器类的结构或目标变量的性质做出任何假设。聚合问题已在序列式和统计式两种情境下得到研究。尽管两个问题之间存在一些重要差异，但经典结果在两种情形中都采用了相同的全局复杂性度量。在本文中，我们通过用更小的局部复杂性替代全局复杂性，重新审视并加强了统计框架下聚合理论的经典结果。部分证明基于Catoni提出的PAC-贝叶斯局部化技术。在其他结果中，我们证明了Leung和Barron提出的指数权重估计量经典界的局部化版本，以及Q-聚合估计量的偏差最优界。这些界改进了Dai、Rigollet和Zhang在固定设计回归中的结果，以及Lecué和Rigollet在随机设计回归中的结果。