In high-dimensional linear models the problem of constructing adaptive confidence sets for the full parameter is known to be generally impossible. We propose re-weighted loss functions under which constructing fully adaptive confidence sets for the parameter is shown to be possible. We give necessary and sufficient conditions on the weights for adaptive confidence sets to exist, and exhibit a concrete rate-optimal procedure in the feasible regime.

翻译：在高维线性模型中，为完整参数构建自适应置信集通常被认为是不可行的。我们提出了一类重新加权的损失函数，在此框架下证明了可以为参数构建完全自适应的置信集。我们给出了自适应置信集存在所需的权重充分必要条件，并在可行域内展示了一个具体的最优速率过程。