Adaptive Minimax Estimation over Sparse lq-Hulls

Given a dictionary of Mn 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 ≤ q ≤ 1) constraint on the linear coefficients. Besides identifying the optimal rates of aggregation for these `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 ≤ q ≤ 1 for general Mn and upper bound tn of the q-norm. Both random and fixed designs, with known or
unknown error variance, are handled, and the `q-aggregations examined in this work cover
major types of aggregation problems previously studied in the literature. Consequences on
minimax-rate adaptive regression under `q-constrained true coefficients (0 ≤ q ≤ 1) are also
provided.