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.
Wang, Z., S., Paterlini, F., Gao e Y., Yang. "Adaptive Minimax Estimation over Sparse lq-Hulls" Working paper, RECENT WORKING PAPER SERIES, Dipartimento di Economia Marco Biagi – Università di Modena e Reggio Emilia, 2012.
Adaptive Minimax Estimation over Sparse lq-Hulls
Paterlini, S.;
2012
Abstract
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.File | Dimensione | Formato | |
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