Hyper-sparse optimal aggregation
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In this paper, we consider the problem of "hyper-sparse aggregation". Namely, given a dictionary F = {f_1, ..., f_M } of functions, we look for an optimal aggregation algorithm that writes f̃ = ∑_j=1^M θ_j f_j with as many zero coefficients θ_j as possible. This problem is of particular interest when F contains many irrelevant functions that should not appear in f̃. We provide an exact oracle inequality for f̃, where only two coefficients are non-zero, that entails f̃ to be an optimal aggregation algorithm. Since selectors are suboptimal aggregation procedures, this proves that 2 is the minimal number of elements of F required for the construction of an optimal aggregation procedures in every situations. A simulated example of this algorithm is proposed on a dictionary obtained using LARS, for the problem of selection of the regularization parameter of the LASSO. We also give an example of use of aggregation to achieve minimax adaptation over anisotropic Besov spaces, which was not previously known in minimax theory (in regression on a random design).
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