  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 $\tilde f = \sum_{j=1}^M \theta_j
f_j$ with as many zero coefficients $\theta_j$ as possible. This problem is of
particular interest when $F$ contains many irrelevant functions that should not
appear in $\tilde{f}$. We provide an exact oracle inequality for $\tilde f$,
where only two coefficients are non-zero, that entails $\tilde f$ 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).
