settings. Recall the definition of a stacked prediction function $\hat Y_w(\cdot) = \sum_{\ell\leq L, t\leq T} w_{\ell, t}\hat Y_t^\ell(\cdot)$ based on a set of SPFs $\hat{\mathcal Y}$ and weights $w\in W$. We indicate as [*oracle*]{} weights $$\begin{gathered}
w_g = \operatorname*{arg\,max}_{w\in W} U_g(w)\\
w_s^{(k)} = \operatorname*{arg\,max}_{w\in W} U_s(w;k).\end{gathered}$$ Note that $W\subset \mathbb R^{TL}$.

Constraints on or penalties applied to select parameters like $w$ can lead to identical results. For example, in several optimization problem constraining an $KL-$dimensional parameter to $W = \{w:\|w\|_2\leq c\}$ is equivalent to the unconstrained optimization with an $L_2$ penalty on the parameter. The use of penalties in the estimation of stacking weights has been discussed in [@breiman1996stacked; @leblanc1996combining]. One of the main arguments is that members of the library of SPFs $\hat{\mathcal Y}$ tend to be correlated, especially those that are trained on the same set $D_t$.

Stacking with data reuse
------------------------

A direct approach to select $w_g$ and $w_s^{(k)}$ consists in optimizing the $\hat U(w; \nu)$ estimates of $U_g(w)$ and $U_s(w; k)$. When the studies are exchangeable $\hat U(w;K^{-1} {\mathbf{1}}_K)$ can be used to select $w_g$. The estimation of stacking weights attempts to provide values close to the oracle solution $w_g$. If instead we develop a 