for every $k$, and $u(\hat y,y) = -(\hat y-y)^2$, then for any $w\in W$, where $W$ is a bounded set in $\mathbb R^{T}$, the following inequality holds $$\label{eq:diff-both}
\begin{aligned}
\sup_{w\in W}\left|\hat U(w;K^{-1}{\mathbf{1}}_K) - \hat U_{\text{ws}}(w;K^{-1}{\mathbf{1}}_K)\right| &\leq O_p(1/n),\\
\sup_{w\in W}\left|\hat U(w;K^{-1}{\mathbf{1}}_K) - \lim_{n\to\infty}\hat U(w;K^{-1}{\mathbf{1}}_K)\right| &\leq O_p(1/\sqrt{n}).
\end{aligned}$$

The above proposition indicates that the difference between utility functions in data reuse stacking and ${\text{CV}_{\text{ws}}}$ is order of magnitude smaller than the random fluctuation in $\hat U(w;K^{-1}{\mathbf{1}}_K)$, and in turn establishes the asymptotic equivalence of utility functions of stacking with data reuse and ${\text{CV}_{\text{ws}}}$. Since the results in Proposition \[prop:cv-eq\] concerns uniform convergence, the near equivalence of $\hat U(w;K^{-1}{\mathbf{1}}_K)$ and $\hat U_{\text{ws}}(w;K^{-1}{\mathbf{1}}_K)$ can translate into asymptotic equivalence of $\hat w$ and $\hat w^{\text{ws}}$, provided the limit of $\hat U(w;K^{-1}{\mathbf{1}}_K)$ has a unique maximizer in $W$.

In Figure \[fig:prop-rate\], we plot the estimated $\left|\hat U(w;K^{-1}{\mathbf{1}}_K) - \hat U_{\text{ws}}(w;K^{-1}{\mathbf{1}}_K)\right|$ and $|\hat U(w;K^{-1}{\mathbf{1}}_K) - \lim_{n\to\infty} \hat U(w;K^{-1}{\mathbf{1}}_K)|$at $w = K^{-1}{\mathbf{1}}_K$ as a function of $n$. We set $K=20$, $p = 10$ and $\beta_k\sim N({\mathbf{1}}_p, I)$. We use a 5-fold ${\text{CV}_{\text{ws}}}$.

### Cross-set CV and stacking with no data reuse

In this section, we focus on leave-one-set-out ${\text{CV}_{\text{cs}}}$ where $T$ iterations are performed. At each iteration a 