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Update crps optimality

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Jonathan Berrisch
2025-06-22 18:07:51 +02:00
parent 74d338b7f1
commit 6b5d7abfea
2 changed files with 8 additions and 8 deletions
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@@ -1336,7 +1336,7 @@ Pointwise can outperform constant procedures
<i class="fa fa-fw fa-arrow-right" style="color:var(--col_grey_10);"></i> $\text{QL}$ is convex: almost optimal convergence w.r.t. *convex aggregation* \eqref{eq_boa_opt_conv} <i class="fa fa-fw fa-check" style="color:var(--col_green_9);"></i> </br>
For almost optimal convergence w.r.t. *selection* \eqref{eq_boa_opt_select} we need:
For almost optimal convergence w.r.t. *selection* \eqref{eq_boa_opt_select} we need [@gaillard2018efficient]:
**A1: Lipschitz Continuity**
@@ -1374,7 +1374,7 @@ for all $x_1,x_2 \in \mathbb{R}$ and $t>0$ that
\mathbb{E}\left[ \left. \left( \alpha(\ell'(x_1, Y_t)(x_1 - x_2))^{2}\right)^{1/\beta} \right|\mathcal{F}_{t-1}\right]
\end{align*}
If $\beta=1$ we get strong-convexity, which implies weak exp-concavity
The strongest case is $\beta=1$ (Strong Convexity)
:::
@@ -1397,9 +1397,9 @@ Let $Y$ be a univariate random variable with (Radon-Nikodym) $\nu$-density $f$,
$\mathcal{Q}_p(x) = \mathbb{E}[ \text{QL}_p(x, Y) ]$
of $Y$ it holds for all $p\in(0,1)$ that
$\mathcal{Q}_p'' = f.$
Additionally, if $f$ is a continuous Lebesgue-density with $f\geq\gamma>0$ for some constant $\gamma>0$ on its support $\text{spt}(f)$ then $\mathcal{Q}_p$ is $\gamma$-strongly convex, which implies satisfaction of condition
Additionally, if $f$ is a continuous Lebesgue-density with $f\geq\gamma>0$ for some constant $\gamma>0$ on its support $\text{spt}(f)$ then $\mathcal{Q}_p$ is $\gamma$-strongly convex.
**A2** with $\beta=1$ <i class="fa fa-fw fa-check" style="color:var(--col_green_9);"></i> @gaillard2018efficient
This implies satisfaction of condition **A2** with $\beta=1$ and $\alpha = \gamma / 2G^2$ <i class="fa fa-fw fa-check" style="color:var(--col_green_9);"></i> [@gaillard2018efficient]
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