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Update crps optimality

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Jonathan Berrisch
2025-06-22 18:07:51 +02:00
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@@ -26350,7 +26350,7 @@ w_{t,k}^{\text{Naive}} = \frac{1}{K}\label{eq:naive_combination}
\end{align}\]</span></p> \end{align}\]</span></p>
<p>Pointwise can outperform constant procedures</p> <p>Pointwise can outperform constant procedures</p>
<p><i class="fa fa-fw fa-arrow-right" style="color:var(--col_grey_10);"></i> <span class="math inline">\(\text{QL}\)</span> is convex: almost optimal convergence w.r.t. <em>convex aggregation</em> <span class="math inline">\(\eqref{eq_boa_opt_conv}\)</span> <i class="fa fa-fw fa-check" style="color:var(--col_green_9);"></i> <br></p> <p><i class="fa fa-fw fa-arrow-right" style="color:var(--col_grey_10);"></i> <span class="math inline">\(\text{QL}\)</span> is convex: almost optimal convergence w.r.t. <em>convex aggregation</em> <span class="math inline">\(\eqref{eq_boa_opt_conv}\)</span> <i class="fa fa-fw fa-check" style="color:var(--col_green_9);"></i> <br></p>
<p>For almost optimal convergence w.r.t. <em>selection</em> <span class="math inline">\(\eqref{eq_boa_opt_select}\)</span> we need:</p> <p>For almost optimal convergence w.r.t. <em>selection</em> <span class="math inline">\(\eqref{eq_boa_opt_select}\)</span> we need <span class="citation" data-cites="gaillard2018efficient">(<a href="#/references" role="doc-biblioref" onclick>Gaillard &amp; Wintenberger, 2018</a>)</span>:</p>
<p><strong>A1: Lipschitz Continuity</strong></p> <p><strong>A1: Lipschitz Continuity</strong></p>
<p><strong>A2: Weak Exp-Concavity</strong></p> <p><strong>A2: Weak Exp-Concavity</strong></p>
<p>QL is Lipschitz continuous with <span class="math inline">\(G=\max(p, 1-p)\)</span>:</p> <p>QL is Lipschitz continuous with <span class="math inline">\(G=\max(p, 1-p)\)</span>:</p>
@@ -26370,7 +26370,7 @@ w_{t,k}^{\text{Naive}} = \frac{1}{K}\label{eq:naive_combination}
&amp; + &amp; +
\mathbb{E}\left[ \left. \left( \alpha(\ell&#39;(x_1, Y_t)(x_1 - x_2))^{2}\right)^{1/\beta} \right|\mathcal{F}_{t-1}\right] \mathbb{E}\left[ \left. \left( \alpha(\ell&#39;(x_1, Y_t)(x_1 - x_2))^{2}\right)^{1/\beta} \right|\mathcal{F}_{t-1}\right]
\end{align*}\]</span></p> \end{align*}\]</span></p>
<p>If <span class="math inline">\(\beta=1\)</span> we get strong-convexity, which implies weak exp-concavity</p> <p>The strongest case is <span class="math inline">\(\beta=1\)</span> (Strong Convexity)</p>
</div></div> </div></div>
</div> </div>
<div id="tabset-2-3"> <div id="tabset-2-3">
@@ -26379,8 +26379,8 @@ w_{t,k}^{\text{Naive}} = \frac{1}{K}\label{eq:naive_combination}
<p>Conditional quantile risk: <span class="math inline">\(\mathcal{Q}_p(x) = \mathbb{E}[ \text{QL}_p(x, Y_t) | \mathcal{F}_{t-1}]\)</span>.</p> <p>Conditional quantile risk: <span class="math inline">\(\mathcal{Q}_p(x) = \mathbb{E}[ \text{QL}_p(x, Y_t) | \mathcal{F}_{t-1}]\)</span>.</p>
<p><i class="fa fa-fw fa-arrow-right" style="color:var(--col_grey_10);"></i> convexity properties of <span class="math inline">\(\mathcal{Q}_p\)</span> depend on the conditional distribution <span class="math inline">\(Y_t|\mathcal{F}_{t-1}\)</span>.</p> <p><i class="fa fa-fw fa-arrow-right" style="color:var(--col_grey_10);"></i> convexity properties of <span class="math inline">\(\mathcal{Q}_p\)</span> depend on the conditional distribution <span class="math inline">\(Y_t|\mathcal{F}_{t-1}\)</span>.</p>
<p><strong>Proposition 2</strong></p> <p><strong>Proposition 2</strong></p>
<p>Let <span class="math inline">\(Y\)</span> be a univariate random variable with (Radon-Nikodym) <span class="math inline">\(\nu\)</span>-density <span class="math inline">\(f\)</span>, then for the second subderivative of the quantile risk <span class="math inline">\(\mathcal{Q}_p(x) = \mathbb{E}[ \text{QL}_p(x, Y) ]\)</span> of <span class="math inline">\(Y\)</span> it holds for all <span class="math inline">\(p\in(0,1)\)</span> that <span class="math inline">\(\mathcal{Q}_p&#39;&#39; = f.\)</span> Additionally, if <span class="math inline">\(f\)</span> is a continuous Lebesgue-density with <span class="math inline">\(f\geq\gamma&gt;0\)</span> for some constant <span class="math inline">\(\gamma&gt;0\)</span> on its support <span class="math inline">\(\text{spt}(f)\)</span> then <span class="math inline">\(\mathcal{Q}_p\)</span> is <span class="math inline">\(\gamma\)</span>-strongly convex, which implies satisfaction of condition</p> <p>Let <span class="math inline">\(Y\)</span> be a univariate random variable with (Radon-Nikodym) <span class="math inline">\(\nu\)</span>-density <span class="math inline">\(f\)</span>, then for the second subderivative of the quantile risk <span class="math inline">\(\mathcal{Q}_p(x) = \mathbb{E}[ \text{QL}_p(x, Y) ]\)</span> of <span class="math inline">\(Y\)</span> it holds for all <span class="math inline">\(p\in(0,1)\)</span> that <span class="math inline">\(\mathcal{Q}_p&#39;&#39; = f.\)</span> Additionally, if <span class="math inline">\(f\)</span> is a continuous Lebesgue-density with <span class="math inline">\(f\geq\gamma&gt;0\)</span> for some constant <span class="math inline">\(\gamma&gt;0\)</span> on its support <span class="math inline">\(\text{spt}(f)\)</span> then <span class="math inline">\(\mathcal{Q}_p\)</span> is <span class="math inline">\(\gamma\)</span>-strongly convex.</p>
<p><strong>A2</strong> with <span class="math inline">\(\beta=1\)</span> <i class="fa fa-fw fa-check" style="color:var(--col_green_9);"></i> <span class="citation" data-cites="gaillard2018efficient">Gaillard &amp; Wintenberger (<a href="#/references" role="doc-biblioref" onclick>2018</a>)</span></p> <p>This implies satisfaction of condition <strong>A2</strong> with <span class="math inline">\(\beta=1\)</span> and <span class="math inline">\(\alpha = \gamma / 2G^2\)</span> <i class="fa fa-fw fa-check" style="color:var(--col_green_9);"></i> <span class="citation" data-cites="gaillard2018efficient">(<a href="#/references" role="doc-biblioref" onclick>Gaillard &amp; Wintenberger, 2018</a>)</span></p>
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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> <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** **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] \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*} \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 $f$,
$\mathcal{Q}_p(x) = \mathbb{E}[ \text{QL}_p(x, Y) ]$ $\mathcal{Q}_p(x) = \mathbb{E}[ \text{QL}_p(x, Y) ]$
of $Y$ it holds for all $p\in(0,1)$ that of $Y$ it holds for all $p\in(0,1)$ that
$\mathcal{Q}_p'' = f.$ $\mathcal{Q}_p'' = f.$
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]
::: :::