BerriJ/PHD-Presentation
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

Update crps optimality

Browse Source
This commit is contained in:
Jonathan Berrisch
2025-06-22 18:07:51 +02:00
parent 74d338b7f1
commit 6b5d7abfea
2 changed files with 8 additions and 8 deletions
Download Patch File Download Diff File Expand all files Collapse all files

8
index.html
View File

@@ -26350,7 +26350,7 @@ w_{t,k}^{\text{Naive}} = \frac{1}{K}\label{eq:naive_combination}
\end{align}\]</span></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>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>A2: Weak Exp-Concavity</strong></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; +
\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>
<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 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><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>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><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>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>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>
</div><div class="column" style="width:4%;">
</div><div class="column" style="width:48%;">