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Improve prop2 proof

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Jonathan Berrisch
2025-06-25 14:37:42 +02:00
parent b18f551a84
commit 1c58c4bf89
2 changed files with 10 additions and 10 deletions
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@@ -26411,20 +26411,20 @@ w_{t,k}^{\text{Naive}} = \frac{1}{K}\label{eq:naive_combination}
<div class="column" style="width:48%;"> <div class="column" style="width:48%;">
<p>First, rewrite <span class="math inline">\(QL_p\)</span> using the check function:</p> <p>First, rewrite <span class="math inline">\(QL_p\)</span> using the check function:</p>
<p><span class="math display">\[\begin{align} <p><span class="math display">\[\begin{align}
\rho_p(z) = z(1(0 &lt; z) - p) \label{eq:check} \rho_p(z) = z(\mathbb{1}(0 &lt; z) - p) \label{eq:check}
\end{align}\]</span></p> \end{align}\]</span></p>
<p><span class="math display">\[\begin{align} <p><span class="math display">\[\begin{align}
QL_p(x, y) &amp;= (\mathbf1(y &lt; x) - p)(x - y) \\ QL_p(x, y) &amp;= (\mathbb{1}(y &lt; x) - p)(x - y) \\
&amp;= \rho_p(x-y) &amp;= \rho_p(x-y)
\end{align}\]</span></p> \end{align}\]</span></p>
<p>Now we can express the quantile risk as:</p> <p>Now we can express the quantile risk as:</p>
<p><span class="math display">\[\begin{align} <p><span class="math display">\[\begin{align}
\mathcal{Q}_p(x) = E[\rho_p(x-y)] = ∫ \rho_p(x-y)f(y)dy \mathcal{Q}_p(x) = \mathbb{E}[\rho_p(x-y)] = ∫ \rho_p(x-y)f(y)dy
\end{align}\]</span></p> \end{align}\]</span></p>
<p>This integral form is where the convolution becomes apparent. A convolution of functions is defined as:</p> <p>This integral form is where the convolution becomes apparent. A convolution of functions is defined as:</p>
<p><span class="math display">\[\begin{align} <p><span class="math display">\[\begin{align}
(g * h)(x) &amp;= ∫ g(z)h(x - z)dz \\ (g * h)(x) &amp;= ∫ g(z)h(x - z)dz \\
&amp;= ∫ h(x - z)g(z)dz &amp;= ∫ g(x - z)h(z)dz
\end{align}\]</span></p> \end{align}\]</span></p>
<p>They are commutative.</p> <p>They are commutative.</p>
</div><div class="column" style="width:4%;"> </div><div class="column" style="width:4%;">
@@ -26449,7 +26449,7 @@ w_{t,k}^{\text{Naive}} = \frac{1}{K}\label{eq:naive_combination}
\end{align}\]</span></p> \end{align}\]</span></p>
<p>The function <span class="math inline">\(\rho&#39;_p(z)\)</span> jumps from <span class="math inline">\(-p\)</span> to <span class="math inline">\(1-p\)</span> at <span class="math inline">\(0\)</span>. So:</p> <p>The function <span class="math inline">\(\rho&#39;_p(z)\)</span> jumps from <span class="math inline">\(-p\)</span> to <span class="math inline">\(1-p\)</span> at <span class="math inline">\(0\)</span>. So:</p>
<p><span class="math display">\[\begin{align} <p><span class="math display">\[\begin{align}
\rho&#39;&#39;_p(x) = \delta_0(z) \quad \text{Dirac Delta} \rho&#39;&#39;_p(x) = \delta_0(z) \quad \text{(Dirac Delta)}
\end{align}\]</span></p> \end{align}\]</span></p>
<p>Now the magical part <i class="fa fa-fw fa-wand-magic-sparkles" style="color:var(--col_amber_6);"></i>:</p> <p>Now the magical part <i class="fa fa-fw fa-wand-magic-sparkles" style="color:var(--col_amber_6);"></i>:</p>
<p><span class="math display">\[\begin{align} <p><span class="math display">\[\begin{align}

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@@ -1422,25 +1422,25 @@ $$\widehat{\mathcal{R}}_{t,\min} = 2\overline{\widehat{\mathcal{R}}}^{\text{QL}}
First, rewrite $QL_p$ using the check function: First, rewrite $QL_p$ using the check function:
\begin{align} \begin{align}
\rho_p(z) = z(1(0 < z) - p) \label{eq:check} \rho_p(z) = z(\mathbb{1}(0 < z) - p) \label{eq:check}
\end{align} \end{align}
\begin{align} \begin{align}
QL_p(x, y) &= (\mathbf1(y < x) - p)(x - y) \\ QL_p(x, y) &= (\mathbb{1}(y < x) - p)(x - y) \\
&= \rho_p(x-y) &= \rho_p(x-y)
\end{align} \end{align}
Now we can express the quantile risk as: Now we can express the quantile risk as:
\begin{align} \begin{align}
\mathcal{Q}_p(x) = E[\rho_p(x-y)] = ∫ \rho_p(x-y)f(y)dy \mathcal{Q}_p(x) = \mathbb{E}[\rho_p(x-y)] = ∫ \rho_p(x-y)f(y)dy
\end{align} \end{align}
This integral form is where the convolution becomes apparent. A convolution of functions is defined as: This integral form is where the convolution becomes apparent. A convolution of functions is defined as:
\begin{align} \begin{align}
(g * h)(x) &= ∫ g(z)h(x - z)dz \\ (g * h)(x) &= ∫ g(z)h(x - z)dz \\
&= ∫ h(x - z)g(z)dz &= ∫ g(x - z)h(z)dz
\end{align} \end{align}
They are commutative. They are commutative.
@@ -1477,7 +1477,7 @@ To find $\mathcal{Q}''_p(x)$ we rewrite \eqref{eq:check}:
The function $\rho'_p(z)$ jumps from $-p$ to $1-p$ at $0$. So: The function $\rho'_p(z)$ jumps from $-p$ to $1-p$ at $0$. So:
\begin{align} \begin{align}
\rho''_p(x) = \delta_0(z) \quad \text{Dirac Delta} \rho''_p(x) = \delta_0(z) \quad \text{(Dirac Delta)}
\end{align} \end{align}
Now the magical part <i class="fa fa-fw fa-wand-magic-sparkles" style="color:var(--col_amber_6);"></i>: Now the magical part <i class="fa fa-fw fa-wand-magic-sparkles" style="color:var(--col_amber_6);"></i>: