Add proof of prop 2

index.qmd

@@ -1297,7 +1297,7 @@ if the loss $\ell$ is $G$-Lipschitz and weak exp-concave in its first coordinate
 
 :::
 
-## Proposition + Conditions
+## Proposition 1 + Conditions
 
 
 :::: {.columns}
@@ -1363,7 +1363,7 @@ The strongest case is $\beta=1$ (Strong Convexity)
 
 ::::
 
-## Proposition + Theorem
+## Proposition 2 + Theorem
 
 :::: {.columns}
 
@@ -1411,6 +1411,89 @@ $$\widehat{\mathcal{R}}_{t,\min} = 2\overline{\widehat{\mathcal{R}}}^{\text{QL}}
 
 ::::
 
+## Proof of P2 {.scrollable}
+
+::: {style="font-size: 85%;"}
+
+:::: {.columns}
+
+::: {.column width="48%"}
+
+First, rewrite $QL_p$ using the check function:
+
+\begin{align}
+  \rho_p(z) = z(1(0 < z) - p) \label{eq:check}
+\end{align}
+
+\begin{align}
+  QL_p(x, y) &= (\mathbf1(y < x) - p)(x - y) \\
+  &= \rho_p(x-y)
+\end{align}
+
+Now we can express the quantile risk as:
+
+\begin{align}
+  \mathcal{Q}_p(x) = E[\rho_p(x-y)] = ∫ \rho_p(x-y)f(y)dy
+\end{align}
+
+This integral form is where the convolution becomes apparent. A convolution of functions is defined as:
+
+\begin{align}
+  (g * h)(x) &= ∫ g(z)h(x - z)dz \\
+  &= ∫ h(x - z)g(z)dz
+\end{align}
+
+They are commutative.
+
+:::
+
+::: {.column width="4%"}
+
+:::
+
+::: {.column width="48%"}
+
+Using exchangability of subgradients in covolutions:
+
+\begin{align}
+  \mathcal{Q}''_p(x) = \rho_p'' * f
+\end{align}
+
+To find $\mathcal{Q}''_p(x)$ we rewrite \eqref{eq:check}:
+
+\begin{align}
+  \rho_p(z) &= 
+  \begin{cases}
+    z(1 - p) = z - zp, & \text{if } z > 0 \\
+    z(0 - p) = -zp,    & \text{if } z \leq 0
+  \end{cases} \\
+  \rho_p'(z) &=
+  \begin{cases}
+    1 - p, & \text{if } z > 0 \\
+    -p,         & \text{if } z < 0
+  \end{cases}
+\end{align}
+
+The function $\rho'_p(z)$ jumps from $-p$ to $1-p$ at $0$. So:
+
+\begin{align}
+  \rho''_p(x) = \delta_0(z) \quad \text{Dirac Delta}
+\end{align}
+
+Now the magical part <i class="fa fa-fw fa-wand-magic-sparkles" style="color:var(--col_amber_6);"></i>:
+
+\begin{align}
+  \mathcal{Q}''_p(x) = \delta_0 * f = f
+\end{align}
+
+Because Dirac Delta is the identity element for convolutions.
+
+:::
+
+::::
+
+::::
+
 ::::
 
 ## A Probabilistic Example