From b18f551a84104a0a99ba81d5edb4eb5eba4adaae Mon Sep 17 00:00:00 2001 From: Jonathan Berrisch Date: Wed, 25 Jun 2025 11:38:11 +0200 Subject: [PATCH] Add proof of prop 2 --- index.html | 434 ++++++++++++++++++++++++++++++----------------------- index.qmd | 87 ++++++++++- 2 files changed, 329 insertions(+), 192 deletions(-) diff --git a/index.html b/index.html index 90151a4..e9fd9b9 100644 --- a/index.html +++ b/index.html @@ -26325,7 +26325,7 @@ w_{t,k}^{\text{Naive}} = \frac{1}{K}\label{eq:naive_combination}

CRPS Learning Optimality

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@@ -26405,6 +26405,60 @@ w_{t,k}^{\text{Naive}} = \frac{1}{K}\label{eq:naive_combination}

BOAG with \(\text{QL}\) satisfies \(\eqref{eq_boa_opt_conv}\) and \(\eqref{eq_boa_opt_select}\)

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First, rewrite \(QL_p\) using the check function:

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\[\begin{align} + \rho_p(z) = z(1(0 < z) - p) \label{eq:check} +\end{align}\]

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\[\begin{align} + QL_p(x, y) &= (\mathbf1(y < x) - p)(x - y) \\ + &= \rho_p(x-y) +\end{align}\]

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Now we can express the quantile risk as:

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\[\begin{align} + \mathcal{Q}_p(x) = E[\rho_p(x-y)] = ∫ \rho_p(x-y)f(y)dy +\end{align}\]

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This integral form is where the convolution becomes apparent. A convolution of functions is defined as:

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\[\begin{align} + (g * h)(x) &= ∫ g(z)h(x - z)dz \\ + &= ∫ h(x - z)g(z)dz +\end{align}\]

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They are commutative.

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Using exchangability of subgradients in covolutions:

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\[\begin{align} + \mathcal{Q}''_p(x) = \rho_p'' * f +\end{align}\]

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To find \(\mathcal{Q}''_p(x)\) we rewrite \(\eqref{eq:check}\):

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\[\begin{align} + \rho_p(z) &= + \begin{cases} + z(1 - p) = z - zp, & \text{if } z > 0 \\ + z(0 - p) = -zp, & \text{if } z \leq 0 + \end{cases} \\ + \rho_p'(z) &= + \begin{cases} + 1 - p, & \text{if } z > 0 \\ + -p, & \text{if } z < 0 + \end{cases} +\end{align}\]

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The function \(\rho'_p(z)\) jumps from \(-p\) to \(1-p\) at \(0\). So:

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\[\begin{align} + \rho''_p(x) = \delta_0(z) \quad \text{Dirac Delta} +\end{align}\]

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Now the magical part :

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\[\begin{align} + \mathcal{Q}''_p(x) = \delta_0 * f = f +\end{align}\]

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Because Dirac Delta is the identity element for convolutions.

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diff --git a/index.qmd b/index.qmd index a1bd76d..aba0b72 100644 --- a/index.qmd +++ b/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 : + +\begin{align} + \mathcal{Q}''_p(x) = \delta_0 * f = f +\end{align} + +Because Dirac Delta is the identity element for convolutions. + +::: + +:::: + +:::: + :::: ## A Probabilistic Example