diff --git a/index.html b/index.html index da8dae0..c5a835e 100644 --- a/index.html +++ b/index.html @@ -26350,7 +26350,7 @@ w_{t,k}^{\text{Naive}} = \frac{1}{K}\label{eq:naive_combination} \end{align}\]

Pointwise can outperform constant procedures

\(\text{QL}\) is convex: almost optimal convergence w.r.t. convex aggregation \(\eqref{eq_boa_opt_conv}\)

-

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 (Gaillard & Wintenberger, 2018):

A1: Lipschitz Continuity

A2: Weak Exp-Concavity

QL is Lipschitz continuous with \(G=\max(p, 1-p)\):

@@ -26370,7 +26370,7 @@ w_{t,k}^{\text{Naive}} = \frac{1}{K}\label{eq:naive_combination} & + \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*}\]

-

If \(\beta=1\) we get strong-convexity, which implies weak exp-concavity

+

The strongest case is \(\beta=1\) (Strong Convexity)

@@ -26379,8 +26379,8 @@ w_{t,k}^{\text{Naive}} = \frac{1}{K}\label{eq:naive_combination}

Conditional quantile risk: \(\mathcal{Q}_p(x) = \mathbb{E}[ \text{QL}_p(x, Y_t) | \mathcal{F}_{t-1}]\).

convexity properties of \(\mathcal{Q}_p\) depend on the conditional distribution \(Y_t|\mathcal{F}_{t-1}\).

Proposition 2

-

Let \(Y\) be a univariate random variable with (Radon-Nikodym) \(\nu\)-density \(f\), then for the second subderivative of the quantile risk \(\mathcal{Q}_p(x) = \mathbb{E}[ \text{QL}_p(x, Y) ]\) of \(Y\) it holds for all \(p\in(0,1)\) that \(\mathcal{Q}_p'' = f.\) 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

-

A2 with \(\beta=1\) Gaillard & Wintenberger (2018)

+

Let \(Y\) be a univariate random variable with (Radon-Nikodym) \(\nu\)-density \(f\), then for the second subderivative of the quantile risk \(\mathcal{Q}_p(x) = \mathbb{E}[ \text{QL}_p(x, Y) ]\) of \(Y\) it holds for all \(p\in(0,1)\) that \(\mathcal{Q}_p'' = f.\) 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.

+

This implies satisfaction of condition A2 with \(\beta=1\) and \(\alpha = \gamma / 2G^2\) (Gaillard & Wintenberger, 2018)

diff --git a/index.qmd b/index.qmd index ac49e5c..68efbe0 100644 --- a/index.qmd +++ b/index.qmd @@ -1336,7 +1336,7 @@ Pointwise can outperform constant procedures $\text{QL}$ is convex: almost optimal convergence w.r.t. *convex aggregation* \eqref{eq_boa_opt_conv}
-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** @@ -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] \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 $f$, $\mathcal{Q}_p(x) = \mathbb{E}[ \text{QL}_p(x, Y) ]$ of $Y$ it holds for all $p\in(0,1)$ that $\mathcal{Q}_p'' = f.$ -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 $\beta=1$ @gaillard2018efficient +This implies satisfaction of condition **A2** with $\beta=1$ and $\alpha = \gamma / 2G^2$ [@gaillard2018efficient] :::