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\begin{appendices}
4 years ago
\counterwithin{lstfloat}{section}
\section{Proofs for sone Lemmata in ...}
In the following there will be proofs for some important Lemmata in
Section~\ref{sec:theo38}. Further proofs not discussed here can be
found in \textcite{heiss2019}
The proves in this section are based on \textcite{heiss2019}. Slight
alterations have been made to accommodate for not splitting $f$ into
$f_+$ and $f_-$.
\begin{Theorem}[Proof of Lemma~\ref{theo38}]
\end{Theorem}
4 years ago
\begin{Lemma}[$\frac{w^{*,\tilde{\lambda}}_k}{v_k}\approx\mathcal{O}(\frac{1}{n})$]
For any $\lambda > 0$ and training data $(x_i^{\text{train}},
y_i^{\text{train}}) \in \mathbb{R}^2, \, i \in
\left\{1,\dots,N\right\}$, we have
\[
\max_{k \in \left\{1,\dots,n\right\}} \frac{w^{*,
\tilde{\lambda}}_k}{v_k} = \po_{n\to\infty}
\]
4 years ago
\end{Lemma}
\begin{Proof}[Proof of Lemma~\ref{lem:s3}]
\[
\sum_{k \in \kappa : \xi_k < T} \varphi(\xi_k, v_k)
h_{k,n} = \sum_{\substack{l \in \mathbb{Z} \\ [\delta l, \delta
(l+1)) \in [C_{g_\xi}^l,\min\{C_{g_\xi}^u, T \}]}}
\left(\sum_{\substack{k \in \kappa \\ \xi_k \in
[\delta l , \delta(l+1))}} \varphi(\xi_k, v_k)
h_{k,n}\right) \approx
\]
\[
\approx \sum_{\substack{l \in \mathbb{Z} \\ [\delta l, \delta
(l+1)) \in [C_{g_\xi}^l,\min\{C_{g_\xi}^u, T \}]}}
\left(\sum_{\substack{k \in \kappa \\ \xi_k \in
[\delta l , \delta(l+1))}} \left(\varphi(\delta l, v_k)
\frac{1}{n g_\xi (\delta l)} \pm \frac{\varepsilon}{n}\right)
\frac{\abs{\left\{m \in \kappa : \xi_m \in [\delta l,
\delta(l+1))\right\}}}{\abs{\left\{m \in \kappa : \xi_m
\in [\delta l, \delta(l+1))\right\}}}\right)
\]
\[
\approx \sum_{\substack{l \in \mathbb{Z} \\ [\delta l, \delta
(l+1)) \in [C_{g_\xi}^l,\min\{C_{g_\xi}^u, T \}]}}
\left(\frac{\sum_{\substack{k \in \kappa \\ \xi_k \in
[\delta l , \delta(l+1))}}\varphi(\delta l,
v_k)}{\abs{\left\{m \in \kappa : \xi_m
\in [\delta l, \delta(l+1))\right\}}}
\frac{\abs{\left\{m \in \kappa : \xi_m \in [\delta l,
\delta(l+1))\right\}}}{n g_\xi (\delta l)}\right) \pm \varepsilon
\]
The amount of kinks in a given interval of length $\delta$ follows a
binomial distribution,
\[
\mathbb{E} \left[\abs{\left\{m \in \kappa : \xi_m \in [\delta l,
\delta(l+1))\right\}}\right] = n \int_{\delta
l}^{\delta(l+1)}g_\xi (x) dx \approx n (\delta g_\xi(\delta l)
\pm \delta \tilde{\varepsilon}),
\]
for any $\delta \leq \delta(\varepsilon, \tilde{\varepsilon})$, since $g_\xi$ is uniformly continuous on its
support by Assumption..
As the distribution of $v$ is continuous as well we get that
$\mathcal{L}(v_k) = \mathcal{L} v| \xi = \delta l) \forall k \in
\kappa : \xi_k \in [\delta l, \delta(l+1))$ for $\delta \leq
\delta(\varepsilon, \tilde{\varepsilon})$. Thus we get with the law of
large numbers
\begin{align*}
&\sum_{k \in \kappa : \xi_k < T} \varphi(\xi_k, v_k)
h_{k,n} \approx\\
&\approx \sum_{\substack{l \in \mathbb{Z} \\ [\delta l, \delta
(l+1)) \in [C_{g_\xi}^l,\min\{C_{g_\xi}^u, T
\}]}}\left(\mathbb{E}[\phi(\xi, v)|\xi=\delta l]
\stackrel{\mathbb{P}}{\pm}\right) \delta \left(1 \pm
\frac{\tilde{\varepsilon}}{g_\xi(\delta l)}\right) \pm \varepsilon
\\
&\approx \left(\sum_{\substack{l \in \mathbb{Z} \\ [\delta
l, \delta
(l+1)) \in [C_{g_\xi}^l,\min\{C_{g_\xi}^u, T
\}]}}\mathbb{E}[\phi(\xi, v)|\xi=\delta l] \delta
\stackrel{\mathbb{P}}{\pm}\tilde{\tilde{\varepsilon}}
\abs{C_{g_\xi}^u - C_{g_\xi}^l}
\right)\\
&\phantom{\approx}\cdot \left(1 \pm
\frac{\tilde{\varepsilon}}{g_\xi(\delta l)}\right) \pm \varepsilon
\end{align*}
\end{Proof}
\begin{Lemma}[($L(f_n) \to L(f)$), Heiss, Teichmann, and
Wutte (2019, Lemma A.11)]
For any data $(x_i^{\text{train}}, y_i^{\text{train}}) \in
\mathbb{R}^2, i \in \left\{1,\dots,N\right\}$, let $(f_n)_{n \in
\mathbb{N}}$ be a sequence of functions that converges point-wise
in probability to a function $f : \mathbb{R}\to\mathbb{R}$, then the
loss $L$ of $f_n$ converges is probability to $L(f)$ as $n$ tends to
infinity,
\[
\plimn L(f_n) = L(f).
\]
\proof Vgl. ...
\end{Lemma}
\begin{Proof}[Step 2]
We start by showing that
\[
\plimn \tilde{\lambda} \norm{\tilde{w}}_2^2 = \lambda g(0)
\left(\int \frac{\left(f_g^{*,\lambda''}\right)^2}{g(x)} dx\right)
\]
With the definitions of $\tilde{w}$, $\tilde{\lambda}$ and
$h$ we have
\begin{align*}
\tilde{\lambda} \norm{\tilde{w}}_2^2
&= \tilde{\lambda} \sum_{k \in
\kappa}\left(f_g^{*,\lambda''}(\xi_k) \frac{h_k
v_k}{\mathbb{E}v^2|\xi = \xi_k]}\right)^2\\
&= \tilde{\lambda} \sum_{k \in
\kappa}\left(\left(f_g^{*,\lambda''}\right)^2(\xi_k) \frac{h_k
v_k^2}{\mathbb{E}v^2|\xi = \xi_k]}\right) h_k\\
& = \lambda g(0) \sum_{k \in
\kappa}\left(\left(f_g^{*,\lambda''}\right)^2(\xi_k)\frac{v_k^2}{g_\xi(\xi_k)\mathbb{E}
[v^2|\xi=\xi_k]}\right)h_k.
\end{align*}
By using Lemma~\ref{lem} with $\phi(x,y) =
\left(f_g^{*,\lambda''}\right)^2(x)\frac{y^2}{g_\xi(\xi)\mathbb{E}[v^2|\xi=y]}$
this converges to
\begin{align*}
&\plimn \tilde{\lambda}\norm{\tilde{w}}_2^2 = \\
&=\lambda
g_\xi(0)\mathbb{E}[v^2|\xi=0]\int_{\supp{g_\xi}}\mathbb{E}\left[
\left(f_g^{*,\lambda''}\right)^2(\xi)\frac{v^2}{
g_\xi(\xi)\mathbb{E}[v^2|\xi=x]^2}\Big{|} \xi = x\right]dx\\
&=\lambda g_\xi(0) \mathbb{E}[v^2|\xi=0] \int_{\supp{g_xi}}
\frac{\left(f_g^{*,\lambda''}\right)^2 (x)}{g_\xi(x)
\mathbb{E}[v^2|\xi=x]} dx \\
&=\lambda g(0) \int_{\supp{g_\xi}} \frac{\left(f_g^{*,\lambda''}\right)^2}{g(x)}dx.
\end{align*}
\end{Proof}
\begin{Lemma}[Heiss, Teichmann, and
Wutte (2019, Lemma A.13)]
Using the notation of Definition .. and ... the following statement
holds:
$\forall \varepsilon \in \mathbb{R}_{>0} : \exists \delta \in
\mathbb{R}_{>0} : \forall \omega \in \Omega : \forall l, l' \in
\left\{1,\dots,N\right\} : \forall n \in \mathbb{N}$
\[
\left(\abs{\xi_l(\omega) - \xi_{l'}(\omega)} < \delta \angle
\text{sign}(v_l(\omega)) = \text{sign}(v_{l'}(\omega))\right)
\implies \abs{\frac{w_l^{*, \tilde{\lambda}}(\omega)}{v_l(\omega)}
- \frac{w_{l'}^{*, \tilde{\lambda}}(\omega)}{v_{l'}(\omega)}} <
\frac{\varepsilon}{n},
\]
if we assume that $v_k$ is never zero.
\proof given in ..
\end{Lemma}
4 years ago
\input{Appendix_code.tex}
\end{appendices}
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