You cannot select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
229 lines
8.8 KiB
TeX
229 lines
8.8 KiB
TeX
|
|
\newpage
|
|
\begin{appendices}
|
|
\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}
|
|
|
|
\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
|
|
\[
|
|
\max_{k \in \left\{1,\dots,n\right\}} \frac{w^{*,
|
|
\tilde{\lambda}}_k}{v_k} = \po_{n\to\infty}
|
|
\]
|
|
|
|
|
|
\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:
|
|
$\forall \varepsilon \in \mathbb{R}_{>0} : \exists \delta \in
|
|
\mathbb{R}_{>0} : \forall \omega \in \Omega : \forall l, l' \in
|
|
\left\{1,\dots,N\right\} : \forall n \in \mathbb{N}$
|
|
\[
|
|
\left(\abs{\xi_l(\omega) - \xi_{l'}(\omega)} < \delta \wedge
|
|
\text{sign}(v_l(\omega)) = \text{sign}(v_{l'}(\omega))\right)
|
|
\implies \abs{\frac{w_l^{*, \tilde{\lambda}}(\omega)}{v_l(\omega)}
|
|
- \frac{w_{l'}^{*, \tilde{\lambda}}(\omega)}{v_{l'}(\omega)}} <
|
|
\frac{\varepsilon}{n},
|
|
\]
|
|
if we assume that $v_k$ is never zero.
|
|
\proof given in ..
|
|
\end{Lemma}
|
|
|
|
\begin{Lemma}[$\frac{w^{*,\tilde{\lambda}}}{v} \approx
|
|
\mathcal{O}(\frac{1}{n})$, Heiss, Teichmann, and
|
|
Wutte (2019, Lemma A.14)]
|
|
For any $\lambda > 0$ and data $(x_i^{\text{train}},
|
|
y_i^{\text{train}}) \in \mathbb{R}^2, i\in
|
|
\left\{1,\dots,\right\}$, we have
|
|
\[
|
|
\forall P \in (0,1) : \exists C \in \mathbb{R}_{>0} : \exists
|
|
n_0 \in \mathbb{N} : \forall n > n_0 : \mathbb{P}
|
|
\left[\max_{k\in \left\{1,\dots,n\right\}}
|
|
\frac{w_k^{*,\tilde{\lambda}}}{v_k} < C
|
|
\frac{1}{n}\right] > P
|
|
% \max_{k\in \left\{1,\dots,n\right\}}
|
|
% \frac{w_k^{*,\tilde{\lambda}}}{v_k} = \plimn
|
|
\]
|
|
\proof
|
|
Let $k^*_+ \in \argmax_{k\in
|
|
\left\{1,\dots,n\right\}}\frac{w^{*,\tilde{\lambda}}}{v_k} : v_k
|
|
> 0$ and $k^*_- \in \argmax_{k\in
|
|
\left\{1,\dots,n\right\}}\frac{w^{*,\tilde{\lambda}}}{v_k} : v_k
|
|
< 0$. W.l.o.g. assume $\frac{w_{k_+^*}^2}{v_{k_+^*}^2} \geq
|
|
\frac{w_{k_-^*}^2}{v_{k_-^*}^2}$
|
|
\begin{align*}
|
|
\frac{F^{\lambda,
|
|
g}\left(f^{*,\lambda}_g\right)}{\tilde{\lambda}}
|
|
\makebox[2cm][c]{$\stackrel{\mathbb{P}}{\geq}$}
|
|
& \frac{1}{2 \tilde{\lambda}}
|
|
F_n^{\tilde{\lambda}}\left(\mathcal{RN}^{*,\tilde{\lambda}}\right)
|
|
= \frac{1}{2 \tilde{\lambda}}\left[\sum ... + \tilde{\lambda} \norm{w}_2^2\right]
|
|
\\
|
|
\makebox[2cm][c]{$\geq$}
|
|
& \frac{1}{2}\left( \sum_{\substack{k: v_k
|
|
> 0 \\\xi_k\in(\xi_{k^*}, \xi_{k^*}
|
|
+ \delta)}} \left(w_k^{*,\tilde{\lambda}}\right)^2 +
|
|
\sum_{\substack{k: v_k < 0 \\\xi_k\in(\xi_{k^*}, \xi_{k^*}
|
|
+ \delta)}} \left(w_k^{*,\tilde{\lambda}}\right)^2\right) \\
|
|
\makebox[2cm][c]{$\overset{\text{Lem. A.6}}{\underset{\delta \text{
|
|
small enough}}{\geq}} $}
|
|
&
|
|
\frac{1}{4}\left(\left(\frac{w_{k_+^*}^{*,\tilde{\lambda}}}
|
|
{v_{k_+^*}}\right)^2\sum_{\substack{k:
|
|
v_k > 0 \\\xi_k\in(\xi_{k^*}, \xi_{k^*} + \delta)}}v_k^2 +
|
|
\left(\frac{w_{k_-^*}^{*,\tilde{\lambda}}}{v_{k_-^*}}\right)^2
|
|
\sum_{\substack{k:
|
|
v_k < 0 \\\xi_k\in(\xi_{k^*}, \xi_{k^*} +
|
|
\delta)}}v_k^2\right)\\
|
|
\makebox[2cm][c]{$\stackrel{\mathbb{P}}{\geq}$}
|
|
& \frac{1}{8}
|
|
\left(\frac{w_{k_+^*}^{*,\tilde{\lambda}}}{v_{k^*}}\right)^2
|
|
n \delta g_\xi(\xi_{k_+^*}) \mathbb{P}(v_k
|
|
>0)\mathbb{E}[v_k^2|\xi_k = \xi_{k^*_+}]
|
|
\end{align*}
|
|
|
|
\end{Lemma}
|
|
|
|
\input{Appendix_code.tex}
|
|
|
|
\end{appendices}
|
|
|
|
|
|
|
|
|
|
|
|
|
|
%%% Local Variables:
|
|
%%% mode: latex
|
|
%%% TeX-master: "main"
|
|
%%% End:
|
|
|