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