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\section{Shallow Neural Networks}

% In order to get a some understanding of the behavior of neural
% networks we study a simplified class of networks called shallow neural
% networks in this chapter.
% We consider shallow neural networks consist of a single
% hidden layer and
In order to get some understanding of the behavior of neural networks
we examine a simple class of networks in this chapter. We consider
networks that contain only one hidden layer and have a single output
node. We call these networks shallow neural networks.
\begin{Definition}[Shallow neural network, Heiss, Teichmann, and
  Wutte (2019, Definition 1.4)]
  For a input dimension $d$ and a Lipschitz continuous activation function $\sigma:
  \mathbb{R} \to \mathbb{R}$ we define a shallow neural network with
  $n$ hidden nodes as
  $\mathcal{NN}_\vartheta : \mathbb{R}^d \to \mathbb{R}$ as
  \[
    \mathcal{NN}_\vartheta \coloneqq \sum_{k=1}^n w_k \sigma\left(b_k +
    \sum_{j=1}^d v_{k,j} x_j\right) + c ~~ \forall x \in \mathbb{R}^d
\]
with
\begin{itemize}
  \item weights $w_k \in \mathbb{R},~k \in \left\{1,\dots,n\right\}$
  \item biases $b_k \in \mathbb{R},~k \in \left\{1, \dots,n\right\}$
  \item weights $v_k \in \mathbb{R}^d,~k\in\left\{1,\dots,n\right\}$
  \item bias $c \in \mathbb{R}$
  \item these weights and biases collected in
  \[
    \vartheta \coloneqq (w, b, v, c) \in \Theta \coloneqq
    \mathbb{R}^{n \times n \times (n \times d) \times 1}
  \]
\end{itemize}
\end{Definition}
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%   \caption{Shallow Neural Network with input- and output-dimension of \(d
%     = 1\)}
%   \label{fig:shallowNN}
% \end{figure}

As neural networks with a large amount of nodes have a large amount of
parameters that can be tuned it can often fit the data quite well. If
a ReLU activation function
\[
  \sigma(x) \coloneqq \max{(0, x)}
\]
is chosen one can easily prove that if the
amount of hidden nodes exceeds the
amount of data points in the training data a shallow network trained
on MSE will perfectly fit the data.
\begin{Theorem}[Shallow neural network can fit data perfectly]
  For training data of size t
  \[
    \left(x_i^{\text{train}}, y_i^{\text{train}}\right) \in \mathbb{R}^d
    \times \mathbb{R},~i\in\left\{1,\dots,t\right\} 
  \]
  a shallow neural network $\mathcal{NN}_\vartheta$ with $n \geq t$
  hidden nodes will perfectly fit the data when 
  minimizing squared error loss.
  \proof
  W.l.o.g. all values $x_{ij}^{\text{train}} \in [0,1],~\forall i \in
  \left\{1,\dots\right\}, j \in \left\{1,\dots,d\right\}$. Now we
  chose $v^*$ in order to calculate a unique value for all
  $x_i^{\text{train}}$:
  \[
    v^*_{k,j} = v^*_{j} = 10^{j-1}, ~ \forall k \in \left\{1,\dots,n\right\}.
  \]
  Assuming $x_i^{\text{train}} \neq x_j^{\text{train}},~\forall i\neq
  j$ we get
  \[
    \left(v_k^*\right)^{\mathrm{T}} x_i^{\text{train}} \neq
    \left(v_k^*\right)^{\mathrm{T}} x_j^{\text{train}}, ~ \forall i
    \neq j.
  \]
  W.l.o.g assume $x_i^{\text{train}}$ are ordered such that
  $\left(v_k^*\right)^{\mathrm{T}} x_i^{\text{train}} <
    \left(v_k^*\right)^{\mathrm{T}} x_j^{\text{train}}, ~\forall j<j$,
  Then we can choose $b^*_k$ such that neuron $k$ is only active for all
  $x_i^{\text{train}}$ with $i \geq k$:
  \begin{align*}
    b^*_1 &> -\left(v^*\right)^{\mathrm{T}} x_1^{\text{train}},\\
    b^*_k &= -\left(v^*\right)^{\mathrm{T}}
            x_{k-1}^{\text{train}},~\forall k \in \left\{2, \dots,
            t\right\}, \\
    b_k^* &\leq -\left(v^*\right)^{\mathrm{T}}
            x_{t}^{\text{train}},~\forall k > t.
  \end{align*}
  With
  \begin{align*}
    w_k^* &= \frac{y_k^{\text{train}} - \sum_{j =1}^{k-1} w^*_j\left(b^*_j +
        x_k^{\text{train}}\right)}{b_k + \left(v^*\right)^{\mathrm{T}}
      x_k^{\text{train}}},~\forall k \in \left\{1,\dots,t\right\}\\
    w_k^* &\in \mathbb{R} \text{ arbitrary, } \forall k > t.
  \end{align*}
  and $\vartheta^* = (w^*, b^*, v^*, c = 0)$ we get
  \[
    \mathcal{NN}_{\vartheta^*} (x_i^{\text{train}}) = \sum_{k =
      1}^{i-1} w_k\left(\left(v^*\right)^{\mathrm{T}}
      x_i^{\text{train}}\right) + w_i\left(\left(v^*\right)^{\mathrm{T}}
      x_i^{\text{train}}\right) = y_i^{\text{train}}.
  \]
  As the squared error of $\mathcal{NN}_{\vartheta^*}$ is zero all
  squared error loss minimizing shallow networks with at least $t$ hidden
  nodes will perfectly fit the data.
  \qed
  \label{theo:overfit}
\end{Theorem}

However this behavior is often not desired as over fit models generally
have bad generalization properties especially if noise is present in
the data. This effect is illustrated in
Figure~\ref{fig:overfit}. Here a shallow neural network that perfectly fits the
training data is
constructed according to the proof of Theorem~\ref{theo:overfit} and
compared to a cubic smoothing spline 
(Definition~\ref{def:wrs}). While the neural network
fits the data better than the spline, the spline represents the
underlying mechanism that was used to generate the data more accurately. The better
generalization of the spline compared to the network is further
demonstrated by the better validation error computed on newly generated
test data.
In order to improve the accuracy of the model we want to reduce
overfitting. A possible way to achieve this is by explicitly
regularizing the network through the cost function as done with
ridge penalized networks  
(Definition~\ref{def:rpnn}) where large weights $w$ are punished. In
Theorem~\ref{theo:main1} we will
prove that this will result in the shallow neural network converging to
regressions splines as the amount of nodes in the hidden layer is
increased.




\begin{figure}
    \pgfplotsset{
      compat=1.11,
legend image code/.code={
  \draw[mark repeat=2,mark phase=2]
plot coordinates {
  (0cm,0cm)
(0.15cm,0cm)        %% default is (0.3cm,0cm)
(0.3cm,0cm)         %% default is (0.6cm,0cm)
};%
}
}
    \begin{tikzpicture}
      \begin{axis}[tick style = {draw = none}, width = \textwidth,
        height = 0.6\textwidth]
        \addplot table
        [x=x, y=y, col sep=comma, only marks,mark options={scale =
          0.7}] {Figures/Data/overfit.csv};  
        \addplot [red, line width=0.8pt] table [x=x_n, y=s_n, col
        sep=comma, forget plot] {Figures/Data/overfit.csv}; 
        \addplot [black, line width=0.8pt] table [x=x_n, y=y_n, col
        sep=comma] {Figures/Data/overfit.csv}; 
        \addplot [black, line width=0.8pt, dashed] table [x=x, y=y, col
        sep=comma] {Figures/Data/overfit_spline.csv};

        \addlegendentry{\footnotesize{data}};
        \addlegendentry{\footnotesize{$\mathcal{NN}_{\vartheta^*}$}};
        \addlegendentry{\footnotesize{spline}};
      \end{axis}
    \end{tikzpicture}
  \caption[Overfitting of shallow neural networks]{For data of the form $y=\sin(\frac{x+\pi}{2 \pi}) +
    \varepsilon,~ \varepsilon \sim \mathcal{N}(0,0.4)$
    (\textcolor{blue}{blue dots}) the neural network constructed 
    according to the proof of Theorem~\ref{theo:overfit} (black) and the
    underlying signal (\textcolor{red}{red}). While the network has no
    bias a cubic smoothing spline (black dashed) fits the data much
    better. For a test set of size 20 with uniformly distributed $x$
    values and responses of the same fashion as the training data the MSE of the neural network is
    0.30, while the MSE of the spline is only 0.14 thus generalizing
    much better.
  }
  \label{fig:overfit}
\end{figure}

\clearpage
\subsection{Convergence Behaviour of 1-dim. Randomized Shallow Neural
  Networks} 


This section is based on \textcite{heiss2019}.

... shallow neural networks with a one dimensional input where the parameters in the
hidden layer are randomized resulting in only the weights is the
output layer being trainable.
Additionally we assume all neurons use a ReLU as activation function
and call such networks randomized shallow neural networks. 

% We will analyze the
% connection between randomized shallow
% Neural Networks with one dimensional input with a ReLU as activation
% function for all neurons and cubic smoothing splines.
% % \[
% %   \sigma(x) = \max\left\{0,x\right\}.
% % \]
% We will see that the punishment of the size of the weights in training
% the randomized shallow
% Neural Network will result in a learned function that minimizes the second
% derivative as the amount of hidden nodes is grown to infinity. In order
% to properly formulate this relation we will first need to introduce
% some definitions, all neural networks introduced in the following will
% use a ReLU as activation at all neurons.

% A randomized shallow network is characterized by only the weight
% parameter of the output layer being trainable, whereas the other
% parameters are random numbers.

\begin{Definition}[Randomized shallow neural network, Heiss, Teichmann, and
  Wutte (2019, Definition 2.1)]
  For an input dimension $d$, let $n \in \mathbb{N}$ be the number of
  hidden nodes and $v(\omega) \in \mathbb{R}^{i \times n}, b(\omega)
  \in \mathbb{R}^n$ randomly drawn weights. Then for a weight vector
  $w$ the corresponding randomized shallow neural network is given by
\[
  \mathcal{RN}_{w, \omega} (x) = \sum_{k=1}^n w_k
  \sigma\left(b_k(\omega) + \sum_{j=1}^d v_{k, j}(\omega) x_j\right).
\]
\label{def:rsnn}
\end{Definition}
% We call a one dimensional randomized shallow neural network were the
%  are penalized in the loss
% function ridge penalized neural networks.


We will prove that ... nodes ..  a randomized shallow neural network will
converge to a function that minimizes the distance to the training
data with .. to its second derivative,
if the $L^2$ norm of the trainable weights $w$ is
penalized in the loss function.
We call such a network that is fitted according to MSE and a penalty term for
the amount of the weights a ridge penalized neural network.
% $\lam$
% We call a randomized shallow neural network trained on MSE and
% punished for the amount of the weights $w$ according to a
% ... $\lambda$ ridge penalized neural networks.

% We call a randomized shallow neural network where the size of the trainable
% weights is punished in the error function a ridge penalized
% neural network. For a tuning parameter $\tilde{\lambda}$ .. the extent
% of penalization we get:
\begin{Definition}[Ridge penalized Neural Network, Heiss, Teichmann, and
  Wutte (2019, Definition 3.2)]
  \label{def:rpnn}
  Let $\mathcal{RN}_{w, \omega}$ be a randomized shallow neural
  network, as introduced in Definition~\ref{def:rsnn} and tuning
  parameter $\tilde{\lambda} \in \mathbb{R}$. Then the optimal ridge
  penalized 
  network is given by
  \[
    \mathcal{RN}^{*, \tilde{\lambda}}_{\omega}(x) \coloneqq
    \mathcal{RN}_{w^{*, \tilde{\lambda}}(\omega), \omega}
  \]
  with
  \[
    w^{*,\tilde{\lambda}}(\omega) :\in \argmin_{w \in
      \mathbb{R}^n} \underbrace{ \left\{\overbrace{\sum_{i = 1}^N \left(\mathcal{RN}_{w,
              \omega}(x_i^{\text{train}}) -
            y_i^{\text{train}}\right)^2}^{L(\mathcal{RN}_{w, \omega})} +
        \tilde{\lambda} \norm{w}_2^2\right\}}_{\eqqcolon F_n^{\tilde{\lambda}}(\mathcal{RN}_{w,\omega})}.
  \]
\end{Definition}
If the amount of hidden nodes $n$ is larger than the amount of
training samples $N$ then for 
$\tilde{\lambda} \to 0$ the network will interpolate the data while
having minimal weights, resulting in the \textit{minimum norm
  network} $\mathcal{RN}_{w^{\text{min}}, \omega}$.
\[
  \mathcal{RN}_{w^{\text{min}}, \omega} \text{ randomized shallow
    Neural network with weights } w^{\text{min}}:
\]
\[
  w^{\text{min}} \in \argmin_{w \in \mathbb{R}^n} \norm{w}, \text{
    s.t. }
  \mathcal{RN}_{w,\omega}(x_i^{\text{train}}) = y_i^{\text{train}}, \, \forall i \in
  \left\{1,\dots,N\right\}. 
\]
For $\tilde{\lambda} \to \infty$ the learned
function will resemble the data less and with the weights
approaching $0$ will converge to the constant $0$ function.

In order to make the notation more convinient in the following the
$\omega$ used to express the realised random parameters will no longer
be explicitly mentioned.

We call a function that minimizes the cubic distance between training points
and the function with regard to the second
derivative of the function a cubic smoothing spline.

\begin{Definition}[Cubic Smoothing Spline]
  Let $x_i^{\text{train}}, y_i^{\text{train}} \in \mathbb{R}, i \in
  \left\{1,\dots,N\right\}$ be trainig data. for a given $\lambda \in
  \mathbb{R}$ the cubic smoothing spline is given by
  \[
    f^{*,\lambda} :\in \argmin_{f \in
      \mathcal{C}^2}\left\{\sum_{i=1}^N
      \left(f\left(x_i^{\text{train}}\right) -
        y_i^{\text{train}}\right)^2 + \lambda \int f^{''}(x)^2dx\right\}.
  \]
\end{Definition}

We will show that for specific hyper parameters the ridge penalized
shallow neural networks converge to a slightly modified variant of the
cubic smoothing spline. We will need to incorporate the densities of the
random parameters in the loss function of the spline to ensure
convergence. Thus we define 
the adapted weighted cubic smoothing spline where the loss for the second
derivative is weighted by a function $g$ and the support of the second
derivative of $f$ has to be a subset the support of $g$. The formal
definition is given in Definition~\ref{def:wrs}.

% We will later ... the converging .. of the ridge penalized shallow
% neural network, in order to do so we will need a slightly modified
% version of the regression
% spline that allows for weighting the penalty term for the second
% derivative with a weight function $g$. This is needed to ...the
% distributions of the random parameters ... We call this the adapted
% weighted cubic smoothing spline.

% Now we take a look at weighted cubic smoothing splines. Later we will prove
% that the ridge penalized neural network as defined in
% Definition~\ref{def:rpnn} converges a weighted cubic smoothing spline, as
% the amount of hidden nodes is grown to inifity.

\begin{Definition}[Adapted weighted cubic smoothing spline]
  \label{def:wrs}
  Let $x_i^{\text{train}}, y_i^{\text{train}} \in \mathbb{R}, i \in
  \left\{1,\dots,N\right\}$ be trainig data. For a given $\lambda \in \mathbb{R}_{>0}$
  and a function $g: \mathbb{R} \to \mathbb{R}_{>0}$ the weighted
  cubic smoothing spline $f^{*, \lambda}_g$ is given by

  \[
    f^{*, \lambda}_g :\in \argmin_{\substack{f \in \mathcal{C}^2(\mathbb{R})
        \\ \supp(f'') \subseteq \supp(g)}} \underbrace{\left\{ \overbrace{\sum_{i =
            1}^N \left(f(x_i^{\text{train}}) - y_i^{\text{train}}\right)^2}^{L(f)} +
        \lambda g(0) \int_{\supp(g)}\frac{\left(f''(x)\right)^2}{g(x)}
        dx\right\}}_{\eqqcolon F^{\lambda, g}(f)}.
  \]
  \todo{Anforderung an Ableitung von f, doch nicht?}
\end{Definition}

Similarly to ridge weight penalized neural networks the parameter
$\lambda$ controls a trade-off between accuracy on the training data
and smoothness or low second dreivative. For $g \equiv 1$ and $\lambda \to 0$ the
resulting function $f^{*, 0+}$ will interpolate the training data while minimizing
the second derivative. Such a function is known as cubic spline
interpolation.

\[
  f^{*, 0+} \text{ smooth spline interpolation: }
\]
\[
  f^{*, 0+} \coloneqq \lim_{\lambda \to 0+} f^{*, \lambda}_1 \in
  \argmin_{\substack{f \in \mathcal{C}^2(\mathbb{R}), \\ f(x_i^{\text{train}}) =
      y_i^{\text{train}}}} = \left( \int _{\mathbb{R}} (f''(x))^2dx\right).
\]

For $\lambda \to \infty$ on the other hand $f_g^{*\lambda}$ converges
to linear regression of the data.

We use two intermediary functions in order to show the convergence of
the ridge penalized shallow neural network to adapted cubic smoothing splines.
% In order to show that ridge penalized shallow neural networks converge
% to adapted cubic smoothing splines for a growing amount of hidden nodes we
% define two intermediary functions.
One being a smooth approximation of
the neural network, and a randomized shallow neural network designed
to approximate a spline.
In order to properly BUILD these functions we need to take the points
of the network into consideration where the TRAJECTORY of the learned
function changes
(or their points of discontinuity).
As we use the ReLU activation the function learned by the
network will possess points of discontinuity where a neuron in the hidden
layer gets activated (goes from 0 -> x>0). We formalize these points
as kinks in Definition~\ref{def:kink}.
\begin{Definition}
  \label{def:kink}
  Let $\mathcal{RN}_w$ be a randomized shallow Neural
  Network according to Definition~\ref{def:rsnn}, then kinks depending
  on the random parameters can
  be observed.
  \[
    \mathcal{RN}_w(x) = \sum_{k = 1}^n w_k \sigma(b_k + v_kx)
  \]
  Because we specified $\sigma(y) \coloneqq \max\left\{0, y\right\}$ a
  kink in $\sigma$ can be observed at $\sigma(0) = 0$. As $b_k + v_kx = 0$ for $x
  = -\frac{b_k}{v_k}$ we define the following:
  \begin{enumerate}[label=(\alph*)]
    \item Let $\xi_k \coloneqq -\frac{b_k}{v_k}$ be the k-th kink of $\mathcal{RN}_w$.
    \item Let $g_{\xi}(\xi_k)$ be the density of the kinks $\xi_k =
    - \frac{b_k}{v_k}$ in accordance to the distributions of $b_k$ and
    $v_k$.
    \item Let $h_{k,n} \coloneqq \frac{1}{n g_{\xi}(\xi_k)}$ be the
    average estmated distance from kink $\xi_k$ to the next nearest
    one.
  \end{enumerate}
\end{Definition}

Using the density of the kinks we construct a kernel and smooth the
network by applying the kernel similar to convolution.

\begin{Definition}[Smooth Approximation of Randomized Shallow Neural
  Network]
  \label{def:srsnn}
  Let $RS_{w}$ be a randomized shallow Neural Network according to
  Definition~\ref{def:rsnn} with weights $w$ and kinks $\xi_k$ with
  corresponding kink density $g_{\xi}$ as given by
  Definition~\ref{def:kink}.
  In order to smooth the RSNN consider following kernel for every $x$:

  \[
    \kappa_x(s) \coloneqq \mathds{1}_{\left\{\abs{s} \leq \frac{1}{2 \sqrt{n}
          g_{\xi}(x)}\right\}}(s)\sqrt{n} g_{\xi}(x), \, \forall s \in \mathbb{R}
  \]

  Using this kernel we define a smooth approximation of
  $\mathcal{RN}_w$ by

  \[
    f^w(x) \coloneqq \int_{\mathds{R}} \mathcal{RN}_w(x-s) \kappa_x(s) ds.
  \]
\end{Definition}

Note that the kernel introduced in Definition~\ref{def:srsnn}
satisfies $\int_{\mathbb{R}}\kappa_x dx = 1$. While $f^w$ looks highly
similar to a convolution, it differs slightly as the kernel $\kappa_x(s)$
is dependent on $x$. Therefore only $f^w = (\mathcal{RN}_w *
\kappa_x)(x)$ is well defined, while $\mathcal{RN}_w * \kappa$ is not.
We use $f^{w^{*,\tilde{\lambda}}}$ to describe the spline
approximating the ridge penalized network
$\mathrm{RN}^{*,\tilde{\lambda}}$.

Next we construct a randomized shallow neural network which
approximates a spline independent from the realization of the random
parameters. In order to achieve this we ...

\begin{Definition}[Spline approximating Randomised Shallow Neural
  Network]
  \label{def:sann}
  Let $\mathcal{RN}$ be a randomised shallow Neural Network according
  to Definition~\ref{def:rsnn} and $f^{*, \lambda}_g$ be the weighted
  cubic smoothing spline as introduced in Definition~\ref{def:wrs}. Then
  the randomised shallow neural network approximating $f^{*,
    \lambda}_g$ is given by 
  \[
    \mathcal{RN}_{\tilde{w}}(x) = \sum_{k = 1}^n \tilde{w}_k \sigma(b_k + v_k x),
  \]
  with the weights $\tilde{w}_k$ defined as
  \[
    \tilde{w}_k \coloneqq \frac{h_{k,n} v_k}{\mathbb{E}[v^2 \vert \xi
      = \xi_k]}  (f_g^{*, \lambda})''(\xi_k).
  \]
\end{Definition}

The approximating nature of the network in
Definition~\ref{def:sann} can be seen by examining the first
derivative of $\mathcal{RN}_{\tilde{w}}(x)$ which is given by
\begin{align}
  \frac{\partial \mathcal{RN}_{\tilde{w}}}{\partial x}
  \Big{|}_{x} &= \sum_k^n \tilde{w}_k \mathds{1}_{\left\{b_k + v_k x >
                0\right\}}(v_k) = \sum_{\substack{k \in \mathbb{N} \\ \xi_k <
  x}} \tilde{w}_k v_k \nonumber \\
              &= \frac{1}{n} \sum_{\substack{k \in \mathbb{N} \\
  \xi_k < x}} \frac{v_k^2}{g_{\xi}(\xi_k) \mathbb{E}[v^2 \vert \xi
  = \xi_k]} (f_g^{*, \lambda})''(\xi_k). \label{eq:derivnn}
\end{align}
As the expression (\ref{eq:derivnn}) behaves similary to a
Riemann-sum for $n \to \infty$ it will converge in probability to the
first derivative of $f^{*,\lambda}_g$. A formal proof of this behaviour
is given in Lemma~\ref{lem:s0}.

In order to ensure the functions used in the proof of the convergence
are well defined we need to assume some properties of the random
parameters and their densities

% In order to formulate the theorem describing the convergence of $RN_w$
% we need to make a couple of assumptions.
% \todo{Bessere Formulierung}

\begin{Assumption}~
  \label{ass:theo38}
  \begin{enumerate}[label=(\alph*)]
    \item The probability density fucntion of the kinks $\xi_k$,
    namely $g_{\xi}$ as defined in Definition~\ref{def:kink} exists
    and is well defined.
    \item The density function $g_\xi$
    has compact support on $\supp(g_{\xi})$.
    \item The density function $g_{\xi}$ is uniformly continuous on $\supp(g_{\xi})$.
    \item $g_{\xi}(0) \neq 0$.
    \item $\frac{1}{g_{\xi}}\Big|_{\supp(g_{\xi})}$ is uniformly
    continous on $\supp(g_{\xi})$.
    \item The conditional distribution $\mathcal{L}(v_k|\xi_k = x)$
    is uniformly continous on $\supp(g_{\xi})$.
    \item $\mathbb{E}\left[v_k^2\right] < \infty$.
  \end{enumerate}
\end{Assumption}

As we will prove the convergence of in the Sobolev space, we hereby
introduce it and the corresponding induced norm.

\begin{Definition}[Sobolev Space]
  For $K \subset \mathbb{R}^n$ open and $1 \leq p \leq \infty$ we
  define the Sobolev space $W^{k,p}(K)$ as the space containing all
  real valued functions $u \in L^p(K)$ such that for every multi-index
  $\alpha \in \mathbb{N}^n$ with $\abs{\alpha} \leq
  k$ the mixed parial derivatives
  \[
    u^{(\alpha)} = \frac{\partial^{\abs{\alpha}} u}{\partial
      x_1^{\alpha_1} \dots \partial x_n^{\alpha_n}}
  \]
  exists in the weak sense and
  \[
    \norm{u^{(\alpha)}}_{L^p} < \infty.
  \]
  \label{def:sobonorm}
  The natural norm of the sobolev space is given by
  \[
    \norm{f}_{W^{k,p}(K)} =
    \begin{cases}
      \left(\sum_{\abs{\alpha} \leq k}
        \norm{f^{(\alpha)}}^p_{L^p}\right)^{\nicefrac{1}{p}},&
      \text{for } 1 \leq p < \infty \\
      max_{\abs{\alpha} \leq k}\left\{f^{(\alpha)}\right\},& \text{for
      } p = \infty
    \end{cases}
    .
  \]
\end{Definition}

With the important definitions and assumptions in place we can now
formulate the main theorem ... the convergence of ridge penalized
random neural networks to adapted cubic smoothing splines when the
parameters are chosen accordingly.

\begin{Theorem}[Ridge weight penaltiy corresponds to weighted cubic smoothing spline]
  \label{theo:main1}
  For $N \in \mathbb{N}$ arbitrary training data
  \(\left(x_i^{\text{train}}, y_i^{\text{train}}
  \right)\) and $\mathcal{RN}^{*, \tilde{\lambda}}, f_g^{*, \lambda}$
  according to Definition~\ref{def:rpnn} and Definition~\ref{def:wrs}
  respectively with Assumption~\ref{ass:theo38} it holds

  \begin{equation}
    \label{eq:main1}
    \plimn \norm{\mathcal{RN^{*, \tilde{\lambda}}} - f^{*,
        \lambda}_{g}}_{W^{1,\infty}(K)} = 0.
  \end{equation}
  
  With
  \begin{align*}
    g(x) & \coloneqq g_{\xi}(x)\mathbb{E}\left[ v_k^2 \vert \xi_k = x
           \right], \forall x \in \mathbb{R}, \\
    \tilde{\lambda} & \coloneqq \lambda n g(0).
  \end{align*}
\end{Theorem}
As mentioned above we will prof Theorem~\ref{theo:main1} utilizing
the ... functions. We show that 
\begin{equation}
  \label{eq:main2}
  \plimn \norm{\mathcal{RN}^{*, \tilde{\lambda}} - f^{w^*}}_{W^{1,
      \infty}(K)} = 0
\end{equation}
and
\begin{equation}
  \label{eq:main3}
  \plimn \norm{f^{w^*} - f_g^{*, \lambda}}_{W^{1,\infty}(K)} = 0
\end{equation}
and then get (\ref{eq:main1}) using the triangle inequality. In
order to prove (\ref{eq:main2}) and (\ref{eq:main3}) we will need to
introduce a number of auxiliary lemmmata, proves of these will be
provided in the appendix.



\begin{Lemma}[Poincar\'e typed inequality]
  \label{lem:pieq}
  Let \(f:\mathbb{R} \to \mathbb{R}\) differentiable with \(f' :
  \mathbb{R} \to \mathbb{R}\) Lesbeque integrable. Then for \(K=[a,b]
  \subset \mathbb{R}\) with \(f(a)=0\) it holds that
  \begin{equation*}
    \label{eq:pti1}
    \exists C_K^{\infty} \in \mathbb{R}_{>0} :
    \norm{f}_{w^{1,\infty}(K)} \leq C_K^{\infty}
    \norm{f'}_{L^{\infty}(K)}.
  \end{equation*}
  If additionaly \(f'\) is differentiable with \(f'': \mathbb{R} \to
  \mathbb{R}\) Lesbeque integrable then additionally
  \begin{equation*}
    \label{eq:pti2}
    \exists C_K^2 \in \mathbb{R}_{>0} : \norm{f}_{W^{1,\infty}(K)} \leq
    C_K^2 \norm{f''}_{L^2(K)}.
  \end{equation*}
  \proof The proof is given in the appendix...
  % With the fundamental theorem of calculus, if
  % \(\norm{f}_{L^{\infty}(K)}<\infty\) we get
  % \begin{equation}
  %   \label{eq:f_f'}
  %   \norm{f}_{L^{\infty}(K)} = \sup_{x \in K}\abs{\int_a^x f'(s) ds} \leq
  %   \sup_{x \in K}\abs{\int_a^x \sup_{y \in K} \abs{f'(y)} ds} \leq \abs{b-a}
  %   \sup_{y \in K}\abs{f'(y)}.
  % \end{equation}
  % Using this we can bound \(\norm{f}_{w^{1,\infty}(K)}\) by
  % \[
  %   \norm{f}_{w^{1,\infty}(K)} \stackrel{\text{Def~\ref{def:sobonorm}}}{=}
  %   \max\left\{\norm{f}_{L^{\infty}(K)},
  %     \norm{f'}_{L^{\infty}(K)}\right\}
  %   \stackrel{(\ref{eq:f_f'})}{\leq} max\left\{\abs{b-a},
  %     1\right\}\norm{f'}_{L^{\infty}(K)}.
  % \]
  % With \(C_k^{\infty} \coloneqq max\left\{\abs{b-a}, 1\right\}\) we
  % get (\ref{eq:pti1}).
  % By using the Hölder inequality, we can proof the second claim.
  % \begin{align*}
  %   \norm{f'}_{L^{\infty}(K)} &= \sup_{x \in K} \abs{\int_a^bf''(y)
  %   \mathds{1}_{[a,x]}(y)dy} \leq \sup_{x \in
  %   K}\norm{f''\mathds{1}_{[a,x]}}_{L^1(K)}\\
  %   &\hspace{-6pt} \stackrel{\text{Hölder}}{\leq} sup_{x
  %   \in
  %   K}\norm{f''}_{L^2(K)}\norm{\mathds{1}_{[a,x]}}_{L^2(K)}
  %   = \abs{b-a}\norm{f''}_{L^2(K)}.
  % \end{align*}
  % Thus (\ref{eq:pti2}) follows with \(C_K^2 \coloneqq
  % \abs{b-a}C_K^{\infty}\).
  % \qed
\end{Lemma}

\begin{Lemma}
  \label{lem:cnvh}
  Let $\mathcal{RN}$ be a shallow Neural network. For \(\varphi :
  \mathbb{R}^2 \to \mathbb{R}\) uniformly continous such that
  \[
    \forall x \in  \supp(g_{\xi}) : \mathbb{E}\left[\varphi(\xi, v)
      \frac{1}{n g_{\xi}(\xi)} \vert \xi = x \right] < \infty,
  \]
  it holds, that
  \[
    \plimn \sum_{k \in \kappa : \xi_k < T} \varphi(\xi_k, v_k)
    h_{k,n}
    =\int_{\min\left\{C_{g_{\xi}}^l, T\right\}}^{min\left\{C_{g_{\xi}}^u,T\right\}}
    \mathbb{E}\left[\varphi(\xi, v) \vert \xi = x \right] dx
  \]
  uniformly in \(T \in K\).
  \proof The proof is given in appendix...
  % For \(T \leq C_{g_{\xi}}^l\) both sides equal 0, so it is sufficient to
  % consider \(T > C_{g_{\xi}}^l\). With \(\varphi\) and
  % \(\nicefrac{1}{g_{\xi}}\) uniformly continous in \(\xi\),
  % \begin{equation}
  %   \label{eq:psi_stet}
  %   \forall \varepsilon > 0 : \exists \delta(\varepsilon) : \forall
  %   \abs{\xi - \xi'} < \delta(\varepsilon) : \abs{\varphi(\xi, v)
  %   \frac{1}{g_{\xi}(\xi)} - \varphi(\xi', v)
  %   \frac{1}{g_{\xi}(\xi')}} < \varepsilon
  % \end{equation}
  % uniformly in \(v\). In order to
  % save space we use the notation \((a \wedge b) \coloneqq \min\{a,b\}\) for $a$ and $b
  % \in \mathbb{R}$. W.l.o.g. assume \(\sup(g_{\xi})\) in an
  % intervall. By splitting the interval in disjoint strips of length \(\delta
  % \leq \delta(\varepsilon)\) we get:
  
  % \[
  %   \underbrace{\sum_{k \in \kappa : \xi_k < T} \varphi(\xi_k, v_k)
  %   \frac{\bar{h}_k}{2}}_{\circled{1}} =
  %   \underbrace{\sum_{l \in \mathbb{Z}:
  %   \left[\delta l, \delta (l + 1)\right] \subseteq
  %   \left[C_{g_{\xi}}^l, C_{g_{\xi}}^u \wedge T
  %   \right]}}_{\coloneqq \, l \in I_{\delta}} \left( \, \sum_{\substack{k \in \kappa\\
  %     \xi_k \in \left[\delta l, \delta (l + 1)\right]}}
  %     \varphi\left(\xi_k, v_k\right)\frac{\bar{h}_k}{2} \right)
  % \]
  % Using (\ref{eq:psi_stet}) we can approximate $\circled{1}$ by
  % \begin{align*}
  %   \circled{1} & \approx \sum_{l \in I_{\delta}} \left( \, \sum_{\substack{k \in \kappa\\
  %     \xi_k \in \left[\delta l, \delta (l + 1)\right]}}
  %     \left(\varphi\left(l\delta, v_k\right)\frac{1}{g_{\xi}(l\delta)}
  %       \pm \varepsilon\right)\frac{1}{n} \underbrace{\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\}}}}_{=
  %     1}\right) \\
  %   \intertext{} 
  %   &= \sum_{l \in I_{\delta}} \left( \frac{ \sum_{ \substack{k \in \kappa\\
  %     \xi_k \in \left[\delta l, \delta (l + 1)\right]}}
  %     \varphi\left(l\delta, v_k\right)}
  %     {\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\}}}{ng_{\xi}(l\delta)}\right) \pm \varepsilon .\\
  %   \intertext{We use the mean to approximate the number of kinks in
  %   each $\delta$-strip, as it follows a bonomial distribution this
  %   amounts to
  %   \[
  %     \mathbb{E}\left[\abs{\left\{m \in \kappa : \xi_m \in [\delta l,
  %         \delta(l + 1)]\right\}\right]} = n \int_{[\delta l, \delta (l +
  %     1)]} g_{\xi}(x)dx \approx n (\delta g_{\xi}(l\delta) \pm
  %     \tilde{\varepsilon}).
  %   \]
  %   Bla Bla Bla $v_k$}
  %   \circled{1} & \approx 
  % \end{align*}
\end{Lemma}

\begin{Lemma}
  For any $\lambda > 0$, training data $(x_i^{\text{train}}
  y_i^{\text{train}}) \in \mathbb{R}^2$, with $ i \in
  \left\{1,\dots,N\right\}$ and subset $K \subset \mathbb{R}$ the spline approximating randomized
  shallow neural network $\mathcal{RN}_{\tilde{w}}$ converges to the
  cubic smoothing spline $f^{*, \lambda}_g$ in
  $\norm{.}_{W^{1,\infty}(K)}$ as the node count $n$ increases,
  \begin{equation}
    \label{eq:s0}
    \plimn \norm{\mathcal{RN}_{\tilde{w}} - f^{*, \lambda}_g}_{W^{1,
        \infty}(K)} = 0
  \end{equation}
  \proof
  Using Lemma~\ref{lem:pieq} it is sufficient to show
  \[
    \plimn \norm{\mathcal{RN}_{\tilde{w}}' - (f^{*,
        \lambda}_g)'}_{L^{\infty}} = 0.
  \]
  This can be achieved by using Lemma~\ref{lem:cnvh} with $\varphi(\xi_k,
  v_k) = \frac{v_k^2}{\mathbb{E}[v^2|\xi = z]} (f^{*, \lambda}_w)''(\xi_k) $
  thus obtaining
  \begin{align*}
    \plimn \frac{\partial \mathcal{RN}_{\tilde{w}}}{\partial x}
    \stackrel{(\ref{eq:derivnn})}{=}
 & \plimn \sum_{\substack{k \in \mathbb{N}                         \\ 
    \xi_k < x}} \frac{v_k^2}{\mathbb{E}[v^2 \vert \xi
    = \xi_k]} (f_g^{*, \lambda})''(\xi_k) h_{k,n}
    \stackrel{\text{Lemma}~\ref{lem:cnvh}}{=}                      \\
    \stackrel{\phantom{(\ref{eq:derivnn})}}{=}
 & 
      \int_{\min\left\{C_{g_{\xi}}^l,T\right\}}^{min\left\{C_{g_{\xi}}^u,T\right\}}
      \mathbb{E}\left[\frac{v^2}{\mathbb{E}[v^2|\xi = z]} (f^{*,
      \lambda}_w)''(\xi) \vert
      \xi = x \right] dx \equals^{\text{Tower-}}_{\text{property}} \\
    \stackrel{\phantom{(\ref{eq:derivnn})}}{=}
 & 
      \int_{\min\left\{C_{g_{\xi}}^l,
      T\right\}}^{min\left\{C_{g_{\xi}}^u,T\right\}}(f^{*,\lambda}_w)''(x)
      dx. 
  \end{align*}
  By the fundamental theorem of calculus and $\supp(f') \subset
  \supp(f)$, (\ref{eq:s0}) follows with Lemma~\ref{lem:pieq}.
  \todo{ist die 0 wichtig?}
  \qed
  \label{lem:s0}
\end{Lemma}

\begin{Lemma}
  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
  \[
    \plimn F^{\tilde{\lambda}}_n(\mathcal{RN}_{\tilde{w}}) =
    F^{\lambda, g}(f^{*, \lambda}_g) = 0.
  \]
  \proof
  The proof is given in the appendix...
  \label{lem:s2}
\end{Lemma}

\begin{Lemma}
  For any $\lambda > 0$ and training data $(x_i^{\text{train}},
  y_i^{\text{train}}) \in \mathbb{R}^2, \, i \in
  \left\{1,\dots,N\right\}$, with $w^*$ as
  defined in Definition~\ref{def:rpnn} and $\tilde{\lambda}$ as
  defined in Theroem~\ref{theo:main1}, it holds
  \[
    \plimn \norm{\mathcal{RN}^{*,\tilde{\lambda}} -
      f^{w*, \tilde{\lambda}}}_{W^{1,\infty}(K)} = 0.
  \]
  \proof The proof is given in Appendix ..
  \label{lem:s3}
\end{Lemma}

\begin{Lemma}
  For any $\lambda > 0$ and training data $(x_i^{\text{train}},
  y_i^{\text{train}}) \in \mathbb{R}^2, \, i \in
  \left\{1,\dots,N\right\}$, with $w^*$ and $\tilde{\lambda}$ as
  defined in Definition~\ref{def:rpnn} and Theroem~\ref{theo:main1}
  respectively, it holds
  \[
    \plimn \abs{F_n^{\tilde{\lambda}}(\mathcal{RN}^{*,\tilde{\lambda}}) -
      F^{\lambda, g}(f^{w*, \tilde{\lambda}})} = 0.
  \]
  \proof The proof is given in appendix...
  \label{lem:s4}
\end{Lemma}

\begin{Lemma}
  For any $\lambda > 0$ and training data $(x_i^{\text{train}},
  y_i^{\text{train}}) \in \mathbb{R}^2, \, i \in
  \left\{1,\dots,N\right\}$, for any sequence of functions $f^n \in
  W^{2,2}$ with
  \[
    \plimn F^{\lambda, g} (f^n) = F^{\lambda, g}(f^{*, \lambda}),
  \]
  it follows
  \[
    \plimn \norm{f^n - f^{*, \lambda}} = 0.
  \]
  \proof The proof is given in appendix ...
  \label{lem:s7}
\end{Lemma}
Using these lemmata we can now proof Theorem~\ref{theo:main1}. We
start by showing that the error measure of the smooth approximation of
the ridge penalized randomized shallow neural network $F^{\lambda,
  g}\left(f^{w^{*,\tilde{\lambda}}}\right)$ 
will converge in probability to the error measure of the adapted weighted regression
spline $F^{\lambda, g}\left(f^{*,\lambda}\right)$ for the specified
parameters.

Using Lemma~\ref{lem:s4} we get that for every $P \in (0,1)$ and
$\varepsilon > 0$ there exists a $n_1 \in \mathbb{N}$ such that
\[
  \mathbb{P}\left[F^{\lambda, g}\left(f^{w^{*,\tilde{\lambda}}}\right) \in
    F_n^{\tilde{\lambda}}\left(\mathcal{RN}^{*,\tilde{\lambda}}\right)
    +[-\varepsilon, \varepsilon]\right] > P, \forall n \in \mathbb{N}_{> n_1}.
\]
As $\mathcal{RN}^{*,\tilde{\lambda}}$ is the optimal network for
$F_n^{\tilde{\lambda}}$ we know that
\[
  F_n^{\tilde{\lambda}}\left(\mathcal{RN}^{*,\tilde{\lambda}}\right)
  \leq F_n^{\tilde{\lambda}}\left(\mathcal{RN}_{\tilde{w}}\right).
\]
Using Lemma~\ref{lem:s2} we get that for every $P \in (0,1)$ and
$\varepsilon > 0$ there exists a $n_2 \in \mathbb{N}$ such that
\[
  \mathbb{P}\left[F_n^{\tilde{\lambda}}\left(\mathcal{RN}_{\tilde{w}}\right)
    \in F^{\lambda, g}\left(f^{*,\lambda}_g\right)+[-\varepsilon,
    \varepsilon]\right] > P, \forall n \in \mathbb{N}_{> n_2}.
\]
If we combine these ... we get that for every $P \in (0,1)$ and
$\varepsilon > 0$ and $n_3 \geq
\max\left\{n_1,n_2\right\}$
\[
  \mathbb{P}\left[F^{\lambda,
      g}\left(f^{w^{*,\tilde{\lambda}}}\right) \leq F^{\lambda,
      g}\left(f^{*,\lambda}_g\right)+2\varepsilon\right] > P, \forall
  n \in \mathbb{N}_{> n_3}.
\]
As ... is in ... and ... is optimal we know that
\[
  F^{\lambda, g}\left(f^{*,\lambda}_g\right) \leq F^{\lambda, g}\left(f^{w^{*,\tilde{\lambda}}}\right)
\]
and thus get with the squeeze theorem
\[
  \plimn F^{\lambda, g}\left(f^{w^{*,\tilde{\lambda}}}\right) = F^{\lambda, g}\left(f^{*,\lambda}_g\right).
\]
We can now use Lemma~\ref{lem:s7} to follow that
\begin{equation}
  \plimn \norm{f^{w^{*,\tilde{\lambda}}} - f^{*,\lambda}_g}
  _{W^{1,\infty}} = 0.
  \label{eq:main4}
\end{equation}
Now by using the triangle inequality with Lemma~\ref{lem:s3} and
(\ref{eq:main4}) we get
\begin{align*}
  \plimn \norm{\mathcal{RN}^{*, \tilde{\lambda}} - f_g^{*,\lambda}}
  \leq& \plimn \bigg(\norm{\mathcal{RN}^{*, \tilde{\lambda}} -
        f_g^{w^{*,\tilde{\lambda}}}}_{W^{1,\infty}}\\
  &+ \norm{f^{w^{*,\tilde{\lambda}}} - f^{*,\lambda}_g} 
    _{W^{1,\infty}}\bigg) = 0
\end{align*}
and thus have proven Theorem~\ref{theo:main1}.
We now know that randomized shallow neural networks behave similar to
spline regression if we regularize the size of the weights during
training. 
\textcite{heiss2019} further explore a connection between ridge penalized
networks and randomized shallow neural networks trained using gradient
descent.
They come to the conclusion that the effect of weight regularization
can be achieved by stopping the training of the randomized shallow
neural network early, with the amount of epochs being proportional to
the punishment for weight size.
This ... that randomized shallow neural networks trained for a certain
amount of iterations converge for a increasing amount of nodes  to
cubic smoothing splines with appropriate weights.
\todo{nochmal nachlesen wie es genau war}

\newpage
\subsection{Simulations}
In the following the behaviour described in Theorem~\ref{theo:main1}
is visualized in a simulated example. For this two sets of training
data have been generated.
\begin{itemize}
  \item $\text{data}_A = (x_{i, A}^{\text{train}},
  y_{i,A}^{\text{train}})$ with 
  \begin{align*}
    x_{i, A}^{\text{train}} &\coloneqq -\pi + \frac{2 \pi}{5} (i - 1),
                              i \in \left\{1, \dots, 6\right\}, \\
    y_{i, A}^{\text{train}} &\coloneqq \sin( x_{i, A}^{\text{train}}). \phantom{(i - 1),
                              i \in \left\{1, \dots, 6\right\}}
  \end{align*}
  \item $\text{data}_B = (x_{i, B}^{\text{train}}, y_{i,
    B}^{\text{train}})$ with
  \begin{align*}
    x_{i, B}^{\text{train}} &\coloneqq \pi\frac{i - 8}{7},
                              i \in \left\{1, \dots, 15\right\}, \\
    y_{i, B}^{\text{train}} &\coloneqq \sin( x_{i, B}^{\text{train}}). \phantom{(i - 1),
                              i \in \left\{1, \dots, 6\right\}}
  \end{align*}
\end{itemize}
For the $\mathcal{RN}$ the random weights are distributed
as follows
\begin{align*}
  \xi_i &\stackrel{i.i.d.}{\sim} \text{Unif}(-5,5), \\
  v_i &\stackrel{i.i.d.}{\sim} \mathcal{N}(0, 5), \\
  b_i &\stackrel{\phantom{i.i.d.}}{\sim} -\xi_i v_i.
\end{align*}
Note that by the choices for the distributions $g$ as defined in
Theorem~\ref{theo:main1} 
would equate to $g(x) = \frac{\mathbb{E}[v_k^2|\xi_k = x]}{10}$. In
order to utilize the
smoothing spline implemented in Mathlab, $g$ has been simplified to $g
\equiv \frac{1}{10}$ instead. For all figures $f_1^{*, \lambda}$ has
been calculated with Matlab's 'smoothingspline', as this minimizes
\[
  \bar{\lambda} \sum_{i=1}^N(y_i^{train} - f(x_i^{train}))^2 + (1 -
  \bar{\lambda}) \int (f''(x))^2 dx
\]
the smoothing parameter used for fittment is $\bar{\lambda} =
\frac{1}{1 + \lambda}$. The parameter $\tilde{\lambda}$ for training
the networks is chosen as defined in Theorem~\ref{theo:main1} and each
one is trained on the full training data for 5000 epoch using
gradient descent. The
results are given in Figure~\ref{fig:rn_vs_rs}, here it can be seen that in
the intervall of the traing data $[-\pi, \pi]$ the neural network and
smoothing spline are nearly identical, coinciding with the proposition.

\input{Figures/RN_vs_RS}


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