\textbf{Motivation}: So far, when looking for $\hat{f}(x)$ the form was $\hat{f}(x)=w^\top x$, or $\hat{f}(x) = w^\top \phi(x)$. 
Note how the features $x, \phi(x)$ are predetermined. Why not learn them?

\textbf{New Optimization Problem}:

The new join-optimization problem, for $w$ and $\phi$:\\
\subtext{$\Theta$ is a set of parameters for $\phi$}
$$
    \hat{w} = \underset{w\in\R^m,\Theta\in\R^{m\times d}}{\text{arg min}}\Biggl( \frac{1}{n}\sum_{i=1}^n l\Bigl( w^\top \phi(x_i;\Theta),y_i \Bigr) \Biggr)
$$
Where $\phi(x,\Theta) = \Bigl( \phi_1(x;\theta_1),\ldots,\phi_m(x;\theta_m) \Bigr)$.\\
\subtext{$\theta_i$ is the $i$th row of $\Theta$, i.e. $\theta_i := (\Theta)_{i,:}$}

\subsection{Definitions}

\definition \textbf{Activation Function}\\
We set $\phi_i(x;\theta_i) = \psi(\theta_i^\top x)$, $\psi$ is the activation function.\\
\subtext{$\theta_i \in \R^d,\quad\psi:\R\to\R$}

{\scriptsize
    \notation More concisely, $\phi(x;\Theta) = \psi(\Theta x)$
}

\begin{center}
    \begin{tabular}{ll}
        \textbf{Activation Function}    & \textbf{Definition}               \\
        \hline
        Identity                        & $\psi(z) = z$                     \\
        Sigmoid                         & $\psi(z) = \frac{1}{1+e^{-z}}$    \\
        Hyperbolic tangent              & $\psi(z) = \tanh(z)$              \\
        Rectified Linear Unit (ReLU)    & $\psi(z) = \max(0,z)$ 
    \end{tabular}
\end{center}

\definition \textbf{Artificial Neural Network}\\
\subtext{The output functions of the above problem take the form:}
$$
    f(x;w,\theta) = \sum_{j=1}^{m}w_j\psi(\theta_j^\top x)
$$
{\scriptsize
    \remark Also called Multi-Layer Perceptron (MLP)
}

\newpage
\textbf{What is happening here?}\\
\smalltext{Explaining the calculation steps for such an $f$ naturally leads to the common pictorial depiction of neural networks.}
\begin{align*}
    \text{(i)}      &\quad x       &=\quad& (x_1,\ldots,x_n) \in \R^d       & \text{(Input Vector)}             \\
    \text{(ii)}     &\quad z       &=\quad& \Theta x                        & \text{(Linear transformation)}    \\
    \text{(iii)}    &\quad h_i     &=\quad& \psi(z_i)                       & \text{(Activation function)}      \\
    \text{(iv)}     &\quad f(x)    &=\quad& \sum_{j=1}^m w_j h_j            & \text{(Output)}
\end{align*}

\definition \textbf{Hidden Layer} $h = \psi(z)$

\definition \textbf{Bias Term} $b \in \R^m$\\
\subtext{Needed, as $f$ might not pass through origin. Similar to using $F_\text{lin}$ in regression, these can also be added by augmenting the input \& hidden layers.}

\textbf{Does this work at all?}\\
\smalltext{Yes, for most functions this does work.}

\definition \textbf{Sigmoidal Function}
$$
    \sigma(t) \text{ s.t. } \begin{cases}
        \sigma: \R \to \R \\
        \underset{t\to\infty}{\lim} = 1 \text{ and } \underset{t\to\infty}{\lim}
    \end{cases}
$$

\theorem \textbf{Universal Approximation Theorem}\\
\smalltext{$\hat{f}$, that uniformly approximates $f$, exists and takes this form:}
$$
    \hat{f}(x) = \textbf{W}^{(2)}\psi\Bigl( \textbf{W}^{(1)}x + b \Bigr)
$$ 
\smalltext{$f: [0,1]^d\to\R$ continuous$,\quad \psi $ sigmoidal}\\
\subtext{$\textbf{W}^{(1)} \in\R^{m\times d},\quad \textbf{W}^{(2)}\in\R^{1\times m},\quad m \in \N$}

Note how $m$ could be very large.\\
\subtext{$m$ can intuitively be understood as the "width" of the ANN}

\newpage
\definition \textbf{Fully Connected Neural Network}

More complex ANNs might have:
\begin{enumerate}
    \item More hidden layers
    \item Multiple outputs
    \item Differen activation functions across layers
\end{enumerate}
These are called \textit{fully connected}, since every node in a layer is connected to every node in the adjacent layers.\\
\subtext{There are also more complex architectures.}
\begin{center}
    \includegraphics[width=0.9\linewidth]{resources/FCANN.png}\\
    \subtext{\textit{Introduction to Machine Learning (2026), p. 183}}
\end{center}

\definition \textbf{Forward Propagation}

How can $\hat{f}$ be evaluated?\\
\subtext{This is just the computation for $1$-layer ANN generalized}

\begin{algorithm}
    \caption{Forward Propagation}
    $h^{(0)}\gets x$\;
    \For{$l=1,\ldots,L$}{
        $z^{(l)} = \textbf{W}^{(l)}h^{(l-1)} + b^{(l)}$ \\
        $h^{(l)} = \psi(z^{(l)})$
    }
    $f \gets \textbf{W}^{(L)}h^{(L-1)}+b^{(L)}$ \\
    \Return f
\end{algorithm}