In regression, we search an $\hat{f}: \R^d \to \R$, i.e. $y,\hat{y} \in \R$.\\
In classification, we want $\hat{y} \in \mathcal{Y} \subset \R$, s.t. $\mathcal{Y}$ is discrete.

\subsection{Binary Classification}

We generally use $\mathcal{Y} = \{+1,-1\}$ and set $\hat{y} = \text{sgn}\bigl(\hat{f}(x)\bigr)$.\\
So, a linear classifier where $\hat{f}(x) = w^\top x$ takes the form:
$$
    x \mapsto \begin{cases}
        1   & w^\top x > 0 \\
        -1  & w^\top x < 0
    \end{cases}
$$
\definition \textbf{Decision Boundary} $\quad \Bigl\{ x \in \R^d \ \Big|\ \hat{f}(x) = 0 \Bigr\}$

Like in regression, using features is again possible.

\subsection{Surrogate Loss}

We'd like to reuse the loss minimization from regression.\\
A natural metric for accuracy is simply checking if $\hat{y} = y$.

\definition \textbf{Zero-One Loss}
$$
    l_{0-1}(\hat{y},y) := \mathbb{I}_{\hat{y}\neq y} = \begin{cases}
        1 & \hat{y} \neq y \\
        0 & \hat{y} = y
    \end{cases}
$$
We could try minimizing this:
$$
    \underset{(x,y) \in \mathcal{D}}{\sum} l_{0-1}\bigl( \hat{y},y \bigr) = \underset{(x,y) \in \mathcal{D}}{\sum} \mathbb{I}_{f_w(x) \cdot y < 0}
$$
Unfortunately, $l_{0-1}$ is non-continuous and non-convex.\\
We introduce \textit{surrogate loss} to still apply GD.

Note how $\mathbb{I}_{\hat{y}\neq y} = \mathbb{I}_{\hat{y}\cdot y < 0}$, so $l_{0-1}$ only depends on $z := \hat{y}\cdot y$.\\
We thus define losses over $z$, that are cont. and convex.

\definition \textbf{Surrogate Loss}
$$
    l_\text{exp} = e^{-z} \qquad l_\text{log} = \log(1+e^{-z})
$$
A notable difference is that $l_\text{exp}'$ is unbounded,\\
while $l_\text{log}' = \frac{1}{1+e^z} \in (-\frac{1}{2}, -1)$ for $z < 0$.\\
This is better for outliers, thus $l_\text{log}$ is usually preferred.

\newpage
\subsection{Logistic Regression}
\subtext{We assume $w_0 = 0$}

We try to minimize $l_\text{log} = \log(1+e^{-z})$, so:
$$
    L(w) = \frac{1}{n}\sum_{i=1}^{n} l_\text{log}(z_i) = \frac{1}{n}\sum_{i=1}^{n}\log\Bigl( 1 + e^{-\overbrace{y_i \cdot w^\top x_i}^{z_i}} \Bigr)
$$
Assume $\{x_i,y_i\}_{i=1}^n$ is linearly seperable, i.e. 
$$
    \exists w \in \R^d:\quad \underbrace{y_i \cdot w^\top x_i}_{z_i} > 0 \quad \forall i \leq n
$$
Then there are multiple valid decision boundaries. Additionally, $L(w)$ is then convex.

the distance $x_0$ to the decision boundary is: $\Vert x_0 \Vert_2 \cdot |\cos(\theta)|$.\\
\subtext{$\theta$ between $w,x_0 \in \R^d$}
$$
    \Vert x_0 \Vert_2 \cdot |\cos(\theta)| = \Vert x_0 \Vert_2 \cdot \frac{|w^\top x_0 |}{\Vert w \Vert_2 \cdot \Vert x_0 \Vert_2} = \frac{|w^\top x_0|}{\Vert w \Vert_2}
$$
\subtext{Note if $w$ is a unit-vector, this is just $|w^\top x_0|$}

\definition \textbf{Margin} $\quad \text{margin}(w) := \underset{1\leq i\leq n}{\min} y_i \cdot w^\top x_i$

\subsection{Solutions}

\definition \textbf{Maximum Margin Solution}
$$
    w_\text{MM} := \underset{\Vert w \Vert_2=1}{\max} \underset{1\leq i\leq n}{\min} \Bigl( y_i \cdot w^\top x_i \Bigr)
$$
If $\mathcal{D}$ is linearly seperable, this is convex.

{\scriptsize
    \remark Also called \textit{hard-margin SVM}, assuming lin.-sep. data
}

\definition \textbf{Support Vector Machine}
$$
    w_\text{SVM} := \underset{w \in \R^d}{\min} \Vert w \Vert_2 \quad\text{s.t.}\quad y_i \cdot w^\top x_i \geq 1 \quad \forall i \leq n
$$

Solving these problems is actually equivalent, up to scaling:

\lemma $\quad\displaystyle\frac{w_\text{SVM}}{\Vert w_\text{SVM} \Vert_2} = w_\text{MM}$
\subtext{(This also holds for the case $w_0 \neq 0$)}

\newpage

By relaxing the SVM constraints, we can use the SVM problem on lin.-insep. data too:
$$
    y_i \cdot w^\top x_i \leq 1 \qquad \to \qquad y_i \cdot w^\top x_i \leq 1 - \zeta
$$
\subtext{$\zeta = (\zeta_1,\ldots,\zeta_n)$ s.t. $\zeta_i \geq 0$}

\definition \textbf{Soft-margin SVM}
$$
    w_\text{SM} = \underset{w\in\R^d,\zeta\in\R^n}{\min}\biggl( \Vert w \Vert^2 + \lambda \sum_{i=1}^{n} \zeta_i \biggr) 
    \text{ s.t. } \begin{cases}
        y_i \cdot w^\top x_i \geq 1-\zeta_i \\
        \zeta_i \geq 0 \quad \forall i \leq n
    \end{cases}
$$
\subtext{$\lambda$ (hyperparam.) intuitively controls how much a violation is penalized}

Another perspective: The optimal $\zeta_i$ are:
$$
    \zeta_i = \begin{cases}
        1 - y_i \cdot w^\top x_i    & \text{if } y_i\cdot w^\top x_i \leq 1 \\
        0                           & \text{else}
    \end{cases}
$$
So the problem can be formulated without $\zeta$ too:

\definition \textbf{$l_2$-penalized Hinge Loss Optimization}
$$
    \underset{w\in\R^d}{\min}\biggl( \Vert w \Vert^2 + \lambda \underbrace{\sum_{i=1}^{n}\max(0, 1-y_i\cdot w^\top x_i)}_\text{Hinge Loss} \biggr)
$$

\newpage

\subsection{Gradient Descent for Classification}

In practice, instead of explicitly solving $w_\text{SVM}$ or $w_\text{MM}$, GD is usually applied on a diff.-able convex surrogate loss.

\subsubsection{On linearly seperable data}

Assuming $\{x_i,y_i\}_{i=1}^n$ is lin. seperable, $L(w) = \frac{1}{n}\sum_{i=1}^{n}l_\text{log}(z_i)$ is convex, but no global optimum exists. 
Using GD, $L(w)$ will approach $0$, but the iterates $\{ w^t \ |\ t \in \N \}$ diverge. However, $w^t$ may converge \textit{in direction}, and interestingly:

\theorem \textbf{GD converges to $w_\text{MM}$ for lin.-sep. data} (log. loss)
$$
    \underset{t\to\infty}{\lim}\frac{w^t}{\Vert w^t \Vert} = w_\text{MM}
$$
\subtext{On $L(w)$ (logistic regression),$\quad \mu=1$}

\subsubsection{On linearly inseperable data}

Assuming $\{x_i,y_i\}_{i=1}^n$ is strictly lin. inseperable, i.e.
$$
    \forall w \neq 0:\quad \exists i \leq n: \quad y_i \cdot w^\top x_i < 0
$$
\theorem \textbf{GD converges on lin.-insep. data} (log. loss)
$$
    \exists \hat{w} \in \R^d: \quad \underset{t\to\infty}{\lim}w^t = \hat{w}
$$
\subtext{On $L(w)$ (logistic regression),$\quad \mu = \frac{4}{\sigma^2_\text{max}(X)}$}

\remark Only holds for $l_\text{log}$. In general, this is a hard problem.

\newpage

\subsection{Multiclass Classification}

What if $|\mathcal{Y}| > 2$? E.g. $\mathcal{Y} = \{\text{cat}, \text{dog}, \text{fish} \}$

Idea: Train $K := |\mathcal{Y}|$ classifiers $\hat{f}_1,\ldots,\hat{f}_K \in F$.\\
\subtext{Why not one $\hat{f}$? E.g. discretize further: $1 \mapsto \text{cat}$, $2 \mapsto \text{dog}$, $3 \mapsto \text{fish}$. \\Problem: this assignment suggests cats are closer to dogs than fish.}

We can then predict the class using these $\hat{f}_k$:
$$
    \hat{y}(x) = \underset{1\leq k\leq K}{\text{arg max}} \hat{f}_k(x)
$$
This leads to one decision boundary per class.

\definition \textbf{One-vs-Rest Training}\\
Train each model seperately by relabeling for each $\hat{f}_k$:
\begin{enumerate}
    \item Define $\mathcal{D}_k = \{x_i, \tilde{y_i}\}$ where $\tilde{y_i} := \begin{cases}
        -1  & y_i=k \\
        1   & y_i\neq k
    \end{cases}$
    \item Run binary classification on $\mathcal{D}_k$ to get $\hat{f}_k$
\end{enumerate}
\subtext{This leads to $K$ classification problems, which might be slow}

Another way to reuse the existing methodology is to use a new loss:

\definition \textbf{Cross-Entropy Loss}
$$
    l_\text{ce}\Bigl( \hat{f}_1(x),\ldots,\hat{f}_K(x),\ y \Bigr) = -\log\Biggl( \frac{\exp\bigl(\hat{f}_y(x)\bigr)}{\sum_{k=1}^{K}\exp\bigl( \hat{f}_k(x) \bigr)} \Biggr)
$$
\subtext{$y \in \{1,\ldots,K\},\quad \hat{f}_i(x) \in \R$}

$l_\text{ce}$ encourages the \textit{true class} $\hat{f}_{y_i}(x_i)$ to be the largest $\hat{f}_k(x_i)$.

{\scriptsize
    \remark For $K=2$, if we use $\mathcal{Y}=\{+1,-1\}$ then $l_\text{ce} = l_\text{log}$.
}

The parametrized optimization problem then is:
$$
    \underset{w_1,\ldots,w_K \in \R^d}{\min}\Biggl( \sum_{i=1}^{n} l_\text{ce}\Bigl( f_w(x_i), y_i \Bigr) \Biggr)
$$
\subtext{Here, $w \in \R^{d \times K}$ is a matrix: $w = (w_1,\ldots,w_K)$}

This then yields $\hat{f}_k = f_{\hat{w}_k}$

{\scriptsize
    \remark These methods may lead to very different decision boundaries!
}

\newpage
\subsection{Generalization}

First, a lot of assumptions:
{\small
    \begin{enumerate}
        \item Inputs $X \in \mathcal{X}$ come from a prob. distribution $\P_X$
        \item Training \& Test set sampled i.i.d. from same distribution\\
        {\color{gray} Note that in general, this is rarely true.}
        \item There exists a ground-truth $y^*$
        \item The observed labels are noisy: $(y \sep x) = \epsilon\cdot y^*(x)$\\
        {\color{gray}\scriptsize A \textit{mutliplicative} noise model, unlike lin.-reg. : $\mathcal{Y} = f^*(X) + \epsilon$}
        \item $\epsilon$ is also from a prob. distribution $\P_\epsilon$\\
        {\color{gray} Not necessarily indep. from $x$!}
    \end{enumerate}
}
Focusing on $y^*(x) \in \{+1,-1\}$, we set $\epsilon \in \{+1,-1\}$.\\
\subtext{Intuitvely: $\epsilon$ just flips the label}

This allows defining a Joint-Distribution 
$$
    \P[x,y] = \P_x[x]\cdot\P[y \sep x]
$$

\subsubsection{Evaluation}

An intuitive metric to check is proximity to $y^*$:\\
\subtext{Which can be done using the 0-1-loss}
$$
    l\Bigl( \hat{y}(x),y^*(x) \Bigr) = \mathbb{I}_{\hat{y}(x)\neq y^*(x)}
$$

Now, we can define the expected classification error:\\
$$
    \E_X\Bigl[ l\bigl( \hat{y}(x), y^*(x) \bigr) \Bigr] = \E_X\Bigl[ \I_{\hat{y}(x) \neq y^*(x)} \Bigr] = \P\Bigl[ \hat{y}(x) \neq y^*(x) \Bigr]
$$

We can't compute or estimate this, since we don't have $y^*$. \\
However, we can find an estimate of $\E_{X,Y}\bigl[ l(\hat{y}(X),Y) \bigr]$ using the observed $X,Y$.

Why is this useful? It approximates the generalisation error:

\definition \textbf{Generalisation Error} (0-1-Loss)
$$
    L\Bigl( \hat{f};\P_{X,Y} \Bigr) = \E_{X,Y}\Bigl[ l\Bigl( \hat{f}(X),Y \Bigr) \Bigr] = \P_{X,Y}\Bigl( \hat{y}\neq y§ \Bigr)
$$

We can empirically evaluate this on a test set:
$$
    \frac{1}{|\mathcal{D}_\text{test}|} \sum_{(x,y)\in\mathcal{D}_\text{test}} \I_{\hat{y}(x)\neq y^*(x)}
$$

\newpage

\subsection{Hypothesis Testing}
Classical statistical methods can also be used.

\definition \textbf{Asymmetric Errors}\\
Misclassifications may be weighted differently.\\
\subtext{Mistakenly classifying $x$ with $y^*(x)=1$ as $2$, may be worse than $3$.}

For binary classification, we may label:
$$
    +1 \mapsto \text{"Positive"} \qquad -1 \mapsto \text{"Negative"}
$$
This leads to the notion of confusion matrices.

\begin{center}
    \begin{tabular}{l|ll}
                        & $y=-1$        & $y=+1$        \\
        \hline
        $\hat{y}= -1$   & TN            & FN (Type II)  \\
        $\hat{y}= +1$   & FP (Type I)   & TP
        
    \end{tabular}
\end{center}

\definition \textbf{Empirical Measure}\\
\smalltext{For an event $A \subset \mathcal X \times \{+1,-1\}$}
$$
    \P_n[A] := \frac{1}{n}\sum_{i=1}^{n}\I_{(x_i,y_i) \in A}
$$
\subtext{$\mathcal{D} = \{ (x_i,y_i) \sep i \leq n \} \subset \mathcal{X} \times \{+1,-1\}$}

$\P_n[A]$ is the percentage of $(x,y) \in \mathcal{D}_\text{train}$ that belong to $A$.

\lemma $\P_n[A]$ \textbf{is an estimate of} $\P_{X,Y}[A]$:
$$
    \underset{n\to\infty}{\lim} \P_n[A] = \P_{X,Y}[A] \qquad \text{\color{gray}\scriptsize (Law of large numbers)}
$$

\remark \textbf{Asymmetric Loss in binary lassification}

We can now weigh FP, FN differently in the 0-1-error:
{\small
$$
    \frac{c_\text{FN}}{|\{ x \sep y=+1 \}|} \underbrace{\sum_{(x,y), y=1} \I_{\hat{y}\neq -1}}_\text{\#FN} + \frac{c_\text{FP}}{|\{x \sep y =-1 \}|}\underbrace{\sum_{(x,y), y=-1} \I_{\hat{y}(x)=+1}}_\text{\#FP}
$$
}
\subtext{Here $c_\text{FP}, c_\text{FN}$ are the weights for penalization}