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a/semester6/iml/parts/00_intro.tex b/semester6/iml/parts/00_intro.tex index 8715867..fdbd303 100644 --- a/semester6/iml/parts/00_intro.tex +++ b/semester6/iml/parts/00_intro.tex @@ -1 +1 @@ -\textbf{placeholder} \ No newline at end of file +\textit{placeholder} \ No newline at end of file diff --git a/semester6/iml/parts/01_regression.tex b/semester6/iml/parts/01_regression.tex index 1301ab0..1708c0b 100644 --- a/semester6/iml/parts/01_regression.tex +++ b/semester6/iml/parts/01_regression.tex @@ -95,7 +95,7 @@ The solution is a stationary point, so: $$ \nabla_w \bigl\Vert y-Xw \bigr\Vert^2 = 2X^\top(X\hat{w}-y) = 0 $$ -Which yields the known \textbf{Normal Equation} +Which yields the \textbf{Normal Equation} from linear algebra. $$ X^\top X\hat{w} = X^\top y $$ diff --git a/semester6/iml/parts/02_classification.tex b/semester6/iml/parts/02_classification.tex new file mode 100644 index 0000000..eab581e --- /dev/null +++ b/semester6/iml/parts/02_classification.tex @@ -0,0 +1,88 @@ +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. + +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. + +\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 $w_0 \neq 0$)} +