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z0n@4{WuXxvQX$f=!}yX@D`P2vx?}1UKMrqKn++b`{!<^QFlJ8-6n~O!v&`*$-yt?$ zsA0rMC3H6-P&Rbqejsf1wa4HQOnte|AxklPo=NF}66KwQNC!=fT*X3yEUQchz=FQV zF5JskcBIR79epQoCJ_BvCsC2bsxC}X-Yj`6SIe-7IN3iJGI#_|YG0~r+W1oK5;Rt! zDWf5k?wqzJFpVIrm6D_5zC>#WVxnh{J6SqgWl817j150(C&jjJTeqiMO#7487N4#u zAoqKyhVwD{!ZO{`7%{s*<(28jRw diff --git a/semester6/iml/parts/01_regression.tex b/semester6/iml/parts/01_regression.tex index 1708c0b..8e1dbdd 100644 --- a/semester6/iml/parts/01_regression.tex +++ b/semester6/iml/parts/01_regression.tex @@ -37,8 +37,8 @@ $$ \textbf{Multiple Linear Regression} directly uses the $x \in \R^d$. \\ Here, $F_\text{affine} = \bigl\{ f(x) = w^\top x + w_0 \big| w \in \R^d, w_0 \in \R \bigr\}$. -Why are we using linear functions instead?\\ -{\scriptsize +\remark Why are we using linear functions instead?\\ +{\footnotesize\color{gray} Any estimator $f \in F_\text{affine}$ can be rewritten as $f\bigl((x,1)\bigr) = (w,w_0)^\top\cdot(x,1)$, thus we can augment the inpurs $x \mapsto (x,1)$ and \\ instead search in $F_\text{linear} = \{ f(x) = \hat{w}^\top x | \hat{w} \in \R^{d+1} \}$ @@ -99,5 +99,3 @@ Which yields the \textbf{Normal Equation} from linear algebra. $$ X^\top X\hat{w} = X^\top y $$ - - diff --git a/semester6/iml/parts/02_classification.tex b/semester6/iml/parts/02_classification.tex index 5c4dc43..8c8dfbd 100644 --- a/semester6/iml/parts/02_classification.tex +++ b/semester6/iml/parts/02_classification.tex @@ -76,6 +76,10 @@ $$ $$ If $\mathcal{D}$ is linearly seperable, this is convex. +{\scriptsize + \remark Also called \textit{hard-margin SVM}, assuming lin.-sep. data +} + \definition \textbf{Support Vector Machine} $$ 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 @@ -86,16 +90,218 @@ 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 $w_0 \neq 0$)} -In practice, instead of explicitly solving $w_\text{SVM}$ or $w_\text{MM}$, GD is usually applied on a diff.-able convex surrogate loss. +\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 -\remark \textbf{Implicit Bias of Gradient Descent} +\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} +\theorem \textbf{GD converges to $w_\text{MM}$ for lin.-sep. data} (log. loss) $$ \underset{t\to\infty}{\lim}\frac{w^t}{\Vert w^t \Vert} = w_\text{MM} -$$ \ No newline at end of file +$$ +\subtext{On $L(w)$ (logistic regression),$\quad \mu=1$} + +\subsubsection{On linearly inseperable data} + +Assuming $\{x_i,y_i\}_{i=1}^n$ is strictly lin. inseperable, i.e. +$$ + \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} \ No newline at end of file diff --git a/semester6/iml/util/helpers.tex b/semester6/iml/util/helpers.tex index 8f9869c..6b32beb 100644 --- a/semester6/iml/util/helpers.tex +++ b/semester6/iml/util/helpers.tex @@ -59,6 +59,8 @@ \def \P{\mathbb{P}} \def \F{\mathcal{F}} +\def \E{\mathbb{E}} +\def \I{\mathbb{I}} % Titles \def \definition{\colorbox{lightgray}{D.} }