From 110c1b72074d6e459e823074941abd096500beee Mon Sep 17 00:00:00 2001 From: RobinB27 Date: Sun, 22 Mar 2026 14:38:47 +0100 Subject: [PATCH] [PS] fixes --- semester4/ps/ps-rb/main.pdf | Bin 260224 -> 265417 bytes semester4/ps/ps-rb/parts/01_variables.tex | 13 +-- semester6/iml/main.pdf | Bin 44418 -> 224310 bytes semester6/iml/main.tex | 3 +- semester6/iml/parts/00_intro.tex | 5 +- semester6/iml/parts/01_regression.tex | 103 ++++++++++++++++++++++ semester6/iml/util/helpers.tex | 6 +- 7 files changed, 117 insertions(+), 13 deletions(-) create mode 100644 semester6/iml/parts/01_regression.tex diff --git a/semester4/ps/ps-rb/main.pdf b/semester4/ps/ps-rb/main.pdf index 5c5e7b96bf1b8c8fd656a2b7c56b02f5e8b69117..d56e5b7ca9242b5b9c9acfbd3b9c960f1cf4af7c 100644 GIT binary patch delta 53358 zcmV(}K+wN{@ej$F5U`381U4}-F_Uq6D1Yr;-E$i^l7IJKq3pvIRas8(4UDsT$X2XW zuI^m9#1}i8_^K$2l4d1QT8eZWzudom-Iy-`?BUFiw5(L!i%m%&252<8fBiWu(?^#_ z4F3J@8N*NbKk>W2zk2r7sgy@~mK$M?UR@k1?lPs-k>D9q`smg9(XSJ6{PxwqlYifb 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z>Um?-_||!Q>2jym+O*<{w1amq3gxH|*V`@x?+=Z;h2o>7WXt3E1j7okDH%eVEQ+i+8+dm=8ndw!5 W@(>^i5jc3*d05~mDa92e;QkN7?-vaK diff --git a/semester4/ps/ps-rb/parts/01_variables.tex b/semester4/ps/ps-rb/parts/01_variables.tex index f217c39..24b6848 100644 --- a/semester4/ps/ps-rb/parts/01_variables.tex +++ b/semester4/ps/ps-rb/parts/01_variables.tex @@ -29,6 +29,12 @@ $$ \forall a \in \R:\qquad F_X(a) = \P[X \leq a] $$ +{\scriptsize + \notation $\underset{x \to a^-}{\lim}F_X(a) = F(a^-) = \P[X < a]$ + + \lemma $\P[a < X \leq b] = F_X(b) - F_X(a)$ +} + \theorem \textbf{Eigenschaftern der Verteilungsfunktion} \begin{tabular}{ll} @@ -43,6 +49,8 @@ $$ \P[X_1 \leq x_q,\cdots, X_n \leq x_n] = \P[X_1 \leq x_1] \cdots \P[X_n \leq x_n] $$ +\newpage + \theorem \textbf{Unabhängigkeit von Gruppierungen}\\ \smalltext{$X_1,\cdots X_n$ sind unabhängig, dann sind auch $Y_1,\cdots,Y_k$ unabhängig:} $$ @@ -50,8 +58,6 @@ $$ $$ \subtext{$1 \leq i_1 < i_2 < \cdots < i_k \leq n \text{ sind Indizes},\quad 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ziu%oaul(ETh8?up>!1#!i^fX}B{3IX7|?_DjJiNb=a&qO9bw*nS>5X3*Rn3{aZ@3; z^ft!7DrS{^4h7a%icXwu=QJ{gz-eh+b(>~B6Iw1lQ`_MYtIyawWWi?t=bPcp%V~!2 zq!-wLC;)*>8U$8xl=+QlpK`mXzX&1mQIuq?*A5& zl=?zEJd&KelHz=PU_=ECdtcuno6xl9;i!HP}~ShLb+R6tu7a1{%B({7-l_%{Jc8ncf+**1@zfC#gUB!-5DqRrG+J6I6=}5p033pI Ar2qf` diff --git a/semester6/iml/main.tex b/semester6/iml/main.tex index a87848c..f01c5ca 100644 --- a/semester6/iml/main.tex +++ b/semester6/iml/main.tex @@ -9,7 +9,8 @@ \begin{document} -\section{Intro} \input{parts/00_intro.tex} +\section{Regression} +\input{parts/01_regression.tex} \end{document} diff --git a/semester6/iml/parts/00_intro.tex b/semester6/iml/parts/00_intro.tex index 9013805..8715867 100644 --- a/semester6/iml/parts/00_intro.tex +++ b/semester6/iml/parts/00_intro.tex @@ -1,4 +1 @@ -\textbf{placeholder} -$$ - a = x\cdot 16 -$$ \ No newline at end of file +\textbf{placeholder} \ No newline at end of file diff --git a/semester6/iml/parts/01_regression.tex b/semester6/iml/parts/01_regression.tex new file mode 100644 index 0000000..1301ab0 --- /dev/null +++ b/semester6/iml/parts/01_regression.tex @@ -0,0 +1,103 @@ +\subsection{Supervised Learning} + +\textbf{Supervised Learning} is the task of 'learning' a function relationship, based on a given set of inputs/outputs. + +Some terminology: + +\begin{tabular}{ll} + $x \in \R^d$ & Inputs (Attributes/Covariates) \\ + $\phi(x) \in \R^p$ & Features \\ + $y \in \R$ & Outputs (Targets/Labels) \\ + $D = \{ (x_i,y_i) \}_{i=1}^n$ & Training Set \\ + $D'$ & Test Set \\ + $f: \R^p \to \R$ & Predictor (Model) \\ + $l(f(x), y)$ & Loss +\end{tabular} + +\textbf{Machine Learning Pipelines} can often be classified using: + +\begin{tabular}{ll} + $F$ & Function Class \\ + $L(f)$ & Training Loss \\ + & Optimization Method +\end{tabular} + +{\small + The function class $F$ is a set of parametrized functions. + We are looking for the $f \in F$ that minimizes $L(f)$. +} + +\definition \textbf{Training Loss} +$$ + L(f) := \frac{1}{n}\sum_{i=1}^{n} l\bigl( f(x_i), y \bigr) +$$ + +\subsection{Multiple Linear Regression} + +\textbf{Multiple Linear Regression} directly uses the $x \in \R^d$. \\ +Here, $F_\text{affine} = \bigl\{ f(x) = w^\top x + w_0 \big| w \in \R^d, w_0 \in \R \bigr\}$. + +Why are we using linear functions instead?\\ +{\scriptsize + Any estimator $f \in F_\text{affine}$ can be rewritten as $f\bigl((x,1)\bigr) = (w,w_0)^\top\cdot(x,1)$, + thus we can augment the inpurs $x \mapsto (x,1)$ and \\ + instead search in $F_\text{linear} = \{ f(x) = \hat{w}^\top x | \hat{w} \in \R^{d+1} \}$ +} + +\newpage + +\textbf{Loss Functions} + +\definition \textbf{Squared Loss} $\quad l\bigl( f(x),y \bigr) := \bigl( f(x) - y \bigr)^2$\\ +\subtext{Most common Loss Function, but sensitive to outliers.} + +\definition \textbf{Absolute Loss} $\quad l_\text{abs}\bigl( f(x),y \bigr) := |f(x)-y|$\\ +\subtext{Less sensitive to outliers, but not differentiable.} + +\definition \textbf{Huber Loss} +$$ + l_\text{huber}\bigl( f(x),y \bigr) := \begin{cases} + \frac{1}{2}\bigl( f(x)-y \bigr)^2 & |f(x)-y| \leq \delta \\ + \delta \bigl( |f(x)-y| - \frac{1}{2}\delta \bigr) & |f(x)-y| > \delta + \end{cases} +$$ +\subtext{Using parameter $\delta$, the penalization of outliers can be controlled} + +\textbf{Assymetric Loss}: In some cases it is desirable to penalize overestimation harder than underestimation, or vice versa. + +\definition \textbf{Quantile Loss} +$$ + l_\tau\bigl( f(x),y \bigr) := \tau \max\Bigl\{ y-f(x),0 \Bigr\} + (1-\tau)\max\Bigl\{ f(x)-y, 0 \Bigr\} +$$ +\subtext{Using parameter $\tau$, over/underestimation can be penalized} + +\newpage + +\textbf{Linear Regression} + +To find $\hat{f} := \underset{f \in F_\text{linear}}{\text{arg min}} L(f)$ we just look for $w \in \R^d$. +$$ + \hat{w} := \underset{w \in \R^d}{\text{arg min}} L(f_w) = \frac{1}{n}\sum_{i=1}^{n}\underbrace{\Bigl( y_i - w^\top x_i \Bigr)^2}_{l\bigl(f(x_i), y_i\bigr)} +$$ +\subtext{A natural abuse of notation here is $L(w) := L(f_w)$.} + +This can be rewritten in matrix notation: +$$ + \sum_{i=1}^{n}\Bigl( y_i - w^\top x_i \Bigr)^2 = \bigl\Vert y-Xw \bigr\Vert^2 +$$ +\subtext{The factor $\frac{1}{n}$ is irrelevant for Optimization, it doesn't depend on $w$} + +So we find the usual problem: +$$ + \hat{w} = \underset{w \in \R^d}{\text{arg min}} \bigl\Vert y-Xw \bigr\Vert^2 +$$ +The solution is a stationary point, so: +$$ + \nabla_w \bigl\Vert y-Xw \bigr\Vert^2 = 2X^\top(X\hat{w}-y) = 0 +$$ +Which yields the known \textbf{Normal Equation} +$$ + X^\top X\hat{w} = X^\top y +$$ + + diff --git a/semester6/iml/util/helpers.tex b/semester6/iml/util/helpers.tex index 60dac48..8f9869c 100644 --- a/semester6/iml/util/helpers.tex +++ b/semester6/iml/util/helpers.tex @@ -61,11 +61,11 @@ \def \F{\mathcal{F}} % Titles -\def \definition{\colorbox{lightgray}{Def} } +\def \definition{\colorbox{lightgray}{D.} } \def \notation{\colorbox{lightgray}{Notation} } -\def \remark{\colorbox{lightgray}{Remark} } +\def \remark{\colorbox{lightgray}{Rmk.} } \def \theorem{\colorbox{lightgray}{Th.} } -\def \lemma{\colorbox{lightgray}{Lem.} } +\def \lemma{\colorbox{lightgray}{L.} } \def \method{\colorbox{lightgray}{Method} }