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zPdYE+dWW51d!2B9YpK9sM!XQ=Uo=&(^BvHwK>A%G^I6M3P!Ps}V1yYH yN0AE?KHmPK<641?!jNUsmW}Ckgmi1f2SUe84>ce?F<25RCJ80vB? diff --git a/semester6/iml/main.tex b/semester6/iml/main.tex index b8ef226..3048283 100644 --- a/semester6/iml/main.tex +++ b/semester6/iml/main.tex @@ -38,4 +38,8 @@ \section{Gaussian Mixture Models} \input{parts/07_GMM.tex} +\newpage +\section{Language Modeling} +\input{parts/08_LLM.tex} + \end{document} diff --git a/semester6/iml/parts/08_LLM.tex b/semester6/iml/parts/08_LLM.tex new file mode 100644 index 0000000..3d150ed --- /dev/null +++ b/semester6/iml/parts/08_LLM.tex @@ -0,0 +1,80 @@ +\textbf{Problem}: How to build ChatGPT? + +\textbf{Assumption}: Producing \& Understanding language are strongly related: If a model can predict how a text continues, it must have understood \textit{something} about language \& meaning. + +\definition \textbf{Language Model}: A prob. distrib. over words/sentences + +The model should assign high prob. to \textit{plausible} texts: +$$ + \P\bigl[\text{"I like to eat pasta"}\bigr] > \P\bigl[\text{"I rocks like eat to"}\bigr] +$$ + +A simple model, exponential in $m$, would be: +$$ + P\bigl( \underbrace{\text{Sentence}}_\text{Length $m$} \bigr) = P\Bigl(\underbrace{X_1=x_1}_\text{1st word}, \underbrace{X_2=x_2}_\text{2nd word},\ldots,X_m=x_m \Bigr) +$$ + +{\footnotesize + \remark In practice, \textit{Tokens} are used instead of words. Tokens are "units of meaning" obtained from a sentence, e.g. seperating "don't" into "do" and "n't", to improve processing. +} + +\subsection{Autoregressive models} + +\textbf{Idea}: Abuse the chain rule of probability. +$$ + P(x_1,x_2,\ldots,x_m) = P(x_1)P(x_2|x_2)\cdots P(x_m|x_1,\ldots,x_{m-1}) +$$ +This lends itself very naturally to sentence generation: +\begin{align*} + x_1 \sim P(X_1) \quad x_2 \sim P(X_2\sep X_1=x_1) \quad \ldots +\end{align*} +Essentially: Sample a token (next word) from the cond. distrib. (sentence so far) at each step. This avoids computing the entire joint distribution. + +{\footnotesize + \remark Model outputs are inputs for next prediction: \textit{Autoregressive}. +} + +\begin{center} + \includegraphics[width=0.6\linewidth]{resources/Autoregressor.png}\\ + \footnotesize\color{gray}\textit{Introduction to Machine Learning (2026), p. 277} +\end{center} + +\subsubsection{Neural Network Language Models} + +Instead of looking at all previous words, only the recent $k$:\\ +\subtext{This is called \textit{Context length}} +\begin{align*} + &P\bigl( X_t=x\sep X_{1:t-1} = x_{1:t-1} \bigr) \\ + \approx &P\bigl(X_t=x\sep X_{t-k:t-1} = x_{t-k:t-1}, \theta\bigr) \\ + := & \text{Cat}\bigl( x \sep \text{softmax}\bigl( f(x_{t-k:t-1}, \theta) \bigr) \bigr) +\end{align*} + +\textbf{Optimization Problem}: +$$ + \underset{\theta}{\min} L(\theta) := \underbrace{-\sum_{t}\log\Bigl( P\bigl( X_t=x_t\sep X_{t-k:t-1} = x_{t-k:t-1}, \theta \bigr) \Bigr)}_\text{Negative Log-Likelihood} +$$ + +This is a form of \textit{Self-Supervision}: No labels are needed.\\ +In Datasets, the next word $x_t$ is the prediction target.\\ +\subtext{Thus: \textit{Any text} can be used as a training set} + +\subsubsection{Input Representation} + +Preparing inputs roughly follows this idea: +$$ + \text{Language} \underset{\text{Tokenization}}{\mapsto} \text{Tokens} \underset{\text{Vectorization}}{\mapsto} \text{Vector} +$$ + +\definition \textbf{Word Embedding Matrix} $\quad \mathbf{W}_e \in \R^{N\times d}, d \ll N$\\ +\subtext{$N$ is the vocabulary size} + +Each token receives a position in $d$-dimensional space: Semantically similar tokens are closer. + +{\footnotesize + \remark \textbf{One-Hot Encoding}\\ + Bad for many reasons: dimension $\propto$ vocabulary size, dot product is always zero (every input equally "dissimilar") +} + +\newpage +\subsection{Transformers \& Attention} + diff --git a/semester6/iml/resources/Autoregressor.png b/semester6/iml/resources/Autoregressor.png new file mode 100644 index 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