From: Francois Fleuret <francois@fleuret.org>
Date: Fri, 26 Nov 2021 23:21:00 +0000 (+0100)
Subject: Initial commit
X-Git-Url: https://fleuret.org/cgi-bin/gitweb/gitweb.cgi?a=commitdiff_plain;h=b94c6b870ce9282f924b043bedac4c62920e7c6a;p=tex.git

Initial commit
---

b94c6b870ce9282f924b043bedac4c62920e7c6a
diff --git a/attention.tex b/attention.tex
new file mode 100644
index 0000000..b6f15dd
--- /dev/null
+++ b/attention.tex
@@ -0,0 +1,153 @@
+% -*- mode: latex; mode: reftex; mode: auto-fill; mode: flyspell; -*-
+
+\documentclass[c,8pt]{beamer}
+
+\usepackage{tikz}
+\newcommand{\transpose}{^{\top}}
+\def\softmax{\operatorname{softmax}}
+
+\setbeamertemplate{navigation symbols}{}
+
+\begin{document}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}[fragile]
+
+Given a query sequence $Q$, a key sequence $K$, and a value sequence
+$V$, compute an attention matrix $A$ by matching $Q$s to $K$s, and
+weight $V$ with it to get $Y$.
+
+\medskip
+
+\[
+\uncover<2,4,6->{
+  A_i = \softmax \left( \frac{Q_i \, K\transpose}{\sqrt{d}} \right)
+}
+%
+\quad \quad \quad
+%
+\uncover<3,5->{
+  Y_i = A_i V
+}
+\]
+
+\medskip
+
+\makebox[\textwidth][c]{
+\begin{tikzpicture}
+
+  \node[cm={0.5, 0.5, 0.0, 1.0, (0.0, 0.0)}] (V) at (-2, 2.35) {
+    \begin{tikzpicture}
+      \draw[fill=green!20] (0, 0) rectangle (4, 1.4);
+      \uncover<3,5>{\draw[fill=yellow] (0, 0) rectangle (4, 1.4);}
+      \foreach \x in { 0.2, 0.4, ..., 3.8 } \draw (\x, 0) -- ++(0, 1.4);
+    \end{tikzpicture}
+  };
+
+  \node[cm={1.0, 0.0, 0.5, 0.5, (0.0, 0.0)}] (A) at (0.5, 1.6) {
+    \begin{tikzpicture}
+      \draw (0, 0) rectangle ++(3, 4);
+    \end{tikzpicture}
+  };
+
+  \uncover<2-3>{
+  \node[cm={0.5, 0.5, 0.0, 1.0, (0.0, 0.0)}] (a1) at (-0.9, 2.1) {
+    \begin{tikzpicture}
+      \draw[draw=none] (0, 0) rectangle (4, 1);
+      \foreach \x/\y in {
+        0.00/0.03, 0.20/0.04, 0.40/0.07, 0.60/0.35, 0.80/0.52,
+        1.00/1.00, 1.20/0.82, 1.40/0.25, 1.60/0.08, 1.80/0.03,
+        2.00/0.15, 2.20/0.24, 2.40/0.70, 2.60/0.05, 2.80/0.03,
+        3.00/0.03, 3.20/0.03, 3.40/0.00, 3.60/0.03, 3.80/0.00 }{
+        \uncover<2>{\draw[black,fill=red] (\x, 0) rectangle ++(0.2, \y);}
+        \uncover<3>{\draw[black,fill=yellow] (\x, 0) rectangle ++(0.2, \y);}
+      };
+    \end{tikzpicture}
+  };
+  }
+
+  \uncover<4-5>{
+  \node[cm={0.5, 0.5, 0.0, 1.0, (0.0, 0.0)}] (a2) at (-0.7, 2.1) {
+    \begin{tikzpicture}
+      \draw[draw=none] (0, 0) rectangle (4, 1);
+      \foreach \x/\y in {
+        0.00/0.03, 0.20/0.04, 0.40/0.07, 0.60/0.03, 0.80/0.03,
+        1.00/0.05, 1.20/0.02, 1.40/0.08, 1.60/0.35, 1.80/0.85,
+        2.00/0.05, 2.20/0.04, 2.40/0.03, 2.60/0.05, 2.80/0.03,
+        3.00/0.03, 3.20/0.03, 3.40/0.00, 3.60/0.03, 3.80/0.00 }{
+        \uncover<4>{\draw[black,fill=red] (\x, 0) rectangle ++(0.2, \y);}
+        \uncover<5>{\draw[black,fill=yellow] (\x, 0) rectangle ++(0.2, \y);}
+      };
+    \end{tikzpicture}
+  };
+  }
+
+  \node[cm={1.0, 0.0, 0.0, 1.0, (0.0, 0.0)}] (Q) at (-0.5, -0.05) {
+    \begin{tikzpicture}
+      \draw[fill=green!20] (0, 0) rectangle (3, 1.0);
+      \foreach \x in { 0.2, 0.4, ..., 2.8 } \draw (\x, 0) -- ++(0, 1.0);
+      \uncover<2>{\draw[fill=yellow] (0.0, 0) rectangle ++(0.2, 1);}
+      \uncover<4>{\draw[fill=yellow] (0.2, 0) rectangle ++(0.2, 1);}
+    \end{tikzpicture}
+    };
+
+  \node[cm={1.0, 0.0, 0.0, 1.0, (0.0, 0.0)}] (Y) at (1.5, 3.45) {
+    \begin{tikzpicture}
+      \uncover<3>{\draw[fill=red] (0.0, 0) rectangle ++(0.2, 1.4);}
+      \uncover<4->{\draw[fill=green!20] (0.0, 0) rectangle ++(0.2, 1.4);}
+      \uncover<6->{\draw[fill=green!20] (0.0, 0) rectangle ++(3, 1.4);}
+      \uncover<5>{\draw[fill=red] (0.2, 0) rectangle ++(0.2, 1.4);}
+      \draw (0, 0) rectangle (3, 1.4);
+      \foreach \x in { 0.2, 0.4, ..., 2.8 } \draw (\x, 0) -- ++(0, 1.4);
+    \end{tikzpicture}
+    };
+
+  \node[cm={0.5, 0.5, 0.0, 1.0, (0.0, 0.0)}] (K) at (3, 1.1) {
+    \begin{tikzpicture}
+      \draw[fill=green!20] (0, 0) rectangle (4, 1);
+      \uncover<2,4>{\draw[fill=yellow] (0, 0) rectangle (4, 1);}
+      \foreach \x in { 0.2, 0.4, ..., 3.8 } \draw (\x, 0) -- ++(0, 1);
+    \end{tikzpicture}
+  };
+
+  \node[left of=V,xshift=0.5cm,yshift=0.7cm] (Vl) {V};
+  \node[left of=Q,xshift=-0.8cm] (Ql) {Q};
+  \node (Al) at (A) {A};
+  \node[right of=K,xshift=-0.6cm,yshift=-0.6cm] (Kl) {K};
+  \node[right of=Y,xshift=0.8cm] (Yl) {Y};
+
+\end{tikzpicture}
+}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}[fragile]
+
+A standard attention layer takes as input two sequences $X$ and $X'$
+and computes
+%
+\begin{align*}
+K & = W^K X \\
+V & = W^V X \\
+Q & = w^Q X' \\
+Y & = \underbrace{\softmax_{row} \left( \frac{Q K\transpose}{\sqrt{d}} \right)}_{A} V
+\end{align*}
+
+When $X = X'$, this is \textbf{self attention}, otherwise \textbf{cross
+  attention.}
+
+\pause
+
+\bigskip
+
+Several such processes can be combined in which case $Y$ is the
+concatenation of the separate results. This is \textbf{multi-head
+  attention}.
+
+\end{frame}
+
+
+\end{document}