From 44c81306a8b515bd621d8437c030a350b1e621fd Mon Sep 17 00:00:00 2001 From: paul-loedige Date: Sun, 20 Feb 2022 18:56:15 +0100 Subject: [PATCH] =?UTF-8?q?Latent=20Variable=20Models=20and=20Variational?= =?UTF-8?q?=20Bayes=20hinzugef=C3=BCgt.?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- Glossary.tex | 25 +++--- ML_Zusammenfassung.tex | 7 +- Makros.tex | 39 ++++++++ Packages.tex | 6 -- Readme.md | 3 +- Style.tex | 3 +- ..._Variable_Models_and_Variational_Bayes.tex | 83 ++++++++++++++++++ ...ecial_case_1:_expectation_maximization.png | Bin 0 -> 1296702 bytes 8 files changed, 142 insertions(+), 24 deletions(-) create mode 100644 Makros.tex create mode 100644 chapters/Classical_Unsupervised_Learning/Latent_Variable_Models_and_Variational_Bayes.tex create mode 100644 images/VAE_special_case_1:_expectation_maximization.png diff --git a/Glossary.tex b/Glossary.tex index 142ef49..412eba6 100644 --- a/Glossary.tex +++ b/Glossary.tex @@ -7,6 +7,16 @@ \shorthandon{"} % {{{ Main glossary% +\newglossaryentry{manifold}{ + name=Mannigfaltigkeit, + description={ + \say{Unter einer Mannigfaltigkeit versteht man in der Mathematik einen topologischen Raum, + der lokal dem euklidischen Raum $\mathbb{R}^n$ gleicht. + Global muss die Mannigfaltigkeit jedoch nicht einem euklidischen Raum gleichen (nicht zu ihm homöomorph sein). + }(\url{https://de.wikipedia.org/wiki/Mannigfaltigkeit}) + } +} + \newglossaryentry{overfitting}{ name=Overfitting, description={ @@ -90,22 +100,9 @@ % }}} % % {{{ acronyms% -% {{{ acronym makros % -\newcommand{\glstopfull}[1]{\texorpdfstring{\glsxtrfull{#1}}{\glsfmtfull{#1}}} -\newcommand{\Glstopfull}[1]{\texorpdfstring{\Glsxtrfull{#1}}{\Glsfmtfull{#1}}} -\newcommand{\glstopfullpl}[1]{\texorpdfstring{\glsxtrfullpl{#1}}{\glsfmtfullpl{#1}}} -\newcommand{\Glstopfullpl}[1]{\texorpdfstring{\Glsxtrfullpl{#1}}{\Glsfmtfullpl{#1}}} -\newcommand{\glstopshort}[1]{\texorpdfstring{\glsxtrshort{#1}}{\glsfmtshort{#1}}} -\newcommand{\Glstopshort}[1]{\texorpdfstring{\Glsxtrshort{#1}}{\Glsfmtshort{#1}}} -\newcommand{\glstopshortpl}[1]{\texorpdfstring{\glsxtrshortpl{#1}}{\glsfmtshortpl{#1}}} -\newcommand{\Glstopshortpl}[1]{\texorpdfstring{\Glsxtrshortpl{#1}}{\Glsfmtshortpl{#1}}} -\newcommand{\glstoplong}[1]{\texorpdfstring{\glsxtrlong{#1}}{\glsfmtlong{#1}}} -\newcommand{\Glstoplong}[1]{\texorpdfstring{\Glsxtrlong{#1}}{\Glsfmtlong{#1}}} -\newcommand{\glstoplongpl}[1]{\texorpdfstring{\glsxtrlongpl{#1}}{\glsfmtlongpl{#1}}} -\newcommand{\Glstoplongpl}[1]{\texorpdfstring{\Glsxtrlongpl{#1}}{\Glsfmtlongpl{#1}}} -% }}} % \setabbreviationstyle[acronym]{long-short} +\newacronym{VAE}{VAE}{Variational Auto-Encoder} \newacronym{KL}{KL}{Kullback-Leibler} \newacronym{EM}{EM}{Expectation Maximization} \newacronym{PCA}{PCA}{Principal Component Analysis} diff --git a/ML_Zusammenfassung.tex b/ML_Zusammenfassung.tex index 0a6f935..6484e5b 100644 --- a/ML_Zusammenfassung.tex +++ b/ML_Zusammenfassung.tex @@ -9,6 +9,7 @@ \input{Packages.tex} \input{Style.tex} +\input{Makros.tex} \def \AUTHOR{Paul Lödige} \def \AUTHOREMAIL{paul.loedige@student.kit.edu} @@ -43,7 +44,8 @@ chapters/Classical_Unsupervised_Learning/Clustering, chapters/Classical_Unsupervised_Learning/Density_Estimation, chapters/Classical_Unsupervised_Learning/Expectation_Maximization, - chapters/Classical_Unsupervised_Learning/Variational_Auto-Encoders, + chapters/Classical_Unsupervised_Learning/Latent_Variable_Models_and_Variational_Bayes, + chapters/Classical_Unsupervised_Learning/Auto-Encoders, % %Mathematische_Grundlagen %chapters/Mathematische_Grundlagen/Lineare_Algebra, %chapters/Mathematische_Grundlagen/Probability_Theory, @@ -102,7 +104,8 @@ \include{chapters/Classical_Unsupervised_Learning/Clustering.tex} \include{chapters/Classical_Unsupervised_Learning/Density_Estimation.tex} \include{chapters/Classical_Unsupervised_Learning/Expectation_Maximization.tex} - \include{chapters/Classical_Unsupervised_Learning/Variational_Auto-Encoders.tex} + \include{chapters/Classical_Unsupervised_Learning/Latent_Variable_Models_and_Variational_Bayes} + \include{chapters/Classical_Unsupervised_Learning/Auto-Encoders.tex} \part{Mathematische Grundlagen} \label{part:Mathematische Grundlagen} diff --git a/Makros.tex b/Makros.tex new file mode 100644 index 0000000..19937b2 --- /dev/null +++ b/Makros.tex @@ -0,0 +1,39 @@ +%-------------------- +% this file contains usefull makros for various functionalities +%-------------------- + +% {{{ reference makros % + +% double reference (name and normal ref) +\newcommand{\dref}[1]{\nameref{#1} (\cref{#1})} +% }}} % + +% {{{ math makros% + +%special operators +\DeclareMathOperator*{\argmax}{arg\,max} +\DeclareMathOperator*{\argmin}{arg\,min} +\DeclareMathOperator*{\lik}{lik} +\DeclareMathOperator*{\loss}{loss} +\DeclareMathOperator*{\loglik}{loglik} +\DeclareMathOperator*{\softmax}{softmax} +\DeclareMathOperator{\dec}{dec} +\DeclareMathOperator{\enc}{enc} +% }}} % + +% {{{ acronym makros % + +% texorpdfstring simplification +\newcommand{\glstopfull}[1]{\texorpdfstring{\glsxtrfull{#1}}{\glsfmtfull{#1}}} +\newcommand{\Glstopfull}[1]{\texorpdfstring{\Glsxtrfull{#1}}{\Glsfmtfull{#1}}} +\newcommand{\glstopfullpl}[1]{\texorpdfstring{\glsxtrfullpl{#1}}{\glsfmtfullpl{#1}}} +\newcommand{\Glstopfullpl}[1]{\texorpdfstring{\Glsxtrfullpl{#1}}{\Glsfmtfullpl{#1}}} +\newcommand{\glstopshort}[1]{\texorpdfstring{\glsxtrshort{#1}}{\glsfmtshort{#1}}} +\newcommand{\Glstopshort}[1]{\texorpdfstring{\Glsxtrshort{#1}}{\Glsfmtshort{#1}}} +\newcommand{\glstopshortpl}[1]{\texorpdfstring{\glsxtrshortpl{#1}}{\glsfmtshortpl{#1}}} +\newcommand{\Glstopshortpl}[1]{\texorpdfstring{\Glsxtrshortpl{#1}}{\Glsfmtshortpl{#1}}} +\newcommand{\glstoplong}[1]{\texorpdfstring{\glsxtrlong{#1}}{\glsfmtlong{#1}}} +\newcommand{\Glstoplong}[1]{\texorpdfstring{\Glsxtrlong{#1}}{\Glsfmtlong{#1}}} +\newcommand{\glstoplongpl}[1]{\texorpdfstring{\glsxtrlongpl{#1}}{\glsfmtlongpl{#1}}} +\newcommand{\Glstoplongpl}[1]{\texorpdfstring{\Glsxtrlongpl{#1}}{\Glsfmtlongpl{#1}}} +% }}} % diff --git a/Packages.tex b/Packages.tex index 922a6dc..430d1cc 100644 --- a/Packages.tex +++ b/Packages.tex @@ -116,12 +116,6 @@ rightsub = \grq% %-------------------- %math symbols and more \usepackage{amsmath} -\DeclareMathOperator*{\argmax}{arg\,max} -\DeclareMathOperator*{\argmin}{arg\,min} -\DeclareMathOperator*{\lik}{lik} -\DeclareMathOperator*{\loss}{loss} -\DeclareMathOperator*{\loglik}{loglik} -\DeclareMathOperator*{\softmax}{softmax} %special symbols \usepackage{fontawesome} diff --git a/Readme.md b/Readme.md index 9433c27..e50bf67 100644 --- a/Readme.md +++ b/Readme.md @@ -1,8 +1,9 @@ # Zusammenfassung Maschinelles Lernen: Grundlagen und Algorithmen ##TODO: -- [ ] Folien aus der Vorlesung, auf die in der Zusammenfassung verwiesen werden einfach in den Anhang packen +- [ ] alle ?? beseitigen - [ ] für alle \nameref prüfen, ob eine richtige Referenz nachfolgen sollte. +- [ ] Folien aus der Vorlesung, auf die in der Zusammenfassung verwiesen werden einfach in den Anhang packen ## Notice Requires you to enable [--shell escape](https://tex.stackexchange.com/questions/516604/how-to-enable-shell-escape-or-write18-visual-studio-code-latex-workshop) diff --git a/Style.tex b/Style.tex index 442afbc..16a51fc 100644 --- a/Style.tex +++ b/Style.tex @@ -10,6 +10,7 @@ \pagestyle{fancy} \fancyhf{} \chead{\textbf{Zusammenfassung \MODULE \\\vspace{1mm}}} -\lhead{\leftmark} +\lhead{\partname~\thepart} +\rhead{\leftmark} \lfoot{\AUTHOR~|~\DATE} \rfoot{\thepage} diff --git a/chapters/Classical_Unsupervised_Learning/Latent_Variable_Models_and_Variational_Bayes.tex b/chapters/Classical_Unsupervised_Learning/Latent_Variable_Models_and_Variational_Bayes.tex new file mode 100644 index 0000000..edc5505 --- /dev/null +++ b/chapters/Classical_Unsupervised_Learning/Latent_Variable_Models_and_Variational_Bayes.tex @@ -0,0 +1,83 @@ +\chapter{Latent Variable Models and Variational Bayes}% +\label{cha:Latent Variable Models and Variational Bayes} + +In \cref{sec:Decomposition in lower-bound and KL-term} wurde bereits gezeigt, +wie die Decomposition für diskrete $\bm z$ aussieht. +Für nicht-diskrete $\bm z$ ergibt sich: +\begin{align} \label{eq:decomposition_in_lower-bound_and_KL-term_nondiscrete} + \underbrace{\log p(\bm x)}_{\text{\glslink{marginal}{marginal} log-like}} + &= \mathcal L(q) + \nomeq{kl_divergence}(q(\bm z)\|p(\bm z|\bm x))\\ + &= \underbrace{\int q(\bm z)\log \frac{p(\bm x|\bm z)p(\bm z)}{q(z)} d\bm z}_{\text{Lower-Bound $\mathcal L(q)$}} + + \underbrace{\int q(\bm z)\log \frac{q(\bm z)}{p(\bm z|\bm x)}}_{\text{\noms{kl_divergence}: $\nomeq{kl_divergence}(q(\bm z)\|p(\bm z|\bm x))$}} +\end{align} +\begin{itemize} + \item $q(\bm z)$ wird hierbei auch als \say{variational\slash\,auxiliary distribution} bezeichnet +\end{itemize} +Bei der Expectation Maximization (\cref{cha:Expectation Maximization}) wurde an dieser Stelle davon ausgegangen, +dass die \gls{KL}-Divergenz eine \gls{closed_form_solution} hat +und daher nach dem Expectation-Step (\cref{sub:Expectation-Step}) $\nomeq{kl_divergence}(q(\bm z)\|p(\bm z|\bm x)) = 0$ gilt. +Mithilfe des Satzes von Bayes (\cref{sub:Bayes Rule}) lässt sich allerdings zeigen, +dass +$$\argmin_q \nomeq{kl_divergence}(q(\bm z)\|p(\bm z|\bm x)) = \argmax_q \mathcal L(q,p)$$ +gilt (Herleitung: {\color{red} Vorlesung 12 Folie 13 und 14}). +Daraus folgt, +dass die Maximierung der \glslink{marginal}{Marginal} Log-Likelihood immer noch sichergestellt ist, +da die \gls{KL}-Divergenz bereits minimal ist ($q(\bm z)\approx p(\bm z|\bm x)$), +wenn die Lower Bound maximiert ist. + +Für einen ganzen Datensatz lässt sich die Lower Bound wie folgt definieren: +\begin{equation} \label{eq:VAE:lower_bound_whole_dataset} + \mathcal L(\{q_i\},p) = \frac{1}{N}\sum_i\int q_i(\bm z)\log p(\bm x_i|\bm z)d\bm z - \nomeq{kl_divergence}(q_i(\bm z)\|p(\bm z)) +\end{equation} +\begin{itemize} + \item $q_i$: individuelle Variational Distribution für jeden Datenpunkt +\end{itemize} +Dies hat zwei Vorteile: +\begin{enumerate} + \item erlaubt direkteres Samplen + \begin{itemize} + \item kein Samplen von $p(\bm z)$ nötig, + stattdessen kann direkt von $q_i(\bm z)\approx p(\bm z|\bm x_i)$ gesampelt werden. + \item nach der Optimierung erzeugt jedes $q_i(\bm z)$ Samples mit hohem $p(\bm x|\bm z)$ + \end{itemize} + \item das Integral ist außerhalb der $\log$-Funktion + \begin{itemize} + \item es wird nur ein Sample von $q_i(\bm z)$ benötigt, um eine Abschätzung ohne Bias zu ermöglichen + \item kann für den \gls{SDG} (\cref{sub:SDG}) verwendet werden, + obwohl die \glslink{marginal}{Marginal} Log-Likelihood hierfür ungeeignet wäre + \end{itemize} +\end{enumerate} +Für die Maximierung dieser Lower Bound gibt es im Grunde genommen zwei Verfahren: + +\paragraph{Special Case 1: \glstoplong{EM}}% +\label{par:Special Case 1: Expectation Maximization} +\mbox{}\\ +\includegraphics[scale=.5]{VAE_special_case_1:_expectation_maximization.png} + +\paragraph{Special Case 2: Amortized Variational Inference}% +\label{par:Special Case 2: Amortized Variational Inference} +\mbox{}\\ +Das \say{Amortized Variational Inference} Verfahren ist das Standardverfahren, +welches in \dref{sec:VAEs} zum Einsatz kommt. +Hierbei wird statt einer auxiliary distribution $q_i(\bm z)$ für jeden Datenpunkt $x_i$ eine amortisierte Verteilung (armortized distribution) $q_\phi(\bm z|\bm x_i)$ verwendet, +welche mittels eines \gls{DNN} (\cref{Deep Neural Networkd}) erstellt wird. +Hierdurch lässt sich die Lower Bound Funktion wie folgt umschreiben: +\begin{equation} \label{eq:amortized_variational_inference_lower_bound} + \mathcal L(q,p) = \frac{1}{N}\sum_i\int q_\phi(\bm z|\bm x_i)\log p_{\bm \varphi}(\bm x_i|\bm z)d\bm z - \nomeq{kl_divergence}(q_\phi(\bm z|\bm x_i)\|p_{\bm\varphi}(\bm z)) +\end{equation} +\begin{itemize} + \item Encoder \tabto{3cm}$q_\phi(\bm z|\bm x)$ + \item Decoder \tabto{3cm}$p_{\bm\varphi}(\bm x|\bm z)$ + \item Latent Prior \tabto{3cm}$p_{\bm\varphi}(\bm z)$ +\end{itemize} +Da die Samples hier nicht vorgegeben sind, +sondern generiert werden, +unterscheidet sich dieses Verfahren vom Maximum-Log-Likelihood (\cref{maximum log-lik}). +Zudem ist die Verwendung von Gradienten sehr ineffizient. +Abhilfe biete der \say{Reparameterization Trick} (siehe {\color{red} Vorlesung 12 Folie 19 und 20}), +welcher es ermöglicht, +die Lower Bound wie folgt umzuschreiben (mehr Details: {\color{red} Vorlesung 12 Folie 21}): +\begin{equation} \label{eq:amortized_variational_inference_reparametrized_lower_bound} + \mathcal L(q,p) = \frac{1}{N} \sum_i\int p(\bm\xi) (\underbrace{\log p_{\bm\varphi}(\bm x_i|\bm h(\bm\xi,\bm x))}_{\text{reconstruction}} + + \underbrace{\log p_{\bm\varphi}(\bm h(\bm xi,\bm x)) - \log q_\phi(\bm h(\bm\xi,\bm x)|\bm x)}_{\text{\glsxtrshort{KL}-term}}) +\end{equation} diff --git a/images/VAE_special_case_1:_expectation_maximization.png b/images/VAE_special_case_1:_expectation_maximization.png new file mode 100644 index 0000000000000000000000000000000000000000..e1bf2445ac90dcd09e428b380f7948abdfd1c0ca GIT binary patch literal 1296702 zcmZU52Ut_t^EL_1P*+3+=@3DbCPg|3NQpFsMS7D61f)wK^a!E?N=Ou>t3*(`RFM{C zrHBLtX`v)a3DSEn-v#%#%l^OT3BpY|IaAKeJMYZ(;WZO|R;DvdG&D4*G&H<3 zG&C&Q#~6TT6dy7y0ly%=2G#*IG5%BN)^Z`JY^xW|p$UEqIndL1Q<6V$pZK?4cc6*eKj z!AcStAsj7g`?ax981~4ctGDkusg@8)vuF7nq((OfZWNzwM+ROSB=S?p$!i+Ee<;!H zux)6f_gNB>FAd~scz8~N_Nu!g=)xGn*uv~c`ULI45HYXECkTxfW;r+aUovi(6823T zqAQnqMH=Uq2_@)8;mE{Y$FZPD`e3uPf4>_MumYkb2vH(Id+N&TE1?v2!fC>ISHoky9Pe%b6+B->$rFIQ^|LU${Z|YSbPJM81Nke}9PQkAI zx_40V6mK^RHzVJxRm{b2rp;P`RXYEE0E+0Fb`N4aBFDjNH?E(=kl5>3cfcFW6g~=< zFXINw2Hyg!FVC~vhNhK#tF-fAO$H(jE&Pi*V4g&SKZfPSy>o{PXqB{zbu;nLhMf@M z1YcAwol)hR#IBJ1ZWeeQ%?`_kp?`fCE@5?=&_WO+R1>1BR{rEEigtCU3Q>-s>TlC* 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