From 59800c8324c466cd2d5924070f7e938ed9d3ebb8 Mon Sep 17 00:00:00 2001 From: ludwigbothmann <46222472+ludwigbothmann@users.noreply.github.com> Date: Tue, 12 Dec 2023 18:16:10 +0100 Subject: [PATCH] Updates from Overleaf --- .../slides-info-kl-ment.tex | 5 +- .../information-theory/slides-info-kl-ml.tex | 3 +- slides/information-theory/slides-info-kl.tex | 6 +- slides/regularization/chapter-order.tex | 3 + .../figure_man/bias-variance-ridge.png | Bin 62867 -> 43656 bytes .../figure_man/data-augmentation-cat.png | Bin 0 -> 24892 bytes .../rsrc/data-augmentation-images-cat.py | 53 ++++++++++++++++++ slides/regularization/slides-regu-intro.tex | 16 ++++++ 8 files changed, 80 insertions(+), 6 deletions(-) create mode 100644 slides/regularization/figure_man/data-augmentation-cat.png create mode 100644 slides/regularization/rsrc/data-augmentation-images-cat.py diff --git a/slides/information-theory/slides-info-kl-ment.tex b/slides/information-theory/slides-info-kl-ment.tex index d4d7b409..a67295a4 100644 --- a/slides/information-theory/slides-info-kl-ment.tex +++ b/slides/information-theory/slides-info-kl-ment.tex @@ -31,13 +31,12 @@ In the following, we derive an alternative measure, namely the KL divergence (relative entropy), that fixes these shortcomings by taking an inductive inference viewpoint. \citebutton{Caticha, 2003}{https://arxiv.org/pdf/physics/0311093.pdf} \end{vbframe} \begin{vbframe}{Inductive inference} - Let's construct a "new" entropy measure $S(p)$ just by desired properties.\\ + We construct a "new" entropy measure $S(p)$ just by desired properties.\\ \lz Let $\mathcal{X}$ be a measurable space with $\sigma$-algebra $\mathcal{F}$ and measure $\mu$ that can be continuous or discrete. \\ We start with a prior distribution $q$ over $\mathcal{X}$ dominated by $\mu$ and a constraint of the form $$\int_D a(\xv) dq(\xv) = c \in \R$$ - with $D \in \mathcal{F}.$ Note that the constraint function $a(\xv)$ is analogous to moment condition functions $g(\cdot)$ in the discrete case. + with $D \in \mathcal{F}.$ The constraint function $a(\xv)$ is analogous to moment condition functions $g(\cdot)$ in the discrete case. We want to update the prior distribution $q$ to a posterior distribution $p$ that fulfills the constraint and is maximal w.r.t. $S(p).$ \\ - \lz For this maximization to make sense, $S$ must be transitive, i.e., $$S(p_1) < S(p_2), S(p_2) < S(p_3) \Rightarrow S(p_1) < S(p_3).$$ \end{vbframe} diff --git a/slides/information-theory/slides-info-kl-ml.tex b/slides/information-theory/slides-info-kl-ml.tex index 7421d76b..f578f63b 100644 --- a/slides/information-theory/slides-info-kl-ml.tex +++ b/slides/information-theory/slides-info-kl-ml.tex @@ -72,7 +72,8 @@ \begin{vbframe}{KL divergence} -Divergences can be used to measure the similarity of distributions. \lz For distributions $p, q$ they are defined such that +Divergences can be used to measure the similarity of distributions.\\ +\lz For distributions $p, q$ they are defined such that \begin{enumerate} \item $D(p, q) \geq 0,$ \item $D(p, q) = 0$ iff $p = q.$ diff --git a/slides/information-theory/slides-info-kl.tex b/slides/information-theory/slides-info-kl.tex index 3ac2f675..707dd8ef 100644 --- a/slides/information-theory/slides-info-kl.tex +++ b/slides/information-theory/slides-info-kl.tex @@ -179,7 +179,8 @@ Because KL quantifies the difference between distributions, it can be used as a loss function between distributions. %to find a good fit for the observed data. \\ \lz -In our example, we can identify an optimal $\sigma$ which minimizes the KL. +In our example, we investigated the KL between $p=N(0, 1)$ and $q=LP(0, \sigma).$ Now, we identify an optimal $\sigma$ which minimizes the KL. + \begin{figure} \includegraphics[width = 6cm ]{figure/kl_norm_lp_sigma.png} \end{figure} @@ -187,7 +188,8 @@ % \begin{figure} % \centering -% \scalebox{0.9}{\includegraphics{figure_man/binom1.png}} +% \scalebox{0.9} +{\includegraphics{figure_man/binom1.png}} % \tiny{\\ Credit: Will Kurt} % \caption{ \footnotesize{\textit{Left}: Histogram of observed frequencies of a random variable $X$ which takes values between 0 and 10. \textit{Right}: The KL divergence between the observed data and Binom(10,p) is minimized when $p \approx 0.57$.}} % \end{figure} diff --git a/slides/regularization/chapter-order.tex b/slides/regularization/chapter-order.tex index 8747ed72..ca1d4f63 100644 --- a/slides/regularization/chapter-order.tex +++ b/slides/regularization/chapter-order.tex @@ -38,6 +38,9 @@ \subsection{Geometric Analysis of L2-Regularization and Weight Decay} \subsection{Geometric Analysis of L1-regularization} \includepdf[pages=-]{../slides-pdf/slides-regu-geom-l1.pdf} +\subsection{Soft-thresholding and L1 regularization deep-dive} +\includepdf[pages=-]{../slides-pdf/slides-regu-lasso-deepdive.pdf} + \subsection{Early Stopping} \includepdf[pages=-]{../slides-pdf/slides-regu-early-stopping.pdf} diff --git a/slides/regularization/figure_man/bias-variance-ridge.png b/slides/regularization/figure_man/bias-variance-ridge.png index 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