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fix plot paths
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ludwigbothmann committed Dec 6, 2023
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4 changes: 2 additions & 2 deletions slides/information-theory/slides-info-kl-ml.tex
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\item \textbf{Probabilistic model fitting}\\
Assume our learner is probabilistic, i.e., we model $p(y| \mathbf{x})$ for example (for example, ridge regression, logistic regression, ...).

\includegraphics[width=0.4\linewidth]{slides/information-theory/figure_man/kl_ml_prob_fit.png}
\includegraphics[width=0.4\linewidth]{figure_man/kl_ml_prob_fit.png}

We want to minimize the difference between $p(y \vert \mathbf{x})$ and the conditional data generating process $\mathbb{P}_{y\vert\mathbf{x}}$ based on the data stemming from $\mathbb{P}_{y, \mathbf{x}}.$

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\end{itemize}
\framebreak

\includegraphics[width=0.6\linewidth]{slides/information-theory/figure_man/kl_ml_fkl_rkl.png} \\
\includegraphics[width=0.6\linewidth]{figure_man/kl_ml_fkl_rkl.png} \\
The asymmetry of the KL has the following implications
\begin{itemize}
\item The forward KL $D_{KL}(p\|q_{\bm{\phi}}) = \E_{\xv \sim p} \log\left(\frac{p(\xv)}{q_{\bm{\phi}}(\xv)}\right)$ is mass-covering since $p(\xv)\log\left(\frac{p(\xv)}{q_{\bm{\phi}}(\xv)}\right) \approx 0$ if $p(\xv) \approx 0$ (as long as both distribution do not extremely differ)
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\input{../../latex-math/basic-math}
\input{../../latex-math/basic-ml}

\newcommand{\titlefigure}{slides/regularization/figure/th_l1_pos.pdf}
\newcommand{\titlefigure}{figure/th_l1_pos.pdf}
\newcommand{\learninggoals}{
\item Understand the relationship between soft-thresholding and L1 regularization
}
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1) $\hat{\theta}_{\text{Lasso},j} > 0:$ \\
\lz
\begin{minipage}{0.4\textwidth}
\includegraphics[width=5cm]{slides/regularization/figure/th_l1_pos.pdf}
\includegraphics[width=5cm]{figure/th_l1_pos.pdf}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
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2) $\hat{\theta}_{\text{Lasso},j} < 0:$ \\
\lz
\begin{minipage}{0.4\textwidth}
\includegraphics[width=5cm]{slides/regularization/figure/th_l1_neg.pdf}
\includegraphics[width=5cm]{figure/th_l1_neg.pdf}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
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\begin{minipage}{0.4\textwidth}
\includegraphics[width=5cm]{slides/regularization/figure/th_l1_zero.pdf}
\includegraphics[width=5cm]{figure/th_l1_zero.pdf}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
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