From 9f0746ce0017161ca743244c066d463f055b5b48 Mon Sep 17 00:00:00 2001 From: ludwigbothmann <46222472+ludwigbothmann@users.noreply.github.com> Date: Wed, 29 Nov 2023 23:28:40 +0100 Subject: [PATCH] update loss surface figure --- .../slides-info-kl-ment.tex | 4 ++-- .../information-theory/slides-info-kl-ml.tex | 6 +++--- .../figure/reg_surfaces_l1_l2.png | Bin 0 -> 599064 bytes .../regularization/rsrc/make_reg_surfaces.py | 18 ++++++------------ slides/regularization/slides-regu-l1l2.tex | 4 ++-- 5 files changed, 13 insertions(+), 19 deletions(-) create mode 100644 slides/regularization/figure/reg_surfaces_l1_l2.png diff --git a/slides/information-theory/slides-info-kl-ment.tex b/slides/information-theory/slides-info-kl-ment.tex index cb43367c..faa9ad81 100644 --- a/slides/information-theory/slides-info-kl-ment.tex +++ b/slides/information-theory/slides-info-kl-ment.tex @@ -44,7 +44,7 @@ \begin{vbframe}{Constructing the KL} \textbf{1) Locality} \\ The constraint must only update the prior distribution in $D, i.e.,$ the region where it is active. \\ -\includegraphics[width=0.3\linewidth]{slides/information-theory/figure_man/kl_me_constraint.png} \\ +\includegraphics[width=0.3\linewidth]{figure_man/kl_me_constraint.png} \\ \lz For this, it can be shown that the non-overlapping domains of $\mathcal{X}$ must contribute additively to the entropy, i.e., $$S(p) = \int F(p(\xv), \xv) d\mu(\xv)$$ @@ -54,7 +54,7 @@ \textbf{2) Invariance to coordinate system} \\ \lz - \includegraphics[width=0.5\linewidth]{slides/information-theory/figure_man/kl_me_cosy.png} \\ + \includegraphics[width=0.5\linewidth]{figure_man/kl_me_cosy.png} \\ Enforcing 2) results in $$S(p) = \int \bm{\Phi}\left(\frac{dp}{dm}(\xv)\right)dm(\xv)$$ where $\bm{\Phi}$ is an unknown function, $m$ is another measure on $\mathcal{X}$ dominated by $\mu$ and $\frac{dp}{dm}$ the Radon–Nikodym derivative which becomes diff --git a/slides/information-theory/slides-info-kl-ml.tex b/slides/information-theory/slides-info-kl-ml.tex index 9fa5d05b..f4397d21 100644 --- a/slides/information-theory/slides-info-kl-ml.tex +++ b/slides/information-theory/slides-info-kl-ml.tex @@ -47,7 +47,7 @@ \item \textbf{Feature selection} In feature selection, we want to choose features the target strongly depends on. -\includegraphics[width=0.6\linewidth]{slides/information-theory/figure_man/kl_ml_mi.png} +\includegraphics[width=0.6\linewidth]{figure_man/kl_ml_mi.png} We can measure dependency by measuring the similarity between $p(\mathbf{x}, y)$ and $p(\mathbf{x})\cdot p(y).$ \\ We will later see that measuring this similarity with KL leads to the concept of mutual information. @@ -72,7 +72,7 @@ \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 $p = q.$ @@ -97,7 +97,7 @@ \\ We have samples from the DGP $p(y|x)$ when we fit our ML model. \\ \lz - \\ + If we have a probabilistic ML model $q_{\bm{\phi}}$ and can specify $p(y|x)$ then the forward KL can be directly applied such that $$\E_{\xv \sim p_{\xv}}D_{KL}(p(\cdot|\xv) \| q_{\bm{\phi}}(\cdot|\xv)) = \E_{\xv \sim p_{\xv}}\E_{y \sim p_{y|\xv}}\log\left(\frac{p(y|\xv)}{q_{\bm{\phi}}(y|\xv)}\right).$$ For example, if $p$ and $q_{\bm{\phi}}$ are Gaussians with the same $\sigma$, minimizing this expression is equivalent to L2 minimization. \\ diff --git a/slides/regularization/figure/reg_surfaces_l1_l2.png b/slides/regularization/figure/reg_surfaces_l1_l2.png new file mode 100644 index 0000000000000000000000000000000000000000..45e9f1acdcd35c9a918a3836efd4073be74112cb GIT binary patch literal 599064 zcmeFZhd-BZ_&5APMhGD!Sy|Z%Np{Fy$=*fw-YeOXm7SH9WRsn}LN*y?lO%g*J;&Ae 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