From 298a2f35d5ff5a8558fd302e8e7cafeecd1bdba2 Mon Sep 17 00:00:00 2001 From: David Widmann Date: Mon, 28 Oct 2019 01:35:26 +0100 Subject: [PATCH] Add initial draft of poster --- poster/UUcolorPantone.sty | 99 + poster/figures/comparison_estimates.tex | 341 ++ poster/figures/comparison_tests.tex | 1962 ++++++++ poster/figures/logos/LiU.pdf | Bin 0 -> 130 bytes poster/figures/logos/UU.pdf | Bin 0 -> 130 bytes poster/figures/triangular_models.tex | 5501 +++++++++++++++++++++++ poster/figures/triangular_pdf_x_y.tex | 1832 ++++++++ poster/neurips.pdf | Bin 0 -> 131 bytes poster/neurips.tex | 332 ++ poster/references.bib | 68 + 10 files changed, 10135 insertions(+) create mode 100644 poster/UUcolorPantone.sty create mode 100644 poster/figures/comparison_estimates.tex create mode 100644 poster/figures/comparison_tests.tex create mode 100644 poster/figures/logos/LiU.pdf create mode 100644 poster/figures/logos/UU.pdf create mode 100644 poster/figures/triangular_models.tex create mode 100644 poster/figures/triangular_pdf_x_y.tex create mode 100644 poster/neurips.pdf create mode 100644 poster/neurips.tex create mode 100644 poster/references.bib diff --git a/poster/UUcolorPantone.sty b/poster/UUcolorPantone.sty new file mode 100644 index 0000000..0f2d5ba --- /dev/null +++ b/poster/UUcolorPantone.sty @@ -0,0 +1,99 @@ +\ifdefined\UseCMYK + \definecolor{uured}{cmyk}{0.07,1.00,0.82,0.26} +\else + \definecolor{uured}{RGB}{166,25,46} +\fi + +\ifdefined\UseCMYK + \definecolor{black}{cmyk}{0.00,0.00,0.00,1.00} +\else + \definecolor{black}{RGB}{0,0,0} +\fi + +\colorlet{uulightgrey}{black!10} +\colorlet{uumidgrey}{black!30} +\colorlet{uudarkgrey}{black!50} + +% Pantone 483 +\ifdefined\UseCMYK + \definecolor{mull}{cmyk}{0.21,0.80,0.81,0.69} +\else + \definecolor{mull}{RGB}{101,48,36} +\fi +\colorlet{mullstark}{mull!30} +\colorlet{mullmellan}{mull!20} +\colorlet{mullsvag}{mull!10} + +% Pantone 1675 +\ifdefined\UseCMYK + \definecolor{sand}{cmyk}{0.05,0.83,0.100,0.27} +\else + \definecolor{sand}{RGB}{169,67,30} +\fi +\colorlet{sandstark}{sand!30} +\colorlet{sandmellan}{sand!20} +\colorlet{sandsvag}{sand!10} + + +% Pantone 131 + +\ifdefined\UseCMYK + \definecolor{blond}{cmyk}{0.02,0.39,1.00,0.10} +\else + \definecolor{blond}{RGB}{204,138,0} +\fi +\colorlet{blondstark}{blond!30} +\colorlet{blondmellan}{blond!20} +\colorlet{blondsvag}{blond!10} + + + +% Pantone 7546 +\ifdefined\UseCMYK + \definecolor{skugga}{cmyk}{0.73,0.45,0.24,0.66} +\else + \definecolor{skugga}{RGB}{37,55,70} +\fi +\colorlet{skuggastark}{skugga!30} +\colorlet{skuggamellan}{skugga!20} +\colorlet{skuggasvag}{skugga!10} + +% Pantone 534 +\ifdefined\UseCMYK + \definecolor{skymning}{cmyk}{0.95,0.75,0.07,0.44} +\else + \definecolor{skymning}{RGB}{37,54,93} +\fi +\colorlet{skymningstark}{skymning!30} +\colorlet{skymningmellan}{skymning!20} +\colorlet{skymningsvag}{skymning!10} + +% Pantone 645 +\ifdefined\UseCMYK + \definecolor{gryning}{cmyk}{0.56,0.21,0.02,0.08} +\else + \definecolor{gryning}{RGB}{125,161,196} +\fi +\colorlet{gryningstark}{gryning!30} +\colorlet{gryningmellan}{gryning!20} +\colorlet{gryningsvag}{gryning!10} + +% Pantone 7484 +\ifdefined\UseCMYK + \definecolor{gronska}{cmyk}{0.91,0.14,0.78,0.60} +\else + \definecolor{gronska}{RGB}{0,87,63} +\fi +\colorlet{gronskastark}{gronska!30} +\colorlet{gronskamellan}{gronska!20} +\colorlet{gronskasvag}{gronska!10} + +% Pantone 7494 +\ifdefined\UseCMYK + \definecolor{skir}{cmyk}{0.35,0.05,0.42,0.14} +\else + \definecolor{skir}{RGB}{156,175,136} +\fi +\colorlet{skirstark}{skir!30} +\colorlet{skirmellan}{skir!20} +\colorlet{skirsvag}{skir!10} \ No newline at end of file diff --git a/poster/figures/comparison_estimates.tex b/poster/figures/comparison_estimates.tex new file mode 100644 index 0000000..bec2e80 --- /dev/null +++ b/poster/figures/comparison_estimates.tex @@ -0,0 +1,341 @@ +\begin{tikzpicture} +\begin{groupplot}[group style={group size={3 by 4}, xlabels at={edge bottom}, ylabels at={edge 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bottom=\tcbposterrowspacing,bottomtitle=\tcbposterrowspacing, + % Subtitles + subtitle style={opacityframe=0,opacityback=0.5,fontupper=\large\bfseries}, + } +} + +% plots +\usepackage{pgfplots,pgfplotstable} +\pgfplotsset{compat=1.16} + +% general settings for plots +\pgfplotsset{grid style=dashed} +\pgfplotsset{enlargelimits=auto} + +\usepgfplotslibrary{groupplots,colorbrewer} +\usetikzlibrary{plotmarks,calc} + +% plotting options +\pgfplotsset{table/search path={data/}} +\pgfplotsset{max space between ticks=150} +\pgfplotsset{every axis/.append style={axis background style={fill=gray!10}},tick label style={font={\footnotesize}}, label style={font={\small}}} +\pgfplotsset{every axis plot/.append style={thick}} +\pgfplotsset{every axis legend/.append style={font=\small, fill=none}} + +% bibliography +\usepackage[backend=biber,sortcites,doi=false,isbn=false,url=false,giveninits=true,maxbibnames=10,style=alphabetic]{biblatex} +\addbibresource{references.bib} +\AtBeginBibliography{\scriptsize} % small font size +\AtEveryBibitem{% +\ifentrytype{inproceedings}{ + \clearfield{review}% + \clearfield{editor}% + \clearfield{series}%% +}{} +} + +% automatic references +\usepackage{cleveref} + +% some abbreviations +\newcommand{\Prob}{\mathbb{P}} +\newcommand*{\E}{\mathbb{E}} +\DeclareMathOperator*{\argmax}{arg\,max} +\DeclareMathOperator{\ECE}{ECE} +\DeclareMathOperator{\biasedskce}{SKCE_b} +\DeclareMathOperator{\unbiasedskce}{SKCE_{uq}} +\DeclareMathOperator{\linearskce}{SKCE_{ul}} +\DeclareMathOperator{\asympunbiasedskce}{aSKCE_{uq}} +\DeclareMathOperator{\asymplinearskce}{aSKCE_{ul}} +\DeclareMathOperator{\measure}{CE} +\DeclareMathOperator{\kernelmeasure}{KCE} +\DeclareMathOperator{\squaredkernelmeasure}{SKCE} +\DeclareMathOperator{\Expect}{\mathbb{E}} +\DeclareMathOperator{\Dir}{Dir} +\DeclareMathOperator{\Categorical}{Cat} + +% Define number of columns and rows +\tcbposterset{poster={columns=2,rows=8}} + +% Define split box +\DeclareTColorBox{splitbox}{ O{} }{standard jigsaw,sharp corners, + sidebyside,sidebyside align=center,opacityfill=0,#1} + +% metadata +\title{Calibration tests in multi-class classification:\\A unifying framework} +\author{David Widmann$^*$ Fredrik Lindsten$^\dagger$ Dave Zachariah$^*$} +\def\institute{$^*$Department of Information Technology, Uppsala University\\ $^\ddagger$Division of Statistics and Machine Learning, Linköping University} +\date{} +\def\contact{david.widmann@it.uu.se fredrik.lindsten@liu.se dave.zachariah@it.uu.se} + +% Default settings +\pagestyle{empty} + +\begin{document} +\begin{tcbposter}[fontsize=26pt] + \makeatletter + \posterbox[blankest,width=\paperwidth,interior engine=path,interior style={color=white},% + top=\tcbposterrowspacing,bottom=\tcbposterrowspacing,left=0.15\paperwidth,right=0.15\paperwidth,% + height=0.15\paperwidth,height plus=0.15\paperwidth,halign=flush center,valign=center, % + underlay={% + \node[right,inner sep=0pt,outer sep=0pt,text width=0.15\paperwidth,align=center] at (frame.west) {\includegraphics[width=0.12\paperwidth]{figures/logos/UU.pdf}};% + \node[left,inner sep=0pt,outer sep=0pt,text width=0.15\paperwidth,align=center] at (frame.east) {\includegraphics[width=0.12\paperwidth]{figures/logos/LiU.pdf}};% + }]{name=title,column=1,below=top,xshift=-\tcbpostercolspacing}{% + \vspace*{2ex}% + {\bfseries\fontsize{4.5ex}{5.2ex}\selectfont{\@title}}\\[2ex]% + {\LARGE{\@author}}\\[1ex]% + {\large{\institute}}% + }% + \makeatother + + % footline + \makeatletter + \posterbox[blankest,width=\paperwidth,% + top=5pt,bottom=5pt,left=\tcbpostercolspacing,right=\tcbpostercolspacing,% + valign=center,fontupper=\ttfamily\small,interior engine=path, interior style={color=uumidgrey}% + ]{name=footline,column=1,above=bottom,xshift=-\tcbpostercolspacing}{% + \@date\hfill\contact% + }% + \makeatother + + % references + \posterbox[adjusted title=References]{name=references,column=1,above=footline,span=2}{ + \printbibliography[heading=none] + } + + \posterbox[adjusted title=Summary of our work]{name=summary,column=1,below=title}{ + \begin{itemize} + \item We propose a \hl{unifying framework} for quantifying calibration error of probabilistic classifiers that encompasses several existing error measures. + \item We introduce a new \hl{kernel calibration error} (KCE), for which we derive \hl{unbiased and consistent estimators}. + \item We show how the transfer of calibration error estimates to \hl{probabilities of false rejection} makes them interpretable and statistically commensurable. + \item We provide \hl{Julia packages} for calibration estimation. + \end{itemize} + } + + \posterbox[adjusted title=General setup]{name=setup,column=1,below=summary}{ + \begin{itemize} + \item Let $X$ be random inputs (features) of $m$ classes $1,\ldots,m$, denoted by $Y$. + \item Consider a \hl{probabilistic model} $g$ that predicts a probability distribution of classes $g(X) \in \Delta^{m}$ for input $X$, where $\Delta^m \coloneqq \{z \in \mathbb{R}^m_{\geq 0} : \|z\|_1 = 1\}$ denotes the $(m-1)$-dimensional probability simplex. + \item Ideally $g$ predicts $g_y(X) = \Prob[Y = y \,|\,X]$ for all classes $y$. + \end{itemize} + } + + \posterbox[adjusted title={Calibration}]{name=calibration,column=1,below=setup}{ + Although in practice the model will never be ideal but at most close to it, we can strive to satisfy other desirable statistical properties such as \hl{calibration}. + \begin{tcolorbox}[colback=blondsvag, halign=center] + Informally, in the long run every prediction should match the relative frequencies of the observed classes. + \end{tcolorbox} + \vspace{-\topsep} + \begin{itemize} + \item Mathematically, a model \(g\) is calibrated if + \begin{equation*} + g_y(X) = \Prob[Y = y \,|\, g(X)] \quad \text{almost surely for all classes } y, + \end{equation*} + or equivalently if + \begin{equation}\label{eq:calibration} + g(X) = r(g(X)) \coloneqq (\mathbb{P}[Y = 1 \,|\, g(X)], \ldots, \mathbb{P}[Y = m \,|\, g(X)]) \quad \text{almost surely}. + \end{equation} + + \item There are many calibrated models \parencite{vaicenavicius19_evaluat}, so there is hope to find one. + \begin{tcolorbox}[colback=blondsvag] + Consider three equally likely classes with triangular distributions of inputs: + \begin{center} + \pgfplotsset{width=0.25\textwidth,height=0.1\textwidth} + \input{figures/triangular_pdf_x_y.tex} + \end{center} + The following models are all calibrated but only the first one is ideal: + \begin{center} + \pgfplotsset{width=0.25\linewidth,height=0.1\textwidth,ytick={0,0.5,1}} + \input{figures/triangular_models.tex} + \end{center} + \end{tcolorbox} + \end{itemize} + } + + \posterbox[adjusted title={Calibration error}]{name=error,column=1,below=calibration}{ + We propose the following general measure of miscalibration which arises naturally from the definition of calibration in \cref{eq:calibration}. + + \begin{tcolorbox}[colback=sandstark] + The \hl{calibration error}~($\measure$) of model $g$ w.r.t.\ a class $\mathcal{F}$ of functions $f \colon \Delta^m \to \mathbb{R}^m$ is + \begin{equation*} + \measure[\mathcal{F}, g] \coloneqq \sup_{f \in \mathcal{F}} \Expect\left[{(r(g(X)) - g(X))}^\intercal f(g(X)) \right]. + \end{equation*} + \end{tcolorbox} + + \vspace{-\topsep} + \begin{itemize} + \item By design, if model $g$ is calibrated then the $\measure$ is zero, regardless of $\mathcal{F}$. + \item The $\measure$ is equal to the common expected calibration error ($\ECE$) + \begin{equation}\label{eq:ece} + \ECE[d, g] = \Expect[d(r(g(X)), g(X))] + \end{equation} + for certain choices of $\mathcal{F}$ and distances $d$ such as the city block distance, the total variation distance, and the squared Euclidean distance. + \item The $\measure$ captures also the maximum mean calibration error \parencite{kumar18_train_calib_measur_neural_networ}. + \end{itemize} + } + + \posterbox[adjusted title={Estimators of the calibration error}]{name=estimator,column=2,below=title}{ + Consider the task of estimating the $\measure$ of model $g$ using a validation set $\{(X_i, Y_i)\}_{i=1}^n$ of i.i.d.\ random pairs of inputs and labels that are distributed according to $(X,Y)$. + + \begin{itemize} + \item Standard estimators of the $\ECE$ are inconsistent and biased in many cases \parencite{vaicenavicius19_evaluat} and can scale poorly to large $m$. + + \begin{tcolorbox}[colback=blondsvag] + The main difficulty is the estimation of the function $r$ in \cref{eq:ece}. \Cref{eq:kce} shows that for $\kernelmeasure$ there is no explicit dependence on $r$! + \end{tcolorbox} + + \item For $i,j \in \{1,\ldots,n\}$ define + \begin{equation*} + h_{i,j} \coloneqq {(e_{Y_i} - g(X_i))}^\intercal k(g(X_i), g(X_j)) (e_{Y_j} - g(X_j)). + \end{equation*} + + \begin{tcolorbox}[colback=sandstark] + If $\mathbb{E}[\|k(g(X),g(X))\|] < \infty$, then the following estimators are \hl{consistent estimators of the squared kernel calibration error} $\squaredkernelmeasure[k, g] \coloneqq \kernelmeasure[k,g]^2$. + \begin{center} + \begin{tabular}{llll} \toprule + Notation & Definition & Properties & Complexity\\ \midrule + $\biasedskce$ & $n^{-2} \sum_{i,j=1}^n h_{i,j}$ & biased & $O(n^2)$ \\ + $\unbiasedskce$ & $ {\binom{n}{2}}^{-1} \sum_{1 \leq i < j \leq n} h_{i,j}$ & unbiased & $O(n^2)$ \\ + $\linearskce$ & $ {\lfloor n/2\rfloor}^{-1} \sum_{i = 1}^{\lfloor n / 2\rfloor} h_{2i-1,2i}$ & unbiased & $O(n)$ \\ \bottomrule + \end{tabular} + \end{center} + \end{tcolorbox} + \end{itemize} + } + + \posterbox[adjusted title={Viewing estimators as test statistics}]{name=statistics,column=2,below=estimator}{ + In general, the $\measure$ does not have a meaningful unit or scale. This renders it difficult to interpret an estimated non-zero error and to compare different models. + + \begin{tcolorbox}[colback=sandstark] + For the consistent and unbiased estimators of the $\squaredkernelmeasure$ we derive \hl{bounds and approximations of the probability of false rejection} of a calibrated model. + \end{tcolorbox} + + \vspace{-\topsep} + \begin{itemize} + \item These results allow us to test the hypothesis that model $g$ is calibrated. + \item The bounds enable us to transfer unintuitive calibration error estimates to an \hl{intuitive and interpretable} probabilistic setting. + \end{itemize} + } + + \posterbox[adjusted title={Kernel calibration error}]{name=kce,column=1,between=error and references}{ + Let $\mathcal{F}$ be the unit ball in a reproducing kernel Hilbert space with matrix-valued kernel $k \colon \Delta^m \times \Delta^m \to \mathbb{R}^{m \times m}$. Then we define the \hl{kernel calibration error} ($\kernelmeasure$) with respect to kernel $k$ as $\kernelmeasure[k, g] \coloneqq \measure[\mathcal{F}, g]$. + + \begin{tcolorbox}[colback=blondsvag] + An example of a matrix-valued kernel is $k(a, b) = M \tilde{k}(a, b)$, where $M \in \mathbb{R}^{m\times m}$ is a positive semi-definite matrix and $\tilde{k} \colon \Delta^m \times \Delta^m \to \mathbb{R}$ is a real-valued kernel. + \end{tcolorbox} + + If $k$ is a universal kernel, then the $\kernelmeasure$ is zero if and only if model $g$ is calibrated. + + \begin{tcolorbox}[colback=sandstark] + If $\Expect[\|k(g(X),g(X))\|] < \infty$, then + \begin{equation}\label{eq:kce} + \kernelmeasure[k,g] = {\bigg(\Expect[{(e_Y - g(X))}^{\intercal} k(g(X), g(X')) {(e_{Y'} - g(X'))}]\bigg)}^{1/2}, + \end{equation} + where $(X', Y')$ is an independent copy of $(X,Y)$ and $e_i$ denotes the $i$th unit vector. + \end{tcolorbox} + } + + \posterbox[adjusted title={Experiments}]{name=experiment,column=2,between=statistics and references}{ + We construct data sets $\{g(X_i), Y_i\}_{i=1}^{250}$ of three models with $10$ classes by sampling predictions $g(X_i) \sim \Dir(0.1, \dots, 0.1)$ and labels $Y_i$ conditionally on $g(X_i)$ from + \begin{align*} + \text{\textbf{M1: }} &\Categorical(g(X_i)), & + \text{\textbf{M2: }} &0.5\Categorical(g(X_i)) + 0.5\Categorical(1,0,\dots,0), & + \text{\textbf{M3: }} &\Categorical(0.1, \dots, 0.1). + \end{align*} + Model \textbf{M1} is calibrated, and models \textbf{M2} and \textbf{M3} are uncalibrated. + + \begin{center} + \input{figures/comparison_estimates.tex} + \captionof{figure}{Calibration error estimates of $10^4$ randomly sampled data sets. The solid black line indicates the mean of the calibration error estimates, and the dashed red line displays the true calibration error of the model.} + \end{center} + + \begin{center} + \input{figures/comparison_tests.tex} + \captionof{figure}{Test errors versus bounds/approximations of the probability of false rejection, evaluated on $500$ ($\asympunbiasedskce$) and $10^4$ (all other test statistics) randomly sampled data sets. For model \textbf{M1} the type I error is shown, for both uncalibrated models the type II error is plotted.} + \end{center} + } +\end{tcbposter} +\end{document} diff --git a/poster/references.bib b/poster/references.bib new file mode 100644 index 0000000..3040415 --- /dev/null +++ b/poster/references.bib @@ -0,0 +1,68 @@ +@InProceedings{kumar18_train_calib_measur_neural_networ, + author = {Kumar, Aviral and Sarawagi, Sunita and Jain, Ujjwal}, + title = {Trainable Calibration Measures for Neural Networks + from Kernel Mean Embeddings}, + booktitle = {Proceedings of the 35th International Conference on + Machine Learning}, + year = 2018, + volume = 80, + pages = {2805--2814}, + abstract = {Modern neural networks have recently been found to + be poorly calibrated, primarily in the direction of + over-confidence. Methods like entropy penalty and + temperature smoothing improve calibration by + clamping confidence, but in doing so compromise the + many legitimately confident predictions. We propose + a more principled fix that minimizes an explicit + calibration error during training. We present MMCE, + a RKHS kernel based measure of calibration that is + efficiently trainable alongside the negative + likelihood loss without careful hyper-parameter + tuning. Theoretically too, MMCE is a sound measure + of calibration that is minimized at perfect + calibration, and whose finite sample estimates are + consistent and enjoy fast convergence + rates. Extensive experiments on several network + architectures demonstrate that MMCE is a fast, + stable, and accurate method to minimize calibration + error while maximally preserving the number of high + confidence predictions.}, + pdf = + {http://proceedings.mlr.press/v80/kumar18a/kumar18a.pdf}, + series = {Proceedings of Machine Learning Research}, +} + +@InProceedings{vaicenavicius19_evaluat, + author = {Vaicenavicius, Juozas and Widmann, David and + Andersson, Carl and Lindsten, Fredrik and Roll, + Jacob and Sch\"{o}n, Thomas B.}, + title = {Evaluating model calibration in classification}, + booktitle = {Proceedings of Machine Learning Research}, + year = 2019, + volume = 89, + pages = {3459--3467}, + abstract = {Probabilistic classifiers output a probability + distribution on target classes rather than just a + class prediction. Besides providing a clear + separation of prediction and decision making, the + main advantage of probabilistic models is their + ability to represent uncertainty about + predictions. In safety-critical applications, it is + pivotal for a model to possess an adequate sense of + uncertainty, which for probabilistic classifiers + translates into outputting probability distributions + that are consistent with the empirical frequencies + observed from realized outcomes. A classifier with + such a property is called calibrated. In this work, + we develop a general theoretical calibration + evaluation framework grounded in probability theory, + and point out subtleties present in model + calibration evaluation that lead to refined + interpretations of existing evaluation + techniques. Lastly, we propose new ways to quantify + and visualize miscalibration in probabilistic + classification, including novel multidimensional + reliability diagrams.}, + month = 4, + series = {Proceedings of Machine Learning Research}, +} \ No newline at end of file