diff --git a/.gitattributes b/.gitattributes index c114f99fa4299aeefa297237bf6a770deb8cce63..bda238879d212dd4d7bd5d8b6dce47640fb106be 100644 --- a/.gitattributes +++ b/.gitattributes @@ -7361,3 +7361,5 @@ SMART/HiddenLayerExperiment/cache/imagenet_resnet50_seed1_vs0.2/test_logits.npy DensityAwareCalibration/Tomani[[:space:]]et[[:space:]]al.[[:space:]]-[[:space:]]2023[[:space:]]-[[:space:]]Beyond[[:space:]]In-Domain[[:space:]]Scenarios[[:space:]]Robust[[:space:]]Density-Aware[[:space:]]Calibration.pdf filter=lfs diff=lfs merge=lfs -text DensityAwareCalibration/classifier_ckpts/cifar10/resnet18.pth filter=lfs diff=lfs merge=lfs -text DensityAwareCalibration/teaser_fig.png filter=lfs diff=lfs merge=lfs -text +icml2026/example_paper.pdf filter=lfs diff=lfs merge=lfs -text +icml2026/figs/logitsgap_vs_confidence_change.pdf filter=lfs diff=lfs merge=lfs -text diff --git a/icml2026/algorithm.sty b/icml2026/algorithm.sty new file mode 100644 index 0000000000000000000000000000000000000000..843e3d5b9a52c705c34255b26177ba6d064630f5 --- /dev/null +++ b/icml2026/algorithm.sty @@ -0,0 +1,79 @@ +% ALGORITHM STYLE -- Released 8 April 1996 +% for LaTeX-2e +% Copyright -- 1994 Peter Williams +% E-mail Peter.Williams@dsto.defence.gov.au +\NeedsTeXFormat{LaTeX2e} +\ProvidesPackage{algorithm} +\typeout{Document Style `algorithm' - floating environment} + +\RequirePackage{float} +\RequirePackage{ifthen} +\newcommand{\ALG@within}{nothing} +\newboolean{ALG@within} +\setboolean{ALG@within}{false} +\newcommand{\ALG@floatstyle}{ruled} +\newcommand{\ALG@name}{Algorithm} +\newcommand{\listalgorithmname}{List of \ALG@name s} + +% Declare Options +% first appearance +\DeclareOption{plain}{ + \renewcommand{\ALG@floatstyle}{plain} +} +\DeclareOption{ruled}{ + \renewcommand{\ALG@floatstyle}{ruled} +} +\DeclareOption{boxed}{ + \renewcommand{\ALG@floatstyle}{boxed} +} +% then numbering convention +\DeclareOption{part}{ + \renewcommand{\ALG@within}{part} + \setboolean{ALG@within}{true} +} +\DeclareOption{chapter}{ + \renewcommand{\ALG@within}{chapter} + \setboolean{ALG@within}{true} +} +\DeclareOption{section}{ + \renewcommand{\ALG@within}{section} + \setboolean{ALG@within}{true} +} +\DeclareOption{subsection}{ + \renewcommand{\ALG@within}{subsection} + \setboolean{ALG@within}{true} +} +\DeclareOption{subsubsection}{ + \renewcommand{\ALG@within}{subsubsection} + \setboolean{ALG@within}{true} +} +\DeclareOption{nothing}{ + \renewcommand{\ALG@within}{nothing} + \setboolean{ALG@within}{true} +} +\DeclareOption*{\edef\ALG@name{\CurrentOption}} + +% ALGORITHM +% +\ProcessOptions +\floatstyle{\ALG@floatstyle} +\ifthenelse{\boolean{ALG@within}}{ + \ifthenelse{\equal{\ALG@within}{part}} + {\newfloat{algorithm}{htbp}{loa}[part]}{} + \ifthenelse{\equal{\ALG@within}{chapter}} + {\newfloat{algorithm}{htbp}{loa}[chapter]}{} + \ifthenelse{\equal{\ALG@within}{section}} + {\newfloat{algorithm}{htbp}{loa}[section]}{} + \ifthenelse{\equal{\ALG@within}{subsection}} + {\newfloat{algorithm}{htbp}{loa}[subsection]}{} + \ifthenelse{\equal{\ALG@within}{subsubsection}} + {\newfloat{algorithm}{htbp}{loa}[subsubsection]}{} + \ifthenelse{\equal{\ALG@within}{nothing}} + {\newfloat{algorithm}{htbp}{loa}}{} +}{ + \newfloat{algorithm}{htbp}{loa} +} +\floatname{algorithm}{\ALG@name} + +\newcommand{\listofalgorithms}{\listof{algorithm}{\listalgorithmname}} + diff --git a/icml2026/algorithmic.sty b/icml2026/algorithmic.sty new file mode 100644 index 0000000000000000000000000000000000000000..ad614783f8b9160eb736ae864a2c6fe6cf7707c5 --- /dev/null +++ b/icml2026/algorithmic.sty @@ -0,0 +1,201 @@ +% ALGORITHMIC STYLE -- Released 8 APRIL 1996 +% for LaTeX version 2e +% Copyright -- 1994 Peter Williams +% E-mail PeterWilliams@dsto.defence.gov.au +% +% Modified by Alex Smola (08/2000) +% E-mail Alex.Smola@anu.edu.au +% +\NeedsTeXFormat{LaTeX2e} +\ProvidesPackage{algorithmic} +\typeout{Document Style `algorithmic' - environment} +% +\RequirePackage{ifthen} +\RequirePackage{calc} +\newboolean{ALC@noend} +\setboolean{ALC@noend}{false} +\newcounter{ALC@line} +\newcounter{ALC@rem} +\newlength{\ALC@tlm} +% +\DeclareOption{noend}{\setboolean{ALC@noend}{true}} +% +\ProcessOptions +% +% ALGORITHMIC +\newcommand{\algorithmicrequire}{\textbf{Require:}} +\newcommand{\algorithmicensure}{\textbf{Ensure:}} +\newcommand{\algorithmiccomment}[1]{\{#1\}} +\newcommand{\algorithmicend}{\textbf{end}} +\newcommand{\algorithmicif}{\textbf{if}} +\newcommand{\algorithmicthen}{\textbf{then}} +\newcommand{\algorithmicelse}{\textbf{else}} +\newcommand{\algorithmicelsif}{\algorithmicelse\ \algorithmicif} +\newcommand{\algorithmicendif}{\algorithmicend\ \algorithmicif} +\newcommand{\algorithmicfor}{\textbf{for}} +\newcommand{\algorithmicforall}{\textbf{for all}} +\newcommand{\algorithmicdo}{\textbf{do}} +\newcommand{\algorithmicendfor}{\algorithmicend\ \algorithmicfor} +\newcommand{\algorithmicwhile}{\textbf{while}} +\newcommand{\algorithmicendwhile}{\algorithmicend\ \algorithmicwhile} +\newcommand{\algorithmicloop}{\textbf{loop}} +\newcommand{\algorithmicendloop}{\algorithmicend\ \algorithmicloop} +\newcommand{\algorithmicrepeat}{\textbf{repeat}} +\newcommand{\algorithmicuntil}{\textbf{until}} + +%changed by alex smola +\newcommand{\algorithmicinput}{\textbf{input}} +\newcommand{\algorithmicoutput}{\textbf{output}} +\newcommand{\algorithmicset}{\textbf{set}} +\newcommand{\algorithmictrue}{\textbf{true}} +\newcommand{\algorithmicfalse}{\textbf{false}} +\newcommand{\algorithmicand}{\textbf{and\ }} +\newcommand{\algorithmicor}{\textbf{or\ }} +\newcommand{\algorithmicfunction}{\textbf{function}} +\newcommand{\algorithmicendfunction}{\algorithmicend\ \algorithmicfunction} +\newcommand{\algorithmicmain}{\textbf{main}} +\newcommand{\algorithmicendmain}{\algorithmicend\ \algorithmicmain} +%end changed by alex smola + +\def\ALC@item[#1]{% +\if@noparitem \@donoparitem + \else \if@inlabel \indent \par \fi + \ifhmode \unskip\unskip \par \fi + \if@newlist \if@nobreak \@nbitem \else + \addpenalty\@beginparpenalty + \addvspace\@topsep \addvspace{-\parskip}\fi + \else \addpenalty\@itempenalty \addvspace\itemsep + \fi + \global\@inlabeltrue +\fi +\everypar{\global\@minipagefalse\global\@newlistfalse + \if@inlabel\global\@inlabelfalse \hskip -\parindent \box\@labels + \penalty\z@ \fi + \everypar{}}\global\@nobreakfalse +\if@noitemarg \@noitemargfalse \if@nmbrlist \refstepcounter{\@listctr}\fi \fi +\sbox\@tempboxa{\makelabel{#1}}% +\global\setbox\@labels + \hbox{\unhbox\@labels \hskip \itemindent + \hskip -\labelwidth \hskip -\ALC@tlm + \ifdim \wd\@tempboxa >\labelwidth + \box\@tempboxa + \else \hbox to\labelwidth {\unhbox\@tempboxa}\fi + \hskip \ALC@tlm}\ignorespaces} +% +\newenvironment{algorithmic}[1][0]{ +\let\@item\ALC@item + \newcommand{\ALC@lno}{% +\ifthenelse{\equal{\arabic{ALC@rem}}{0}} +{{\footnotesize \arabic{ALC@line}:}}{}% +} +\let\@listii\@listi +\let\@listiii\@listi +\let\@listiv\@listi +\let\@listv\@listi +\let\@listvi\@listi +\let\@listvii\@listi + \newenvironment{ALC@g}{ + \begin{list}{\ALC@lno}{ \itemsep\z@ \itemindent\z@ + \listparindent\z@ \rightmargin\z@ + \topsep\z@ \partopsep\z@ \parskip\z@\parsep\z@ + \leftmargin 1em + \addtolength{\ALC@tlm}{\leftmargin} + } + } + {\end{list}} + \newcommand{\ALC@it}{\addtocounter{ALC@line}{1}\addtocounter{ALC@rem}{1}\ifthenelse{\equal{\arabic{ALC@rem}}{#1}}{\setcounter{ALC@rem}{0}}{}\item} + \newcommand{\ALC@com}[1]{\ifthenelse{\equal{##1}{default}}% +{}{\ \algorithmiccomment{##1}}} + \newcommand{\REQUIRE}{\item[\algorithmicrequire]} + \newcommand{\ENSURE}{\item[\algorithmicensure]} + \newcommand{\STATE}{\ALC@it} + \newcommand{\COMMENT}[1]{\algorithmiccomment{##1}} +%changes by alex smola + \newcommand{\INPUT}{\item[\algorithmicinput]} + \newcommand{\OUTPUT}{\item[\algorithmicoutput]} + \newcommand{\SET}{\item[\algorithmicset]} +% \newcommand{\TRUE}{\algorithmictrue} +% \newcommand{\FALSE}{\algorithmicfalse} + \newcommand{\AND}{\algorithmicand} + \newcommand{\OR}{\algorithmicor} + \newenvironment{ALC@func}{\begin{ALC@g}}{\end{ALC@g}} + \newenvironment{ALC@main}{\begin{ALC@g}}{\end{ALC@g}} +%end changes by alex smola + \newenvironment{ALC@if}{\begin{ALC@g}}{\end{ALC@g}} + \newenvironment{ALC@for}{\begin{ALC@g}}{\end{ALC@g}} + \newenvironment{ALC@whl}{\begin{ALC@g}}{\end{ALC@g}} + \newenvironment{ALC@loop}{\begin{ALC@g}}{\end{ALC@g}} + \newenvironment{ALC@rpt}{\begin{ALC@g}}{\end{ALC@g}} + \renewcommand{\\}{\@centercr} + \newcommand{\IF}[2][default]{\ALC@it\algorithmicif\ ##2\ \algorithmicthen% +\ALC@com{##1}\begin{ALC@if}} + \newcommand{\SHORTIF}[2]{\ALC@it\algorithmicif\ ##1\ + \algorithmicthen\ {##2}} + \newcommand{\ELSE}[1][default]{\end{ALC@if}\ALC@it\algorithmicelse% +\ALC@com{##1}\begin{ALC@if}} + \newcommand{\ELSIF}[2][default]% +{\end{ALC@if}\ALC@it\algorithmicelsif\ ##2\ \algorithmicthen% +\ALC@com{##1}\begin{ALC@if}} + \newcommand{\FOR}[2][default]{\ALC@it\algorithmicfor\ ##2\ \algorithmicdo% +\ALC@com{##1}\begin{ALC@for}} + \newcommand{\FORALL}[2][default]{\ALC@it\algorithmicforall\ ##2\ % +\algorithmicdo% +\ALC@com{##1}\begin{ALC@for}} + \newcommand{\SHORTFORALL}[2]{\ALC@it\algorithmicforall\ ##1\ % + \algorithmicdo\ {##2}} + \newcommand{\WHILE}[2][default]{\ALC@it\algorithmicwhile\ ##2\ % +\algorithmicdo% +\ALC@com{##1}\begin{ALC@whl}} + \newcommand{\LOOP}[1][default]{\ALC@it\algorithmicloop% +\ALC@com{##1}\begin{ALC@loop}} +%changed by alex smola + \newcommand{\FUNCTION}[2][default]{\ALC@it\algorithmicfunction\ ##2\ % + \ALC@com{##1}\begin{ALC@func}} + \newcommand{\MAIN}[2][default]{\ALC@it\algorithmicmain\ ##2\ % + \ALC@com{##1}\begin{ALC@main}} +%end changed by alex smola + \newcommand{\REPEAT}[1][default]{\ALC@it\algorithmicrepeat% + \ALC@com{##1}\begin{ALC@rpt}} + \newcommand{\UNTIL}[1]{\end{ALC@rpt}\ALC@it\algorithmicuntil\ ##1} + \ifthenelse{\boolean{ALC@noend}}{ + \newcommand{\ENDIF}{\end{ALC@if}} + \newcommand{\ENDFOR}{\end{ALC@for}} + \newcommand{\ENDWHILE}{\end{ALC@whl}} + \newcommand{\ENDLOOP}{\end{ALC@loop}} + \newcommand{\ENDFUNCTION}{\end{ALC@func}} + \newcommand{\ENDMAIN}{\end{ALC@main}} + }{ + \newcommand{\ENDIF}{\end{ALC@if}\ALC@it\algorithmicendif} + \newcommand{\ENDFOR}{\end{ALC@for}\ALC@it\algorithmicendfor} + \newcommand{\ENDWHILE}{\end{ALC@whl}\ALC@it\algorithmicendwhile} + \newcommand{\ENDLOOP}{\end{ALC@loop}\ALC@it\algorithmicendloop} + \newcommand{\ENDFUNCTION}{\end{ALC@func}\ALC@it\algorithmicendfunction} + \newcommand{\ENDMAIN}{\end{ALC@main}\ALC@it\algorithmicendmain} + } + \renewcommand{\@toodeep}{} + \begin{list}{\ALC@lno}{\setcounter{ALC@line}{0}\setcounter{ALC@rem}{0}% + \itemsep\z@ \itemindent\z@ \listparindent\z@% + \partopsep\z@ \parskip\z@ \parsep\z@% + \labelsep 0.5em \topsep 0.2em% + \ifthenelse{\equal{#1}{0}} + {\labelwidth 0.5em } + {\labelwidth 1.2em } + \leftmargin\labelwidth \addtolength{\leftmargin}{\labelsep} + \ALC@tlm\labelsep + } + } + {\end{list}} + + + + + + + + + + + + + + diff --git a/icml2026/appendix.tex b/icml2026/appendix.tex new file mode 100644 index 0000000000000000000000000000000000000000..94adb7e136a4d22bd0a37af635488180269bfea4 --- /dev/null +++ b/icml2026/appendix.tex @@ -0,0 +1,1227 @@ +\section{Theoretical Proofs} +\label{Appendix:Theoretical_Proofs} + + +\subsection{Temperature--Confidence Relation and Margin Bounds} +\label{subsec:confidence-relation} + +\textbf{Problem and observation.} To reach a target top-class confidence $\hat p\in(0,1)$, how constrained is $T$? Empirically, fixing only $z_M$ leaves $T$ ill-determined; using the \emph{Margin} $m$ yields tight bounds. We now prove this, step by step. + +\paragraph{Target confidence equation.} Requiring $p_{\phi,\max}=\hat p$ is equivalent to +\begin{align} +\frac{e^{z_M/T}}{\sum_{j=1}^K e^{z_j/T}}&=\hat p +&&\Longleftrightarrow& +\sum_{j\neq M} e^{(z_j-z_M)/T}&=\frac{1}{\hat p}-1\ :=\ S. +\label{eq:temp-conf} +\end{align} +Because $\hat p$ is the \emph{maximum} softmax probability, $\hat p\ge 1/K$, hence $S\le K{-}1$; moreover $S>0$ since $\hat p<1$. Thus $S\in(0,K{-}1]$, and if we assume a strict top-1 margin $m>0$ (no top-2 ties) then $\hat p>1/K$ and $S\in(0,K{-}1)$. + +\paragraph{Unboundedness if only $z_M$ is known.} +Assume $z_j-z_M=-\delta$ for all $j\neq M$ with $\delta>0$. Then +\begin{align} +\sum_{j\neq M} e^{(z_j-z_M)/T} +&=\sum_{j\neq M} e^{-\delta/T} +=(K-1)\,e^{-\delta/T} +=S, \label{eq:unb-1}\\ +\Rightarrow\quad T&=-\frac{\delta}{\log\!\left(\frac{S}{K-1}\right)}. +\label{eq:unb-2} +\end{align} +When $S1/K$), $\log(S/(K{-}1))<0$, so as $\delta\in(0,\infty)$ varies, \eqref{eq:unb-2} sweeps $T\in(0,\infty)$. Thus +\[ +\boxed{\text{Fixing }z_M\text{ alone leaves the feasible }T\text{ unbounded: }(0,\infty).} +\] + +\paragraph{$m$-boundedness.} +Let $\mu\in\arg\max_{j\neq M} z_j$ denote a runner-up index and define the (nonnegative) margin $m:=z_M-z_\mu$. For any $j\notin\{M,\mu\}$, +\[ +z_j-z_M\ \le\ z_\mu-z_M\ =\ -m +\quad\Longrightarrow\quad +e^{(z_j-z_M)/T}\ \le\ e^{-m/T}. +\] +Since $e^{(z_\mu-z_M)/T}=e^{-m/T}$, we have +\begin{align} +\sum_{j\neq M} e^{(z_j-z_M)/T} +&= e^{-m/T} + \sum_{j\notin\{M,\mu\}} e^{(z_j-z_M)/T} \nonumber\\ +&\ge e^{-m/T}, \label{eq:gb-lower}\\ +\sum_{j\neq M} e^{(z_j-z_M)/T} +&\le e^{-m/T} + (K-2)\,e^{-m/T} += (K-1)\,e^{-m/T}. \label{eq:gb-upper} +\end{align} +Combining \eqref{eq:temp-conf}, \eqref{eq:gb-lower}, and \eqref{eq:gb-upper} yields +\begin{equation} +e^{-m/T}\ \le\ S\ \le\ (K-1)\,e^{-m/T}, +\qquad S=\tfrac{1}{\hat p}-1\in(0,K{-}1]. +\label{eq:gb-ineq-1} +\end{equation} +Equivalently, +\begin{equation} +-\log\!\Big(\tfrac{S}{K-1}\Big)\ \ge\ \frac{m}{T}\ \ge\ -\log S. +\label{eq:gb-ineq-5} +\end{equation} +Solving \eqref{eq:gb-ineq-5} for $T>0$ gives the $m$-bounded feasible set: +\[ +\boxed{ +\begin{cases} +\displaystyle +T\ \in\ \Bigl[\,\frac{m}{-\log\!\bigl(S/(K-1)\bigr)}\,,\,\frac{m}{-\log S}\,\Bigr], +& \text{if } 01/2),\\[1.2ex] +\displaystyle +T\ \in\ \Bigl[\,\frac{m}{-\log\!\bigl(S/(K-1)\bigr)}\,,\,\infty\Bigr), +& \text{if } 1\le S1/2$). In particular, for $K=2$ the bounds coincide and $T=\frac{m}{-\log S}$ is uniquely determined. + + + + +\subsection{Charbonnier--SoftECE Upper-Bounds smCE} +\label{subsec:Charbonnier-soft-controls-smce} + +\paragraph{Setup and goal.} +Let $p(x)\in[0,1]$ denote the predicted probability of correctness (top-class confidence) and +$a(x):=\mathbb{1}\{\hat y(x)=y\}\in\{0,1\}$ the correctness indicator. +We measure calibration via the \emph{smooth calibration error} (smCE), +the worst-case correlation between the residual $a(X)-p(X)$ and any $1$-Lipschitz probe of the prediction $p(X)$ (cf.\ forecasting(\!\cite{kakade2004deterministic}) and the ML calibration view of \cite{blasiok2023does}): +\begin{align*} +\mathrm{smCE}(f) &:= \sup_{\varphi\in\mathcal H}\;\Big|\;\mathbb{E}\big[(a(X)-p(X))\,\varphi\!\big(p(X)\big)\big]\;\Big|, \\ +\mathcal H &:= \big\{\varphi:[0,1]\to[-1,1]\ \text{s.t.}\ \mathrm{Lip}(\varphi)\le 1\big\}. +\end{align*} +We study the \emph{Charbonnier--SoftECE} objective (a smoothed, Huberized absolute calibration error): +\[ +\mathcal{H}_{\lambda,\delta}(f) +\;:=\; +\E_{X}\!\left[ + \int_{0}^{1} + K_\lambda\!\big(p(X),u\big)\,\phi_\delta\!\big(a(X)-u\big)\;\rho(u)\,du +\right], +\qquad +\phi_\delta(r):=\sqrt{r^2+\delta^2}, +\] +where $\rho$ is a reference density on $[0,1]$ and +\begin{equation} +\label{eq:kernel-normalization-rev} +K_\lambda(p,u)\;=\;\frac{k_\lambda(p-u)}{\int_0^1 k_\lambda(p-v)\,\rho(v)\,dv}, +\qquad +k_\lambda(t):=e^{-\lambda t^2},\ \ \lambda>0, +\end{equation} +so that $\int_0^1 K_\lambda(p,u)\,\rho(u)\,du=1$ for every $p\in[0,1]$. +Assume \emph{boundedness and bounded-away-from-zero} of $\rho$: there exist constants $0<\rho_{\min}\le \rho(u)\le\rho_{\max}<\infty$ for all $u\in[0,1]$, and write $\kappa:=\rho_{\max}/\rho_{\min}$. + +\paragraph{Main result.} +\begin{theorem}[Charbonnier--SoftECE upper-bounds smCE] +\label{thm:Charbonnier-soft-controls-smce-rev} +Under the assumptions above, for all classifiers $f$ and all $\delta\ge 0$, +\begin{equation} +\label{eq:master-bound-rev} +\mathrm{smCE}(f)\ \le\ \mathcal{H}_{\lambda,\delta}(f)\ +\ 2\,B_\lambda, +\qquad +B_\lambda\;:=\;\sup_{p\in[0,1]}\int_0^1 |p-u|\,K_\lambda(p,u)\,\rho(u)\,du. +\end{equation} +Moreover, for the Gaussian kernel $k_\lambda(t)=e^{-\lambda t^2}$, +\begin{equation} +\label{eq:Blambda-explicit} +B_\lambda\ \le\ \min\!\left\{\,1,\ \frac{2\,\kappa}{\sqrt{\pi}}\cdot\frac{1}{\sqrt{\lambda}\ \mathrm{erf}(\sqrt{\lambda})}\right\}, +\end{equation} +and in particular for $\lambda\ge 1$, +\begin{equation} +\label{eq:Blambda-simplified} +B_\lambda\ \le\ \frac{C_\kappa}{\sqrt{\lambda}}, +\qquad +C_\kappa\;:=\;\frac{2\,\kappa}{\sqrt{\pi}\ \mathrm{erf}(1)}\ \approx\ 1.339\,\kappa. +\end{equation} +\end{theorem} + +\paragraph{Proof.} +For brevity write $p:=p(X)$ and $a:=a(X)$. +Fix any $\varphi\in\mathcal H$ with $\|\varphi\|_\infty\le 1$. +Introduce the (normalized) kernel smoothing operator +\[ +(\mathsf T_\lambda\varphi)(p)\ :=\ \int_0^1 K_\lambda(p,u)\,\varphi(u)\,\rho(u)\,du. +\] + +\emph{Decomposition.} +\begin{align} +\label{eq:decomp-rev} +\mathbb{E}[(a-p)\,\varphi(p)] +\;=\; +\mathbb{E}[(a-p)\,(\mathsf T_\lambda\varphi)(p)] +\;+\; +\mathbb{E}\big[(a-p)\,\{\varphi(p)-(\mathsf T_\lambda\varphi)(p)\}\big]. +\end{align} + +\emph{Approximation (mollification) error.} +By $\mathrm{Lip}(\varphi)\le 1$ and the triangle inequality, +\begin{align*} +\big|\varphi(p)-(\mathsf T_\lambda\varphi)(p)\big| +&=\left|\int_0^1 K_\lambda(p,u)\,\{\varphi(p)-\varphi(u)\}\,\rho(u)\,du\right| +\ \le\ \int_0^1 K_\lambda(p,u)\,|p-u|\,\rho(u)\,du. +\end{align*} +Taking the supremum over $p$ and $\varphi$ yields +\begin{equation} +\label{eq:epsLEB} +\sup_{\varphi\in\mathcal H}\ \sup_{p\in[0,1]}\big|\varphi(p)-(\mathsf T_\lambda\varphi)(p)\big|\ \le\ B_\lambda. +\end{equation} +Hence, using $|a-p|\le 1$, +\[ +\Big|\mathbb{E}\big[(a-p)\,\{\varphi(p)-(\mathsf T_\lambda\varphi)(p)\}\big]\Big| +\ \le\ B_\lambda. +\] + +\emph{Aligned main term.} +By Fubini/Tonelli (bounded integrands) and normalization $\int K_\lambda(p,u)\rho(u)\,du=1$, +\[ +\mathbb{E}[(a-p)\,(\mathsf T_\lambda\varphi)(p)] +=\int_0^1 \varphi(u)\,\mathbb{E}\big[(a-p)\,K_\lambda(p,u)\big]\ \rho(u)\,du, +\] +and since $|\varphi(u)|\le 1$, +\begin{equation} +\label{eq:main-aligned} +\big|\mathbb{E}[(a-p)\,(\mathsf T_\lambda\varphi)(p)]\big| +\ \le\ \int_0^1 \Big|\,\mathbb{E}\big[(a-p)\,K_\lambda(p,u)\big]\,\Big|\,\rho(u)\,du. +\end{equation} +For each $u$, using $|x+y|\le |x|+|y|$ and $|a-p|\le |a-u|+|p-u|$, together with $\phi_\delta(r)\ge |r|$, +\begin{align} +\label{eq:triangle-rev} +\Big|\,\mathbb{E}\big[(a-p)\,K_\lambda(p,u)\big]\,\Big| +&\le \mathbb{E}\big[\,|a-p|\,K_\lambda(p,u)\big]\notag\\ +&\le\ \mathbb{E}\big[\,\phi_\delta(a-u)\,K_\lambda(p,u)\big]\ +\ \mathbb{E}\big[\,|p-u|\,K_\lambda(p,u)\big]. +\end{align} +Integrating \eqref{eq:triangle-rev} against $\rho(u)\,du$ and applying Fubini, +\begin{align*} +\int_0^1 \Big|\,\mathbb{E}\big[(a-p)\,K_\lambda(p,u)\big]\,\Big|\,\rho(u)\,du +&\ \le\ \mathbb{E}\!\left[\int_0^1 K_\lambda(p,u)\,\phi_\delta(a-u)\,\rho(u)\,du\right]\\ +&\quad\ +\ \sup_{p}\int_0^1 |p-u|\,K_\lambda(p,u)\,\rho(u)\,du, +\end{align*} +i.e., +\begin{equation} +\label{eq:aligned-bdd} +\big|\mathbb{E}[(a-p)\,(\mathsf T_\lambda\varphi)(p)]\big| +\ \le\ \mathcal{H}_{\lambda,\delta}(f)\ +\ B_\lambda. +\end{equation} + +\emph{Conclusion.} +Combining \eqref{eq:epsLEB}, \eqref{eq:aligned-bdd} with \eqref{eq:decomp-rev} and taking the supremum over $\varphi\in\mathcal H$ gives +$\mathrm{smCE}(f)\le \mathcal{H}_{\lambda,\delta}(f)+2B_\lambda$. + +\medskip +\noindent\emph{Explicit bounds for $B_\lambda$.} +By definition, +\[ +B_\lambda=\sup_{p\in[0,1]}\ \frac{\int_0^1 |p-u|\,k_\lambda(p-u)\,\rho(u)\,du}{\int_0^1 k_\lambda(p-v)\,\rho(v)\,dv}. +\] +Using $\rho(u)\le \rho_{\max}$ in the numerator and $\rho(v)\ge \rho_{\min}$ in the denominator, and changing variables $t=p-u$ or $t=p-v$, we obtain for all $p\in[0,1]$: +\[ +B_\lambda +\ \le\ \kappa\cdot +\frac{\int_{\mathbb{R}} |t|\,e^{-\lambda t^2}\,dt} + {\int_{p-1}^{p} e^{-\lambda t^2}\,dt}. +\] +Since $p\in[0,1]$, the denominator integrates over a length-$1$ interval contained in $[-1,1]$; by symmetry and unimodality of $t\mapsto e^{-\lambda t^2}$, the minimum over such intervals is attained at an endpoint, e.g.\ $[0,1]$. Hence +\[ +B_\lambda +\ \le\ \kappa\cdot +\frac{\int_{\mathbb{R}} |t|\,e^{-\lambda t^2}\,dt} + {\int_{0}^{1} e^{-\lambda t^2}\,dt} +\ =\ \kappa\cdot +\frac{\tfrac{1}{\lambda}} + {\tfrac{\sqrt{\pi}}{2\sqrt{\lambda}}\ \mathrm{erf}(\sqrt{\lambda})} +\ =\ \frac{2\,\kappa}{\sqrt{\pi}}\cdot\frac{1}{\sqrt{\lambda}\ \mathrm{erf}(\sqrt{\lambda})}. +\] +Since $|p-u|\le 1$ and $\int K_\lambda\rho=1$, we also have $B_\lambda\le 1$. +For $\lambda\ge 1$, $\mathrm{erf}(\sqrt{\lambda})\ge \mathrm{erf}(1)$, yielding \eqref{eq:Blambda-simplified}. +\hfill$\square$ + +\paragraph{Interpretation and guidance.} +The guarantee \eqref{eq:master-bound-rev} decomposes into a \emph{model-dependent} term $\mathcal{H}_{\lambda,\delta}(f)$ and a \emph{design-only} kernel bias $B_\lambda$, the average \emph{soft-bin radius} around $p$. The Charbonnier envelope obeys $\phi_\delta(r)\ge |r|$, so replacing $|a-u|$ with $\phi_\delta(a-u)$ never weakens control of smCE and yields smooth gradients near $r=0$. For Gaussian kernels, $B_\lambda=\mathcal{O}(\kappa/\sqrt{\lambda})$ as in \eqref{eq:Blambda-simplified}, so increasing $\lambda$ monotonically tightens the bound; the cap $B_\lambda\le 1$ ensures uniform validity for all $\lambda>0$. +(Discrete soft-binned implementations—via Riemann-sum quadrature of the $u$-integral—inherit the same inequality up to a standard design-only quadrature error that vanishes as the grid is refined.) + + + + +\subsection{Charbonnier--SoftECE vs.\ NLL} +\label{subsec:Charbonnier-vs-nll} + +We compare negative log-likelihood (NLL) with Charbonnier--SoftECE within the SMART family +$T(x)=h(m(x))$ that scales by the \emph{margin} +$m(x):=z_{(1)}(x)-z_{(2)}(x)\in\R_{\ge 0}$. +Throughout, assume +(i) $T(x)\in[T_{\min},T_{\max}]$ with $00$, define the local scaling +\begin{align*} +T_s(x) &:= \begin{cases}T(x)/s,& x\in A,\\ T(x),& x\notin A,\end{cases} \\ +t_s(x) &:= \frac{z(x)}{T_s(x)} = \begin{cases}s\,t(x),& x\in A,\\ t(x),& x\notin A,\end{cases} \\ +q^{(s)} &:= \softmax(t_s), \quad p_s := q^{(s)}_{M}. +\end{align*} +Because uniform multiplication by $s>0$ preserves coordinate ordering, $M$ is unchanged for all $s>0$; (iii) rules out measure-zero ties at the boundary. + +\begin{lemma}[Directional derivatives under local margin-dependent scaling] +\label{lem:dir-derivatives-nll} +Let $L_{\mathrm{nll}}(h):=\mathbb{E}[-\log q_Y(X)]$. +For any $C^1$ probe $\psi:[0,1]\to\R$ with $\mathrm{Lip}(\psi)\le 1$ and $\|\psi\|_\infty\le 1$, +the Gâteaux derivatives at $s=1$ exist and +\begin{align} +\left.\frac{d}{ds}L_{\mathrm{nll}}(h_s)\right|_{s=1} +&=\ \mathbb{E}\!\big[\mathbf{1}_A\,(\langle t\rangle_q - t_Y)\big], +\label{eq:dir-NLL}\\[-0.3em] +\left.\frac{d}{ds}\,\mathbb{E}\big[(Y^{\top}-p_s)\psi(p_s)\big]\right|_{s=1} +&=\ \mathbb{E}\!\Big[\mathbf{1}_A\;p\,\big(t_M-\langle t\rangle_q\big)\, +\big(\psi'(p)\,(r_X-p)-\psi(p)\big)\Big], +\label{eq:dir-calib-nll} +\end{align} +where $\langle t\rangle_q:=\sum_k q_k\,t_k$ and $r_X:=r_X(X)$. +\end{lemma} + +\begin{proof} +On $A$, $\partial_s q_k^{(s)}=q_k^{(s)}(t_k-\langle t\rangle_{q^{(s)}})$, hence +$\partial_s(-\log q_Y^{(s)})=\langle t\rangle_{q^{(s)}}-t_Y$. Outside $A$ the derivative vanishes. +Dominated convergence applies since $\big|\partial_s(-\log q_Y^{(s)})\big|\le 2\|t\|_\infty$ and $\mathbb{E}\|t\|_\infty\le \mathbb{E}\|z\|_\infty/T_{\min}<\infty$, yielding \eqref{eq:dir-NLL}. +For $F_\psi(s):=\mathbb{E}[(Y^{\top}-p_s)\psi(p_s)]$, with $M$ fixed, +$\partial_s p_s=\partial_s q_M^{(s)}=q_M^{(s)}(t_M-\langle t\rangle_{q^{(s)}})=p_s(t_M-\langle t\rangle_{q^{(s)}})$. +Thus +\[ +\partial_s\big((Y^{\top}-p_s)\psi(p_s)\big) +=\big(-\psi(p_s)+(Y^{\top}-p_s)\psi'(p_s)\big)\,\partial_s p_s. +\] +Conditioning on $X$ replaces $Y^{\top}$ by $r_X(X)$, whence \eqref{eq:dir-calib-nll} at $s=1$ after integration; dominated convergence holds because $p\,|t_M-\langle t\rangle_q|\le 2\|t\|_\infty$ and $|\psi'|\le 1$, $|\psi|\le 1$. +\end{proof} + +\begin{lemma}[Margin lower bound for the top-logit advantage] +\label{lem:margin-advantage-nll} +On $\{M=\arg\max t\}$, +\begin{equation} +t_M-\langle t\rangle_q \ \ge\ (1-p)\,\big(t_M-t_{(2)}\big) +\ =\ (1-p)\,\frac{m}{T}. +\label{eq:adv-lb-margin-nll} +\end{equation} +\end{lemma} + +\begin{proof} +$\langle t\rangle_q=p\,t_M+\sum_{j\neq M}q_j t_j\le p\,t_M+(1-p)\,t_{(2)}$; rearrange. +\end{proof} + +\paragraph{A correct NLL directional upper bound (multi-class).} +Define the \emph{runner-up gap} $g(x):=t_{(1)}(x)-t_{(2)}(x)=m(x)/T(x)\ge 0$ +and the \emph{non-top spread} $\Delta(x):=t_{(2)}(x)-t_{(K)}(x)\ge 0$. +For any $x$ with predicted index $M$ and confidence $p=q_M(x)$, +\begin{equation} +\label{eq:NLL-upper-correct} +\mathbb{E}\!\left[\langle t\rangle_q-t_Y\mid X=x\right] +\ \le\ \big(p-r_X(x)\big)\,g(x)\ +\ \big(1-r_X(x)\big)\,\Delta(x). +\end{equation} +In particular, for binary classification ($K=2$) one has $\Delta\equiv 0$ and \eqref{eq:NLL-upper-correct} reduces to +$\mathbb{E}[\langle t\rangle_q-t_Y\mid X]\ =\ (p-r_X)\,g$ (exact). + +\begin{proof}[Derivation of \eqref{eq:NLL-upper-correct}] +With $\eta_k(x):=\mathbb{P}(Y{=}k\mid X{=}x)$, +\[ +\mathbb{E}\!\left[\langle t\rangle_q-t_Y\mid X=x\right] +=\sum_k (q_k-\eta_k)\,t_k +=(p-r_X)\,(t_M-t_{(2)})+\sum_{j\neq M}(q_j-\eta_j)\,(t_j-t_{(2)}). +\] +Since $t_j-t_{(2)}\le 0$ and $\sum_{j\neq M}(\eta_j-q_j)_+\le \sum_{j\neq M}\eta_j=1-r_X$, the last sum is $\le (1-r_X)\,(t_{(2)}-t_{(K)})=(1-r_X)\,\Delta$. +\end{proof} + +\paragraph{Consequences and a mild spread control.} +On a slice $A=\{m\in G\}$, assume the empirically checkable \emph{spread control} +\begin{equation} +\label{eq:spread-control-nll} +\Delta(x)\ \le\ \Delta_G\ <\ \infty\qquad \text{for all }x\in A. +\end{equation} +Then, combining \eqref{eq:NLL-upper-correct} with Lemma~\ref{lem:dir-derivatives-nll}, +\begin{align} +\left.\frac{d}{ds}L_{\mathrm{nll}}(h_s)\right|_{s=1} +&=\mathbb{E}\big[\mathbf{1}_A(\langle t\rangle_q-t_Y)\big] +\ \le\ \mathbb{E}\!\big[\mathbf{1}_A \,(p-r_X)\,g\big]\ +\ \Delta_G\,\mu_A, +\label{eq:NLL-signed-with-spread} +\end{align} +where $\mu_A:=\mathbb{P}\{X\in A\}$. +In the binary case $\Delta\equiv 0$ and \eqref{eq:NLL-signed-with-spread} holds with equality. + +\paragraph{Two-slice \emph{mismatch} under mild, empirically observed heterogeneity.} +We next give conditions under which a single SMART-feasible local move reduces NLL yet \emph{increases} smCE. + +\emph{Assumptions (empirically checkable).} +Fix a \emph{compact} margin slice $G\subset[m_{\min},m_{\max}]$ and set $A:=\{x:\ m(x)\in G\}$. +Let $\gamma_{\min}:=\inf_{x\in A}\frac{m(x)}{T(x)}$ and $\gamma_{\max}:=\sup_{x\in A}\frac{m(x)}{T(x)}$ (finite and positive by $G$ compact and $T\in[T_{\min},T_{\max}]$). +Assume there exist disjoint compact intervals $J_{\mathrm{U}},J_{\mathrm{O}}\subset(p_0,1)$ with gap $\Delta>0$ and constants $\rho_{\mathrm{U}},\rho_{\mathrm{O}}>0$ such that +\[ +r(p)-p\ \ge\ \rho_{\mathrm{U}}\quad\text{for }p\in J_{\mathrm{U}},\qquad +r(p)-p\ \le\ -\,\rho_{\mathrm{O}}\quad\text{for }p\in J_{\mathrm{O}}. +\] +Write +$\mu_{\mathrm{U}}:=\mathbb{P}\{\,p\in J_{\mathrm{U}},\ x\in A\,\}$, +$\mu_{\mathrm{O}}:=\mathbb{P}\{\,p\in J_{\mathrm{O}},\ x\in A\,\}$, +$\mu_{\mathrm{gap}}:=\mathbb{P}\{\,x\in A,\ p\notin J_{\mathrm{U}}\cup J_{\mathrm{O}}\,\}$, +and assume additionally: +\begin{itemize} +\item[(a)] (\emph{bounded conditional density of $p$ on $A$}) the conditional distribution of $p$ given $X\in A$ has a density $f_{p|A}$ on $(0,1)$ with $\|f_{p|A}\|_\infty\le D_G<\infty$. In particular, for any interval $I\subset(0,1)$, $\mathbb{P}\{p\in I,\,X\in A\}\le D_G\,|I|$. +\item[(b)] (\emph{slice-bounded advantage}) there exists $C_G<\infty$ with +$p(x)\big(t_M(x)-\langle t(x)\rangle_{q(x)}\big)\le C_G$ for $x\in A$ (e.g., it holds with $C_G:=2\operatorname*{ess\,sup}_{x\in A}\|t(x)\|_\infty$ whenever $t$ is essentially bounded on $A$). +\item[(c)] the spread control \eqref{eq:spread-control-nll} holds on $A$ with constant $\Delta_G$. +\end{itemize} + +\begin{proposition}[Two-slice mismatch: NLL $\downarrow$ but smCE $\uparrow$ (multi-class)] +\label{prop:two-slice-mismatch-nll} +Consider the \emph{sharpening} direction $s\uparrow 1$ applied on the SMART-feasible set $A=\{m\in G\}$. +If +\begin{equation} +\label{eq:NLL-mismatch-condition} +\rho_{\mathrm{U}}\,\gamma_{\min}\,\mu_{\mathrm{U}} +\ >\ \gamma_{\max}\,\big(\rho_{\mathrm{O}}\,\mu_{\mathrm{O}}+\mu_{\mathrm{gap}}\big)\ +\ \Delta_G\,\mu_A, +\end{equation} +then $\left.\tfrac{d}{ds}L_{\mathrm{nll}}(h_s)\right|_{s=1}<0$ (NLL strictly decreases). +Moreover, for any $c\in(0,\min\{1,\Delta\})$ there exists a $1$-Lipschitz probe $\psi$ with +$\psi\equiv 0$ on $J_{\mathrm{U}}$, $\psi\equiv -c$ on $J_{\mathrm{O}}$, and with transitions confined to a band whose $A$-mass is at most $\varepsilon>0$, such that +\begin{equation} +\label{eq:calib-mismatch-deriv-nll} +\left.\frac{d}{ds}\mathbb{E}[(Y^{\top}-p_s)\psi(p_s)]\right|_{s=1} +\ \ge\ c\,\underline{p}_{\mathrm{O}}\,(1-\overline{p}_{\mathrm{O}})\,\gamma_{\min}\,\mu_{\mathrm{O}} +\ -\ (1+c)\,C_G\,\varepsilon, +\end{equation} +where $\underline{p}_{\mathrm{O}}:=\inf J_{\mathrm{O}}$ and $\overline{p}_{\mathrm{O}}:=\sup J_{\mathrm{O}}$. +Choosing $\varepsilon<\tfrac{c\,\underline{p}_{\mathrm{O}}(1-\overline{p}_{\mathrm{O}})\gamma_{\min}}{(1+c)\,C_G}\,\mu_{\mathrm{O}}$ makes the right-hand side strictly positive. Because on $J_{\mathrm{O}}$ one has $(r(p)-p)\psi(p)\ge c\,\rho_{\mathrm{O}}$ while $\psi\equiv 0$ on $J_{\mathrm{U}}$, the signed functional at $s=1$ obeys +\begin{equation} +\label{eq:baseline-positive-nll} +\mathbb{E}[(Y^{\top}-p)\psi(p)] +\ =\ \mathbb{E}[(r(p)-p)\psi(p)] +\ \ge\ c\,\rho_{\mathrm{O}}\,\mu_{\mathrm{O}}\ -\ c\,\varepsilon\ >\ 0, +\end{equation} +so a positive derivative implies a strict increase of its absolute value. Hence smCE strictly increases along $s\uparrow 1$. +\end{proposition} + +\begin{proof} +By \eqref{eq:NLL-signed-with-spread} and splitting $A$ into the three regions, +\begin{align*} +\left.\frac{d}{ds}L_{\mathrm{nll}}(h_s)\right|_{s=1} &\le \mathbb{E}\!\big[\mathbf{1}_{A\cap\{p\in J_{\mathrm{U}}\}}(p-r_X)g\big] \\ +&\quad + \mathbb{E}\!\big[\mathbf{1}_{A\cap\{p\in J_{\mathrm{O}}\}}(p-r_X)g\big] \\ +&\quad + \mathbb{E}\!\big[\mathbf{1}_{A\cap\{p\notin J_{\mathrm{U}}\cup J_{\mathrm{O}}\}}(p-r_X)g\big] \\ +&\quad + \Delta_G\,\mu_A. +\end{align*} +On $A\cap\{p\in J_{\mathrm{U}}\}$, $g\ge \gamma_{\min}$ and +$\mathbb{E}[p-r_X\mid p]=p-r(p)\le -\,\rho_{\mathrm{U}}$, +hence the contribution is $\le -\,\rho_{\mathrm{U}}\gamma_{\min}\mu_{\mathrm{U}}$. +On $A\cap\{p\in J_{\mathrm{O}}\}$, $g\le \gamma_{\max}$ and $\mathbb{E}[p-r_X\mid p]\ge \rho_{\mathrm{O}}$, giving at most $\gamma_{\max}\rho_{\mathrm{O}}\mu_{\mathrm{O}}$. +On the gap region, $|p-r_X|\le 1$ and $g\le \gamma_{\max}$, giving at most $\gamma_{\max}\,\mu_{\mathrm{gap}}$. +This yields strict negativity under \eqref{eq:NLL-mismatch-condition}. +For the probe, on $J_{\mathrm{O}}$ we have $\psi'(p)=0$ and $-\psi(p)=c$, +so by \eqref{eq:dir-calib-nll} and Lemma~\ref{lem:margin-advantage-nll}, +\[ +\left.\frac{d}{ds}\mathbb{E}[(Y^{\top}-p_s)\psi(p_s)]\right|_{s=1,\,p\in J_{\mathrm{O}}} +\ \ge\ c\,\underline{p}_{\mathrm{O}}\,(1-\overline{p}_{\mathrm{O}})\,\gamma_{\min}. +\] +On $J_{\mathrm{U}}$ the contribution is $0$ since $\psi\equiv 0$. +On the transition band (of $A$-mass $\varepsilon$), $|\psi'|\le 1$ and $|\psi|\le c$, +hence $|\psi'(p)(r_X-p)-\psi(p)|\le (1+c)$ while $p(t_M-\langle t\rangle_q)\le C_G$ on $A$ by (b). +Thus the transition contribution is at most $(1+c)\,C_G\,\varepsilon$ in magnitude, giving \eqref{eq:calib-mismatch-deriv-nll}. +Finally, \eqref{eq:baseline-positive-nll} holds since $\psi$ depends only on $p$ and $\mathbb{E}[Y^{\top}-p\mid p]=r(p)-p$. +By (a), we can realize the $1$-Lipschitz $\psi$ with linear ramps of total width at most $2c$, whence $\varepsilon\le 2D_G c$; shrinking $c$ if needed makes the stated choice of $\varepsilon$ feasible. +\end{proof} + +\begin{lemma}[Small-$s$ realization for the mismatch] +\label{lem:small-s-nll} +Under Proposition~\ref{prop:two-slice-mismatch-nll}, there exists $s^\uparrow>1$ arbitrarily close to $1$ with +\[ +L_{\mathrm{nll}}(h_{s^\uparrow})\ <\ L_{\mathrm{nll}}(h) +\qquad\text{and}\qquad +\mathrm{smCE}(f_{h_{s^\uparrow}})\ >\ \mathrm{smCE}(f_{h}). +\] +\end{lemma} + +\begin{proof} +$L_{\mathrm{nll}}(h_s)$ is $C^1$ at $s=1$ by Lemma~\ref{lem:dir-derivatives-nll}, with strictly negative derivative; hence $L_{\mathrm{nll}}(h_{s^\uparrow})1$ sufficiently close to $1$. +For smCE, fix the $\psi$ from Proposition~\ref{prop:two-slice-mismatch-nll}; then $F_\psi(s):=\mathbb{E}[(Y^{\top}-p_s)\psi(p_s)]$ is $C^1$ with $F_\psi(1)>0$ and $F_\psi'(1)>0$, so $|F_\psi(s^\uparrow)|>|F_\psi(1)|$ for all $s^\uparrow>1$ close enough to $1$. Since $\mathrm{smCE}(f_{h_s})\ge |F_\psi(s)|$, it follows that $\mathrm{smCE}(f_{h_{s^\uparrow}})>\mathrm{smCE}(f_{h})$. +\end{proof} + +\paragraph{Takeaway.} +Along SMART-feasible local scalings of the temperature map $T(x)=h(m(x))$, Charbonnier--SoftECE continues to control smCE via \eqref{eq:smce-Charbonnier-master-again-nll}, whereas NLL can be \emph{locally improved} (decreased) while smCE \emph{deteriorates} (increases) under mild, empirically checkable heterogeneity of confidence slices (Proposition~\ref{prop:two-slice-mismatch-nll}). The NLL directional formula is exact in binary classification; in multi-class settings the same conclusion holds under a weak spread control on non-top logits. + + + +\section{The Use of Large Language Models} + +During the preparation of this work, we utilized a Large Language Model (LLM) to assist with editorial refinement of the manuscript. The model's application was limited exclusively to improving textual quality and presentation, not for generating substantive research content. The LLM's contributions included: + +\begin{itemize} + \item Enhancing sentence structure and paragraph organization to improve clarity, brevity, and scholarly tone. + \item Identifying and correcting errors in grammar, spelling, and punctuation. + \item Strengthening coherence and smoothing transitions throughout the text. +\end{itemize} + + +\section{Runtime Efficiency} +To verify the time efficiency of our method, we compare the runtime with baseline methods. The result is reported in Table~\ref{tab:imagenet_resnet_runtime}. TS optimizes a single scalar temperature via a few gradient steps or closed‐form updates, then applies this same factor to every logit, resulting in a negligible overhead (2.42\,s). SMART yields a small per‐sample inference cost and hence a modest total runtime (23.03\,s). Logits are input into PTS's small neural network for each sample to predict a bespoke temperature, incurring a larger computational cost than SMART. CTS is the most expensive at more than 1 hour with the highest variance, as it conducts an exhaustive grid search for 5 epochs over a dense temperature grid (e.g. 0.1 -10) for each of the 1 000 classes, leading to $O(C\times G\times N)$ evaluations (classes $\times$ grid points $\times$ samples). The spline‐based calibrator precomputes a monotonic mapping on the validation set and then applies a fast piecewise‐linear transform at test time, yielding intermediate overhead. These differences illustrate the trade‐off between expressive power and efficiency: TS is almost instantaneous, SMART adds only a small network‐forward cost per sample, PTS trades per‐sample flexibility for moderate cost, and CTS’s brute‐force search becomes prohibitive at scale. + +\begin{table*}[ht] +\caption{\textbf{Average Runtime (s) on ImageNet} over 10 runs on a ResNet-50 model.} +\footnotesize +\label{tab:imagenet_resnet_runtime} +\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}}lccccc@{}} +\toprule +\textbf{Method} & TS & Spline & PTS & CTS & SMART \\ +\midrule +Runtime (s) & +$2.42 \pm 0.1$ & +$28.51 \pm 0.9$ & +$1050.44 \pm 37.8$ & +$5457.55 \pm 125.5$ & +$23.03 \pm 0.41$ \\ +\bottomrule +\end{tabular*} +% \vspace{-0.2in} +\end{table*} + + +% \section{margin Perspective on Calibration} +% \label{sec:logit_gap_perspective} + +% Traditional calibration analysis evaluates models from an overall perspective, potentially masking important sample-specific miscalibration patterns. By examining calibration behavior across the continuous spectrum of margin values, we uncover fundamental insights about how neural networks distribute confidence and why existing calibration methods often fall short. This analysis provides crucial validation for our theoretical framework established in Section~\ref{subsec:theoretical_evidence} and design choices made in Section~\ref{subsec:our_scaling}. + +% \begin{figure*}[ht] +% \centering +% \begin{subfigure}[b]{0.32\textwidth} +% \includegraphics[width=\textwidth]{figs/overall_reliability_imagenet_vit_b_16.pdf} +% \caption{Overall calibration for ImgNet} +% \label{fig:overall_imagenet_vit} +% \end{subfigure} +% \hfill +% \begin{subfigure}[b]{0.32\textwidth} +% \includegraphics[width=\textwidth]{figs/group_3_reliability_imagenet_vit_b_16.pdf} +% \caption{High margin samples} +% \label{fig:high_gap_imagenet_vit} +% \end{subfigure} +% \hfill +% \begin{subfigure}[b]{0.32\textwidth} +% \includegraphics[width=\textwidth]{figs/group_1_reliability_imagenet_vit_b_16.pdf} +% \caption{Low margin samples} +% \label{fig:low_gap_imagenet_vit} +% \end{subfigure} + +% \begin{subfigure}[b]{0.32\textwidth} +% \includegraphics[width=\textwidth]{figs/overall_reliability_ciafr100_resnet50.pdf} +% \caption{Overall calibration for Cifar-100} +% \label{fig:overall_cifar_resnet} +% \end{subfigure} +% \hfill +% \begin{subfigure}[b]{0.32\textwidth} +% \includegraphics[width=\textwidth]{figs/group_3_reliability_ciafr100_resnet50.pdf} +% \caption{High margin samples Cifar-100} +% \label{fig:high_gap_cifar_resnet} +% \end{subfigure} +% \hfill +% \begin{subfigure}[b]{0.32\textwidth} +% \includegraphics[width=\textwidth]{figs/logitsgap_vs_confidence_change.pdf} +% \caption{margin vs. confidence change} +% \label{fig:gap_vs_change} +% \end{subfigure} + +% \caption{\textbf{margin reveals hidden calibration patterns across the confidence spectrum.} Top row: ImageNet ViT-B/16 shows almost perfect but slightly under-confident overall calibration (a) but reveals systematic under-confidence in high margin samples (b) and almost well-calibrated but slightly under-confident low margin samples (c). Bottom row: CIFAR-100 ResNet-50 demonstrates that even with overall over-confidence (d), high margin samples remain under-confident (e). Panel (f) shows how different methods adjust confidence across margin values, with SMART providing targeted adjustments while TS and PTS show suboptimal patterns.} +% \label{fig:logit_gap_analysis} +% \end{figure*} + +% \paragraph{Revealing Hidden Calibration Heterogeneity Through margin Groupings} Our reliability diagram analysis unveils a crucial insight that traditional overall calibration metrics obscure: models exhibit fundamentally different calibration behaviors across margin groups. This heterogeneity validates our theoretical framework from Section~\ref{subsec:theoretical_evidence} while revealing why sample-specific calibration approaches are essential for achieving truly reliable uncertainty estimates. + +% Figure~\ref{fig:logit_gap_analysis} demonstrates this heterogeneity across different scenarios. For ImageNet with ViT-B/16, while overall calibration appears near-perfect (Figure~\ref{fig:overall_imagenet_vit}), decomposing by margin reveals distinct patterns: low gap samples achieve almost perfect but slightly under-confident calibration (Figure~\ref{fig:low_gap_imagenet_vit}), while high gap samples suffer from systematic miscalibration (Figure~\ref{fig:high_gap_imagenet_vit}). This pattern persists even under different overall calibration conditions, as shown in CIFAR-100 with ResNet-50 (Figures~\ref{fig:overall_cifar_resnet} and~\ref{fig:high_gap_cifar_resnet}), indicating that margin-based groupings reveal fundamental calibration characteristics that transcend dataset-specific or architecture-specific behaviors. + +% \paragraph{The Under-Confidence Paradox in High margin Samples} Perhaps the most counterintuitive finding emerges from examining high margin samples across our reliability diagrams. Despite representing "easy" classifications with substantial margins between top predictions, these samples consistently exhibit under-confidence rather than the expected over-confidence. In Figure~\ref{fig:high_gap_imagenet_vit}, high gap samples from ImageNet ViT-B/16 show systematic under-confidence, with predicted confidence consistently lower than empirical accuracy. This pattern holds even when overall model behavior differs dramatically, as demonstrated in Figure~\ref{fig:high_gap_cifar_resnet} where CIFAR-100 ResNet-50 maintains under-confidence in high gap samples despite overall over-confidence. + +% This paradox fundamentally challenges conventional intuitions about neural network confidence and validates our theoretical analysis from Section~\ref{subsec:theoretical_evidence}. High margin samples, despite being far from decision boundaries, require confidence enhancement rather than reduction. This phenomenon suggests that modern neural networks, when trained with standard cross-entropy objectives, develop conservative confidence estimates for samples with clear classification margins. The consistency of this pattern across different architectures and datasets indicates that this represents a fundamental characteristic of neural network training dynamics rather than an artifact of specific experimental conditions. + +% \paragraph{Method-Specific Failures from the margin Perspective} The confidence adjustment patterns revealed in Figure~\ref{fig:gap_vs_change} expose fundamental limitations in existing calibration approaches when viewed through the margin lens. Temperature Scaling's uniform adjustment completely ignores the heterogeneous calibration needs we identified, applying identical modifications regardless of sample characteristics. This uniform approach proves particularly problematic because it fails to provide the targeted adjustments different gap groups require. + +% More critically, PTS demonstrates the exact problems we highlighted in Section~\ref{subsec:our_scaling} regarding high-dimensional parameterization introducing noise. The method makes substantial, often counterproductive adjustments to low margin samples that, in cases like ImageNet ViT-B/16, already achieve good calibration and require minimal intervention. This unnecessary manipulation exemplifies how "the increased dimensionality introduces substantial noise for precise temperature parameterization," leading to degraded performance where careful preservation of existing calibration would be more appropriate. + +% In contrast, SMART's learned adjustment pattern directly validates our design principles. For ImageNet ViT-B/16, SMART provides minimal adjustments to low gap samples that are already well-calibrated, while delivering targeted confidence increases to high gap samples suffering from under-confidence. This adaptive behavior emerges naturally from our lightweight scalar-to-scalar mapping, demonstrating how principled architectural choices translate into appropriate calibration strategies without requiring explicit programming of these patterns. + +% \paragraph{Theoretical Validation and Design Implications} These empirical findings provide compelling validation for our theoretical framework established in Propositions~\ref{prop:unbound} and~\ref{prop:g-bound}. The systematic under-confidence in high margin samples directly supports our argument that temperature values should be bounded by margins rather than determined by global statistics or high-dimensional features. The fact that SMART automatically learns to provide confidence enhancement for high gap samples while preserving the calibration of well-adjusted low gap samples demonstrates the practical effectiveness of our theoretical bounds. + + + + + + + +\section{The proposed SMART framework} +\label{Appendix:SMART_framework} + +This section presents the detailed algorithmic implementation of SMART, providing a step-by-step procedure for applying margin-based temperature scaling with soft-binned ECE optimization. + +% \begin{algorithm}[h] +% \caption{The proposed SMART framework} +% \label{alg:SMART} +% \begin{algorithmic}[1] +% \State \textbf{Input:} Validation logits and labels $\{\mathbf{z}_i, y_i\}_{i=1}^{N}$, temperature parameterization network $h_\phi(\cdot)$. + +% \State Compute Margin $m_i = \mathbf{z}_{i,\text{max}} - \mathbf{z}_{i,\text{2nd}}$ for each sample $i$. +% \State Normalize: $\hat{m}_i = \frac{m_i - \mu_m}{\sigma_m}$ + +% \For {each epoch} %{$e = 1$ to $\text{epochs}$} +% \State Compute sample-wise temperature $T = h_\phi(\hat{m}_i)$ for each sample $i$. +% \State Compute calibrated logits $\hat{\mathbf{z}}_i = \mathbf{z}_i / T$ for each sample $i$. +% \State Compute loss $\mathcal{L}_i = \text{CharbonnierSmoothedSoftECE}(\hat{\mathbf{z}}_i, y_i)$ for each sample $i$. +% \State Update parameters $\phi$ using gradient descent from the overall loss $\mathcal{L}_{calib}=\sum_{i=1}^N\mathcal{L}_i$ +% \EndFor + +% \end{algorithmic} +% \end{algorithm} + +\begin{algorithm}[ht] +\caption{SMART: Sample Margin-Aware Recalibration of Temperature} +\label{alg:smart} +\begin{algorithmic}[1] +\STATE \textbf{Input:} Validation logits and labels $\{\mathbf{z}_i, y_i\}_{i=1}^{N_{\text{val}}}$, temperature network $h_\phi(\cdot)$ +\STATE Compute margins: $m_i = z_{i,\max} - z_{i,\text{2nd}}$ for each $i \in \{1, \ldots, N_{\text{val}}\}$ +\FOR{epoch $= 1, \ldots, N_{\text{epochs}}$} + \STATE Predict temperatures: $T_i = h_\phi(m_i)$ for each $i$ + \STATE Scale logits: $\tilde{\mathbf{z}}_i = \mathbf{z}_i / T_i$ for each $i$ + \STATE Compute loss: $\mathcal{L}_i = \text{CharbonnierSoftECE}(\tilde{\mathbf{z}}_i, y_i)$ for each $i$ (Equation~\ref{eq:charbonnier_softece}) + \STATE Update: $\phi \leftarrow \phi - \eta \nabla_\phi \sum_{i=1}^{N_{\text{val}}} \mathcal{L}_i$ via SGD +\ENDFOR +\STATE \textbf{Return:} Trained temperature network $h_\phi$ +\end{algorithmic} +\end{algorithm} + + +\section{Full Calibration Performance} + +Full calibration performance for Table \ref{table:comparison_ECE} is in Table \ref{table:comparison_ECE_full}. + +\label{Appendix:full_calibration_performance} +\begin{table*}[ht] +\caption{\textbf{Comparison of Post-Hoc Calibration Methods Using ECE (\%, $\downarrow$, 15 bins) Across Various Datasets and Models} (mean $\pm$ std across 5 seeds). The best-performing method for each dataset-model combination is in bold, and our method is highlighted.} +\centering +\tiny +\setlength{\tabcolsep}{2pt} +{\fontsize{5pt}{6pt}\selectfont +\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}cccccccccccc} +\toprule +\textbf{Dataset} & \textbf{Model} & \textbf{Vanilla} & \textbf{TS} & \textbf{PTS} & \textbf{CTS} & \textbf{Spline} & \textbf{GC} & \textbf{ProCal} & \textbf{FC} & \textbf{SMART (ours)} \\ +\midrule +\multirow{2}{*}{CIFAR-10} + & ResNet-50 & $4.34$ & $1.38 \pm 0.26$ & $1.10 \pm 0.21$ & $0.83 \pm 0.15$ & $1.52 \pm 0.03$ & $1.37 \pm 0.08$ & $4.17 \pm 0.12$ & $1.66 \pm 0.09$ & $\cellgray\mathbf{0.76 \pm 0.02}$ \\ + & Wide-ResNet & $3.24$ & $0.93 \pm 0.20$ & $0.90 \pm 0.19$ & $0.81 \pm 0.17$ & $1.74 \pm 0.01$ & $0.89 \pm 0.06$ & $2.81 \pm 0.11$ & $1.12 \pm 0.07$ & $\cellgray\mathbf{0.43 \pm 0.05}$ \\ +\midrule +\multirow{2}{*}{CIFAR-100} + & ResNet-50 & $17.53$ & $5.61 \pm 1.39$ & $1.96 \pm 0.48$ & $3.67 \pm 0.88$ & $3.48 \pm 0.00$ & $5.70 \pm 0.15$ & $9.71 \pm 0.18$ & $2.91 \pm 0.12$ & $\cellgray\mathbf{1.37 \pm 0.27}$ \\ + & Wide-ResNet & $15.34$ & $4.50 \pm 0.62$ & $1.96 \pm 0.27$ & $3.01 \pm 0.42$ & $3.76 \pm 0.00$ & $4.55 \pm 0.13$ & $9.44 \pm 0.16$ & $4.49 \pm 0.14$ & $\cellgray\mathbf{1.80 \pm 0.10}$ \\ +\midrule +\multirow{6}{*}{ImageNet-1K} + & ResNet-50 & $3.65$ & $2.17 \pm 0.03$ & $0.95 \pm 0.36$ & $2.17 \pm 0.78$ & $0.62 \pm 0.18$ & $2.44 \pm 0.12$ & $1.08 \pm 0.14$ & $1.71 \pm 0.08$ & $\cellgray\mathbf{0.52 \pm 0.12}$ \\ + & DenseNet-121 & $2.53$ & $1.85 \pm 0.04$ & $1.02 \pm 0.46$ & $1.86 \pm 0.81$ & $0.81 \pm 0.35$ & $2.20 \pm 0.25$ & $1.52 \pm 0.21$ & $1.35 \pm 0.29$ & $\cellgray\mathbf{0.57 \pm 0.03}$ \\ + & Wide-ResNet & $5.43$ & $2.89 \pm 0.11$ & $1.14 \pm 0.24$ & $3.27 \pm 0.69$ & $0.66 \pm 0.10$ & $3.66 \pm 0.16$ & $1.57 \pm 0.10$ & $1.62 \pm 0.09$ & $\cellgray\mathbf{0.52 \pm 0.07}$ \\ + & Swin-B & $5.05$ & $3.91 \pm 0.07$ & $1.05 \pm 0.05$ & $1.53 \pm 0.08$ & $0.88 \pm 0.14$ & $4.95 \pm 0.17$ & $1.00 \pm 0.15$ & $5.05 \pm 0.06$ & $\cellgray\mathbf{0.46 \pm 0.03}$ \\ + & ViT-B-16 & $5.62$ & $3.60 \pm 0.19$ & $1.23 \pm 0.29$ & $4.65 \pm 1.02$ & $0.91 \pm 0.31$ & $4.39 \pm 0.25$ & $0.97 \pm 0.30$ & $5.65 \pm 0.06$ & $\cellgray\mathbf{0.48 \pm 0.13}$ \\ + & ViT-B-32 & $6.39$ & $3.93 \pm 0.02$ & $1.27 \pm 0.97$ & $2.12 \pm 1.59$ & $0.81 \pm 0.12$ & $4.67 \pm 0.13$ & $0.88 \pm 0.32$ & $6.39 \pm 0.06$ & $\mathbf{\cellgray{0.71 \pm 0.18}}$ \\ +\midrule +\multirow{5}{*}{ImageNet-C} + & ResNet-50 & $13.82$ & $1.97 \pm 0.02$ & $1.12 \pm 0.13$ & $1.69 \pm 0.20$ & $5.61 \pm 0.15$ & $2.69 \pm 0.11$ & $5.79 \pm 0.19$ & $2.51 \pm 0.13$ & $\cellgray\mathbf{0.62 \pm 0.03}$ \\ + & DenseNet-121 & $12.57$ & $1.58 \pm 0.00$ & $1.19 \pm 0.15$ & $1.44 \pm 0.19$ & $5.18 \pm 0.13$ & $2.01 \pm 0.09$ & $9.88 \pm 0.24$ & $9.44 \pm 0.31$ & $\cellgray\mathbf{0.63 \pm 0.01}$ \\ + & Swin-B & $12.03$ & $5.82 \pm 0.05$ & $1.53 \pm 0.00$ & $3.05 \pm 0.01$ & $2.58 \pm 0.21$ & $6.92 \pm 0.18$ & $2.53 \pm 0.12$ & $5.18 \pm 0.17$ & $\cellgray\mathbf{1.23 \pm 0.04}$ \\ + & ViT-B-16 & $8.28$ & $5.24 \pm 0.01$ & $1.27 \pm 0.05$ & $2.76 \pm 0.10$ & $1.71 \pm 0.22$ & $5.95 \pm 0.15$ & $1.96 \pm 0.14$ & $5.37 \pm 0.20$ & $\cellgray\mathbf{1.06 \pm 0.02}$ \\ + & ViT-B-32 & $7.69$ & $5.10 \pm 0.00$ & $1.07 \pm 0.08$ & $2.97 \pm 0.24$ & $1.43 \pm 0.24$ & $6.40 \pm 0.16$ & $1.55 \pm 0.11$ & $5.50 \pm 0.18$ & $\cellgray\mathbf{0.96 \pm 0.01}$ \\ +\midrule +\multirow{6}{*}{ImageNet-LT} + & ResNet-50 & $3.63$ & $2.01 \pm 0.02$ & $0.99 \pm 0.32$ & $2.17 \pm 0.68$ & $0.56 \pm 0.10$ & $2.20 \pm 0.17$ & $1.12 \pm 0.20$ & $1.80 \pm 0.23$ & $\cellgray\mathbf{0.56 \pm 0.04}$ \\ + & DenseNet-121 & $2.50$ & $1.80 \pm 0.06$ & $1.20 \pm 0.26$ & $1.88 \pm 0.41$ & $\mathbf{0.79 \pm 0.07}$ & $2.05 \pm 0.11$ & $1.79 \pm 0.09$ & $1.76 \pm 0.50$ & $\cellgray{0.81 \pm 0.01}$ \\ + & Wide-ResNet & $5.40$ & $2.99 \pm 0.05$ & $1.21 \pm 0.77$ & $2.87 \pm 1.79$ & $0.81 \pm 0.24$ & $3.59 \pm 0.18$ & $1.28 \pm 0.06$ & $1.68 \pm 0.10$ & $\cellgray\mathbf{0.53 \pm 0.02}$ \\ + & Swin-B & $4.69$ & $3.98 \pm 0.12$ & $1.21 \pm 0.45$ & $1.50 \pm 0.56$ & $0.79 \pm 0.17$ & $4.79 \pm 0.27$ & $0.95 \pm 0.16$ & $4.82 \pm 0.10$ & $\cellgray\mathbf{0.58 \pm 0.01}$ \\ + & ViT-B-16 & $5.58$ & $3.73 \pm 0.13$ & $1.14 \pm 0.47$ & $1.43 \pm 0.58$ & $0.66 \pm 0.05$ & $4.34 \pm 0.14$ & $0.77 \pm 0.14$ & $5.72 \pm 0.08$ & $\cellgray\mathbf{0.56 \pm 0.14}$ \\ + & ViT-B-32 & $6.28$ & $3.98 \pm 0.06$ & $1.35 \pm 0.41$ & $2.12 \pm 0.63$ & $0.72 \pm 0.23$ & $4.76 \pm 0.08$ & $0.83 \pm 0.12$ & $6.26 \pm 0.03$ & $\mathbf{\cellgray{0.60 \pm 0.11}}$ \\ +\midrule +\multirow{5}{*}{ImageNet-S} + & ResNet-50 & $22.32$ & $2.06 \pm 0.06$ & $1.69 \pm 0.27$ & $1.48 \pm 0.23$ & $9.76 \pm 0.22$ & $1.99 \pm 0.16$ & $9.52 \pm 0.31$ & $12.58 \pm 1.35$ & $\cellgray\mathbf{0.92 \pm 0.09}$ \\ + & DenseNet-121 & $20.13$ & $1.67 \pm 0.28$ & $1.93 \pm 0.19$ & $1.16 \pm 0.11$ & $9.20 \pm 0.32$ & $1.77 \pm 0.15$ & $12.93 \pm 0.23$ & $22.67 \pm 1.07$ & $\cellgray\mathbf{0.59 \pm 0.25}$ \\ + & Swin-B & $24.61$ & $6.50 \pm 0.05$ & $1.53 \pm 0.19$ & $3.62 \pm 0.45$ & $8.66 \pm 0.15$ & $6.92 \pm 0.35$ & $8.05 \pm 0.30$ & $1.70 \pm 0.06$ & $\cellgray\mathbf{1.26 \pm 0.05}$ \\ + & ViT-B-16 & $16.57$ & $5.75 \pm 0.08$ & $1.33 \pm 0.21$ & $2.84 \pm 0.43$ & $5.70 \pm 0.19$ & $6.36 \pm 0.29$ & $5.67 \pm 0.38$ & $1.93 \pm 0.18$ & $\cellgray\mathbf{0.98 \pm 0.08}$ \\ + & ViT-B-32 & $14.22$ & $4.99 \pm 0.15$ & $1.67 \pm 0.27$ & $3.25 \pm 0.50$ & $4.07 \pm 0.21$ & $6.23 \pm 0.16$ & $4.44 \pm 0.23$ & $1.56 \pm 0.09$ & $\cellgray\mathbf{0.87 \pm 0.18}$ \\ +\bottomrule +\end{tabular*} +} +\label{table:comparison_ECE_full} +% \vspace{-0.2in} +\end{table*} + +\section{Calibration Performance on Other Metrics} +\label{Appendix:Calibration_Performance_on_Other_Metrics} + + +\subsection{Accuracy Performance} + +\begin{table*}[ht] +\caption{\textbf{Comparison of Classification Accuracy (\%) Across Calibration Methods (Seed 1–5 Averaged).}} +\centering +\scriptsize +\setlength{\tabcolsep}{3pt} +\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}ccccccccc} +\toprule +\textbf{Dataset} & \textbf{Model} & \textbf{Vanilla} & \textbf{TS} & \textbf{PTS} & \textbf{CTS} & \textbf{Spline} & \textbf{SMART} \\ +\midrule +\multirow{2}{*}{CIFAR-10} + & ResNet-50 & 95.05\% & 95.05\% & 95.05\% & 94.88\% & 95.05\% & \cellgray{95.05\%} \\ + & Wide-ResNet & 96.13\% & 96.13\% & 96.13\% & 96.09\% & 96.13\% & \cellgray{96.13\%} \\ +\midrule +\multirow{2}{*}{CIFAR-100} + & ResNet-50 & 76.69\% & 76.69\% & 76.69\% & 76.38\% & 76.69\% & \cellgray{76.69\%} \\ + & Wide-ResNet & 79.29\% & 79.29\% & 79.29\% & 79.28\% & 79.29\% & \cellgray{79.29\%} \\ +\midrule +\multirow{6}{*}{ImageNet-1K} + & ResNet-50 & 76.16\% & 76.16\% & 76.16\% & 75.32\% & 76.17\% & \cellgray{76.16\%} \\ + & DenseNet-121 & 74.44\% & 74.44\% & 74.44\% & 73.71\% & 74.43\% & \cellgray{74.44\%} \\ + & Wide-ResNet & 78.46\% & 78.46\% & 78.46\% & 77.70\% & 78.46\% & \cellgray{78.46\%} \\ + & Swin-B & 83.17\% & 83.17\% & 83.17\% & 82.80\% & 83.17\% & \cellgray{83.17\%} \\ + & ViT-B-16 & 81.12\% & 81.12\% & 81.12\% & 79.64\% & 80.86\% & \cellgray{81.12\%} \\ + & ViT-B-32 & 75.95\% & 75.95\% & 75.95\% & 75.14\% & 75.94\% & \cellgray{75.95\%} \\ +\midrule +\multirow{5}{*}{ImageNet-C} + & ResNet-50 & 19.16\% & 19.16\% & 19.16\% & 19.34\% & 19.16\% & \cellgray{19.16\%} \\ + & DenseNet-121 & 21.25\% & 21.25\% & 21.25\% & 21.36\% & 21.25\% & \cellgray{21.25\%} \\ + & Swin-B & 40.83\% & 40.83\% & 40.83\% & 41.22\% & 40.83\% & \cellgray{40.83\%} \\ + & ViT-B-16 & 41.07\% & 41.07\% & 41.07\% & 41.28\% & 41.07\% & \cellgray{41.07\%} \\ + & ViT-B-32 & 37.82\% & 37.82\% & 24.56\% & 37.96\% & 37.85\% & \cellgray{37.82\%} \\ +\midrule +\multirow{6}{*}{ImageNet-LT} + & ResNet-50 & 76.04\% & 76.04\% & 76.04\% & 75.43\% & 76.04\% & \cellgray{76.04\%} \\ + & DenseNet-121 & 74.34\% & 74.34\% & 74.34\% & 73.88\% & 74.40\% & \cellgray{74.34\%} \\ + & Wide-ResNet & 78.39\% & 78.39\% & 78.39\% & 77.67\% & 78.40\% & \cellgray{78.39\%} \\ + & Swin-B & 82.95\% & 82.95\% & 82.95\% & 82.55\% & 82.94\% & \cellgray{82.95\%} \\ + & ViT-B-16 & 80.95\% & 80.95\% & 80.95\% & 80.58\% & 81.00\% & \cellgray{80.95\%} \\ + & ViT-B-32 & 75.89\% & 75.89\% & 75.89\% & 75.14\% & 75.92\% & \cellgray{75.89\%} \\ +\midrule +\multirow{5}{*}{ImageNet-S} + & ResNet-50 & 24.09\% & 24.09\% & 24.09\% & 23.88\% & 24.09\% & \cellgray{24.09\%} \\ + & DenseNet-121 & 24.30\% & 24.30\% & 24.30\% & 23.87\% & 31.55\% & \cellgray{24.30\%} \\ + & Swin-B & 31.54\% & 31.54\% & 31.54\% & 31.65\% & 31.55\% & \cellgray{31.54\%} \\ + & ViT-B-16 & 29.37\% & 29.37\% & 29.37\% & 29.51\% & 29.39\% & \cellgray{29.37\%} \\ + & ViT-B-32 & 27.77\% & 27.77\% & 27.77\% & 27.76\% & 27.75\% & \cellgray{27.77\%} \\ +\bottomrule +\end{tabular*} +\label{table:accuracy_comparison} +\end{table*} + + +\paragraph{Accuracy Preservation Analysis} Table \ref{table:accuracy_comparison} confirms that SMART achieves superior calibration while perfectly preserving classification accuracy—a fundamental advantage of post-hoc methods. Unlike CTS, which suffers accuracy drops up to 1.48 percentage points due to class-specific boundary alterations, or Spline's variable impacts on transformers, SMART's design ensures zero accuracy loss. By operating exclusively on the margin rather than full logit vectors, SMART focuses solely on confidence scaling without disturbing the relative ordering that determines predictions. This preservation holds even under severe distribution shifts like ImageNet-C and ImageNet-Sketch, where SMART simultaneously maintains base model accuracy while dramatically improving calibration. This dual guarantee makes SMART uniquely suitable for safety-critical applications requiring both correct predictions and reliable uncertainty estimates. + + + + + +\subsection{AdaECE Performance} +\label{appendix:detailed_adaece_analysis} + + +\begin{table*}[ht] +\caption{\textbf{Comparison of AdaECE Calibration Methods Using AdaECE($\downarrow$, \%, 15bins) Across Various Datasets and Models (Seed 1--5 Averaged).}} +\centering +\scriptsize +\setlength{\tabcolsep}{3pt} +\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}cccccccc} +\toprule +\textbf{Dataset} & \textbf{Model} & \textbf{Vanilla} & \textbf{TS} & \textbf{PTS} & \textbf{CTS} & \textbf{Spline} & \textbf{SMART} \\ +\midrule +\multirow{2}{*}{CIFAR-10} + & ResNet-50 & $4.33\pm0.0\%$ & $2.14\pm0.0\%$ & $0.83\pm28.6\%$ & $1.56\pm26.2\%$ & $2.14\pm1.1\%$ & $\cellgray\mathbf{0.99\pm4.3\%}$ \\ + & Wide-ResNet & $3.24\pm0.0\%$ & $1.71\pm0.0\%$ & $0.89\pm21.9\%$ & $1.47\pm19.7\%$ & $2.30\pm0.4\%$ & $\cellgray\mathbf{0.50\pm12.2\%}$ \\ +\midrule +\multirow{2}{*}{CIFAR-100} + & ResNet-50 & $17.53\pm0.0\%$ & $5.66\pm0.0\%$ & $1.91\pm35.3\%$ & $3.43\pm32.0\%$ & $3.55\pm0.0\%$ & $\cellgray\mathbf{2.27\pm25.2\%}$ \\ + & Wide-ResNet & $15.34\pm0.0\%$ & $4.41\pm0.0\%$ & $1.69\pm13.0\%$ & $2.95\pm11.6\%$ & $3.95\pm0.1\%$ & $\cellgray\mathbf{1.83\pm2.1\%}$ \\ +\midrule +\multirow{6}{*}{ImageNet-1K} + & ResNet-50 & $3.68\pm1.3\%$ & $2.13\pm0.5\%$ & $0.92\pm44.1\%$ & $2.21\pm39.8\%$ & $0.81\pm28.7\%$ & $\cellgray\mathbf{0.79\pm8.7\%}$ \\ + & DenseNet-121 & $2.52\pm1.4\%$ & $1.74\pm1.8\%$ & $1.05\pm41.3\%$ & $1.78\pm38.0\%$ & $0.77\pm28.0\%$ & $\cellgray\mathbf{0.65\pm10.2\%}$ \\ + & Wide-ResNet & $5.31\pm0.3\%$ & $2.87\pm2.8\%$ & $1.04\pm20.6\%$ & $3.24\pm18.0\%$ & $\mathbf{0.83\pm36.3\%}$ & $\cellgray{0.87\pm14.3\%}$\\ + & Swin-B & $4.86\pm0.6\%$ & $4.50\pm1.0\%$ & $1.05\pm4.6\%$ & $1.59\pm5.1\%$ & $1.04\pm5.3\%$ & $\cellgray\mathbf{0.74\pm12.2\%}$\\ + & ViT-B-16 & $5.57\pm1.2\%$ & $4.10\pm2.3\%$ & $1.09\pm29.7\%$ & $4.85\pm27.4\%$ & $1.07\pm29.2\%$ & $\cellgray\mathbf{0.79\pm15.4\%}$\\ + & ViT-B-32 & $6.41\pm0.4\%$ & $3.92\pm1.7\%$ & $1.27\pm71.9\%$ & $1.90\pm66.4\%$ & $0.96\pm15.3\%$ & $\cellgray\mathbf{0.78\pm3.6\%}$ \\ +\midrule +\multirow{5}{*}{ImageNet-C} + & ResNet-50 & $13.84\pm0.2\%$ & $2.02\pm1.7\%$ & $1.06\pm0.7\%$ & $1.76\pm0.6\%$ & $5.49\pm2.8\%$ & $\cellgray\mathbf{0.74\pm8.0\%}$ \\ + & DenseNet-121 & $12.57\pm0.1\%$ & $1.64\pm0.7\%$ & $1.17\pm9.9\%$ & $1.48\pm8.2\%$ & $2.57\pm7.9\%$ & $\cellgray\mathbf{0.70\pm3.6\%}$ \\ + & Swin-B & $11.98\pm0.1\%$ & $5.83\pm1.0\%$ & $1.58\pm0.0\%$ & $3.07\pm0.2\%$ & $5.13\pm2.3\%$ & $\cellgray\mathbf{1.31\pm2.9\%}$\\ + & ViT-B-16 & $8.24\pm0.3\%$ & $5.25\pm0.9\%$ & $1.27\pm5.9\%$ & $2.77\pm5.3\%$ & $2.57\pm7.9\%$ & $\cellgray\mathbf{1.09\pm4.0\%}$\\ + & ViT-B-32 & $7.66\pm0.2\%$ & $5.11\pm0.0\%$ & $1.07\pm4.3\%$ & $2.97\pm3.7\%$ & $1.45\pm16.8\%$ & $\cellgray\mathbf{1.01\pm4.2\%}$\\ +\midrule +\multirow{6}{*}{ImageNet-LT} + & ResNet-50 & $3.54\pm0.9\%$ & $2.02\pm1.2\%$ & $0.92\pm35.5\%$ & $2.17\pm33.0\%$ & $0.71\pm20.7\%$ & $\cellgray\mathbf{0.67\pm3.3\%}$\\ + & DenseNet-121 & $2.37\pm3.4\%$ & $1.74\pm2.1\%$ & $1.17\pm23.6\%$ & $1.86\pm21.3\%$ & $\mathbf{0.73\pm26.4\%}$ & $\cellgray{0.76\pm0.7\%}$\\ + & Wide-ResNet & $5.22\pm0.4\%$ & $2.98\pm0.9\%$ & $1.22\pm62.4\%$ & $2.83\pm58.1\%$ & $0.79\pm18.1\%$ & $\cellgray\mathbf{0.98\pm4.4\%}$\\ + & Swin-B & $4.69\pm0.6\%$ & $4.48\pm1.2\%$ & $1.43\pm19.1\%$ & $1.23\pm18.0\%$ & $0.95\pm6.7\%$ & $\cellgray\mathbf{0.74\pm31.3\%}$\\ + & ViT-B-16 & $5.57\pm0.8\%$ & $4.18\pm2.9\%$ & $1.13\pm43.4\%$ & $1.06\pm40.1\%$ & $0.95\pm12.9\%$ & $\cellgray\mathbf{0.85\pm15.1\%}$\\ + & ViT-B-32 & $6.26\pm0.6\%$ & $3.97\pm1.6\%$ & $1.30\pm31.1\%$ & $2.04\pm28.2\%$ & $0.86\pm26.5\%$ & $\cellgray\mathbf{0.84\pm10.1\%}$\\ +\midrule +\multirow{5}{*}{ImageNet-S} + & ResNet-50 & $22.31\pm0.3\%$ & $2.01\pm2.9\%$ & $1.64\pm16.4\%$ & $1.51\pm14.7\%$ & $9.51\pm2.4\%$ & $\cellgray\mathbf{0.90\pm15.8\%}$\\ + & DenseNet-121 & $20.15\pm0.5\%$ & $1.67\pm17.0\%$& $1.93\pm9.6\%$ & $1.16\pm8.3\%$ & $8.7\pm1.92\%$ & $\cellgray\mathbf{0.76\pm32.3\%}$\\ + & Swin-B & $24.62\pm0.0\%$ & $6.40\pm0.5\%$ & $1.53\pm12.2\%$ & $3.57\pm11.1\%$ & $9.06\pm4.2\%$ & $\cellgray\mathbf{1.53\pm3.8\%}$\\ + & ViT-B-16 & $16.57\pm0.2\%$ & $5.62\pm0.7\%$ & $1.33\pm8.7\%$ & $2.98\pm7.3\%$ & $8.66\pm1.9\%$ & $\cellgray\mathbf{1.08\pm4.3\%}$\\ + & ViT-B-32 & $14.19\pm0.3\%$ & $4.98\pm2.9\%$ & $1.66\pm16.0\%$ & $3.23\pm14.1\%$ & $5.64\pm3.3\%$ & $\cellgray\mathbf{1.07\pm19.9\%}$\\ +\bottomrule +\end{tabular*} +\label{table:adaece_comparison_updated} +\end{table*} + + +This section provides an in-depth analysis of calibration performance using AdaECE across different datasets and model architectures, complementing the results presented in Section~\ref{Sec:Calibration_performance}. Adaptive-ECE is a measure of calibration performance that addresses the bias of equal-width binning scheme of ECE. It adapts the bin-size to the number of samples and ensures that each bin is evenly distributed with samples. The formula for Adaptive-ECE is as follows: +\begin{equation} +\text{Adaptive-ECE} = \sum_{i=1}^{\mathbb{B}} \frac{|B_i|}{N}|I_i - C_i| \text{ s.t. } \forall i, j \cdot |B_i| = |B_j| +\end{equation} +AdaECE offers a more rigorous assessment of calibration quality than standard ECE by adapting bin boundaries to ensure uniform sample distribution, preventing calibration errors from being masked in sparsely populated confidence regions. Table \ref{table:adaece_comparison_updated} presents comprehensive AdaECE results across all evaluated datasets and architectures. SMART consistently outperforms competing methods under this metric, achieving the lowest AdaECE on 24 of 26 dataset-architecture combinations. + + +\begin{figure}[ht] +\centering +\includegraphics[width=\textwidth]{figs/adaece_cifar_figure.pdf} +\caption{\textbf{AdaECE comparison on CIFAR datasets.} SMART consistently achieves superior calibration on both CIFAR-10 and CIFAR-100 across multiple architectures. From left to right are Cifar10 ResNet-50/Wide-ResNet, Cifar100 ResNet-50/Wide-ResNet.} +\label{fig:adaece_cifar} +\end{figure} + + +\paragraph{CIFAR Performance Analysis.} SMART demonstrates exceptional calibration on CIFAR datasets in Figure~\ref{fig:adaece_cifar}, achieving the lowest AdaECE with notably stable variance compared to competitors. The key insight emerges when comparing CIFAR-10 to CIFAR-100: while global methods like TS suffer dramatic degradation as class count increases, SMART maintains robust performance. PTS shows competitive results but with substantially higher variance, indicating reliability issues. Spline struggles particularly with CIFAR-100's complex confidence landscape, revealing how non-parametric methods become less effective as classification complexity increases. + +\begin{figure}[ht] +\centering +\includegraphics[width=\textwidth]{figs/adaece_imagenet_figure.pdf} +\caption{\textbf{AdaECE($\downarrow$, \%, 15bins) comparison on ImageNet-1K.} SMART delivers consistent calibration across diverse architectures, from CNNs to vision transformers. From left to right are ResNet-50, DenseNet-121, Wide-ResNet, Swin-B, ViT-B-16, ViT-B-32.} +\label{fig:adaece_imagenet} +\end{figure} + +\paragraph{Large-Scale Classification on ImageNet.} In Figure~\ref{fig:adaece_imagenet}, The ImageNet results reveal a crucial architectural insight: SMART maintains consistent performance across both CNN and transformer designs, while traditional methods like TS and CTS show pronounced degradation on transformers. This architectural robustness highlights SMART's ability to capture fundamental uncertainty signals through the margin regardless of model inductive biases. PTS exhibits extreme variance, confirming that high-dimensional parameterizations struggle with reliability when learning complex temperature mappings, particularly on large-scale datasets. + +\begin{figure}[ht] +\centering +\includegraphics[width=\textwidth]{figs/adaece_imagenet_c_figure.pdf} +\caption{\textbf{AdaECE($\downarrow$, \%, 15bins) comparison on ImageNet-C.} SMART maintains exceptional calibration under corruption, while Spline and TS-based methods demonstrate significant degradation. From left to right are ResNet-50, DenseNet-121, Swin-B, ViT-B-16, ViT-B-32.} +\label{fig:adaece_imagenet_c} +\end{figure} + +\paragraph{Robustness to Input Corruption.} As shown in Figure~\ref{fig:adaece_imagenet_c}, SMART's resilience under corruption provides compelling evidence for the stability of decision boundary information. While Spline performs competitively on clean ImageNet, it deteriorates dramatically under corruption with values 5-7$\times$ higher than SMART. This collapse reveals a fundamental limitation: non-parametric methods overfit to validation distributions and fail when input characteristics change. SMART's focus on decision boundary uncertainty via the margin remains informative even when input distributions shift substantially. + +\begin{figure}[ht] +\centering +\includegraphics[width=\textwidth]{figs/adaece_imagenet_lt_figure.pdf} +\caption{\textbf{AdaECE($\downarrow$, \%, 15bins) comparison on ImageNet-LT.} SMART maintains strong calibration under long-tailed class distributions, particularly on CNN architectures. From left to right are ResNet-50, DenseNet-121, Wide-ResNet, Swin-B, ViT-B-16, ViT-B-32.} +\label{fig:adaece_imagenet_lt} +\end{figure} + +\paragraph{Long-Tailed Distribution Calibration.} As shown in Figure~\ref{fig:adaece_imagenet_lt}, The ImageNet-LT results reveal that class imbalance presents a fundamentally different calibration challenge than input corruption. Interestingly, Spline performs competitively here, suggesting non-parametric methods can handle statistical imbalances better than distributional shifts. However, CTS underperforms despite being explicitly designed for per-class variations, demonstrating that simply applying different temperatures per class is insufficient for complex imbalanced scenarios. + +\begin{figure}[ht] +\centering +\includegraphics[width=\textwidth]{figs/adaece_imagenet_sketch_figure.pdf} +\caption{\textbf{AdaECE($\downarrow$, \%, 15bins) comparison on ImageNet-Sketch.} SMART maintains exceptional calibration under extreme domain shift, while Spline struggles significantly. From left to right are ResNet-50, DenseNet-121, Swin-B, ViT-B-16, ViT-B-32.} +\label{fig:adaece_imagenet_sketch} +\end{figure} + +\paragraph{Extreme Domain Shift Calibration.} The sketch-based domain shift represents the most challenging calibration scenario in Figure~\ref{fig:adaece_imagenet_sketch} , where SMART demonstrates its most dramatic advantage. Spline's collapse here reinforces the brittleness of non-parametric methods under distribution shifts, while SMART's consistent performance across all architectures provides strong evidence that margin information captures robust uncertainty signals that transcend specific input characteristics or domains. + + + +% \section{In-training Bin Size} + +% \begin{figure}[ht] +% \centering +% \includegraphics[width=0.7\textwidth]{figs/bin_count_smart.pdf} +% \caption{\textbf{Ablation Study for in‐training Soft‐ECE bin count}} +% \captionsetup{justification=raggedright,singlelinecheck=false} +% \centering +% \label{fig:bin_count_sensitivity} +% \end{figure} + + +% \paragraph{Effect of in-training Bin Size on SMART Calibration.} +% We examine how the choice of histogram bin count during SMART's in-training ECE estimation influences final calibration performance. SMART exhibits good robustness to this hyperparameter across these architectures and datasets. For CIFAR-10 with ResNet-50, the calibration error remains virtually unchanged across all tested bin counts (5-25), demonstrating exceptional stability. ImageNet with ResNet-50 shows modest sensitivity, with performance gradually improving as bin count increases, before stabilizing around 20 bins. ViT-B/16 on ImageNet displays more pronounced variability, exhibiting a performance peak at 10 bins before recovering at 15 bins. This architecture-dependent behavior aligns with our theoretical understanding of the bias-variance trade-off in the Soft-ECE objective: convolutional architectures demonstrate greater stability to binning discretization compared to transformer models, likely due to their inherently different confidence distribution characteristics. The overall bin-count insensitivity of SMART reinforces our method's practical utility—practitioners can generally select a moderate bin count (15-20) without extensive tuning. This robustness stems from the Soft-ECE's smoothing effect through membership functions, which distributes calibration signals across neighboring bins and creates a more continuous optimization landscape that remains stable across different discretization levels. + +\section{Comparison Of Various Training-Time Calibration Methods On Other Metrics} +\label{appendix:training_time_calibration} + +This section presents a comprehensive evaluation of SMART when combined with various training-time calibration methods across multiple metrics, extending the ECE analysis provided in Section~\ref{sec:training_time_calibration}. We examine SMART's performance using AdaECE, Classwise ECE (CECE), Negative Log-Likelihood (NLL), and classification accuracy. + +\subsection{Accuracy Preservation} + +\begin{table*}[ht] +\centering +\caption{\textbf{Comparison of Train-time Calibration Methods Using Accuracy($\uparrow$, \%) Across Various Datasets and Models.} Results demonstrate that SMART preserves the original model accuracy across all training methods. Results are from the best run of 5 seeds.} +\label{table:accuracy_compare_with_training_time_methods} +\scriptsize +\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}cccccccccccccc} +\toprule +\textbf{Dataset} & \textbf{Model} &\multicolumn{2}{c}{\textbf{NLL}} &\multicolumn{2}{c}{\textbf{Brier Loss}} &\multicolumn{2}{c}{\textbf{MMCE}} &\multicolumn{2}{c}{\textbf{LS-0.05}} &\multicolumn{2}{c}{\textbf{FLSD-53}} &\multicolumn{2}{c}{\textbf{FL-3}} \\ + & & base & ours & base & ours & base & ours & base & ours & base & ours & base & ours \\ +\midrule +\multirow{4}{*}{CIFAR10} + & ResNet-50 & 95.1 & \cellgray{95.1} & 95.0 & \cellgray{95.0} & 95.0 & \cellgray{95.0} & 94.7 & \cellgray{94.7} & 95.0 & \cellgray{95.0} & 94.8 & \cellgray{94.8} \\ + & ResNet-110 & 95.1 & \cellgray{95.1} & 94.5 & \cellgray{94.5} & 94.6 & \cellgray{94.6} & 94.5 & \cellgray{94.5} & 94.6 & \cellgray{94.6} & 94.9 & \cellgray{94.9} \\ + & DenseNet-121 & 95.0 & \cellgray{95.0} & 94.9 & \cellgray{94.9} & 94.6 & \cellgray{94.6} & 94.9 & \cellgray{94.9} & 94.6 & \cellgray{94.6} & 94.7 & \cellgray{94.7} \\ + & Wide-ResNet & 96.1 & \cellgray{96.1} & 95.9 & \cellgray{95.9} & 96.1 & \cellgray{96.1} & 95.8 & \cellgray{95.8} & 96.0 & \cellgray{96.0} & 95.9 & \cellgray{95.9} \\ +\midrule +\multirow{4}{*}{CIFAR100} + & ResNet-50 & 76.7 & \cellgray{76.7} & 76.6 & \cellgray{76.6} & 76.8 & \cellgray{76.8} & 76.6 & \cellgray{76.6} & 76.8 & \cellgray{76.8} & 77.3 & \cellgray{77.3} \\ + & ResNet-110 & 77.3 & \cellgray{77.3} & 74.9 & \cellgray{74.9} & 76.9 & \cellgray{76.9} & 76.6 & \cellgray{76.6} & 77.5 & \cellgray{77.5} & 77.1 & \cellgray{77.1} \\ + & DenseNet-121 & 75.5 & \cellgray{75.5} & 76.3 & \cellgray{76.3} & 76.0 & \cellgray{76.0} & 75.9 & \cellgray{75.9} & 77.3 & \cellgray{77.3} & 76.8 & \cellgray{76.8} \\ + & Wide-ResNet & 79.3 & \cellgray{79.3} & 79.4 & \cellgray{79.4} & 79.3 & \cellgray{79.3} & 78.8 & \cellgray{78.8} & 79.9 & \cellgray{79.9} & 80.3 & \cellgray{80.3} \\ +\bottomrule +\end{tabular*} +\vspace{-0.1in} +\end{table*} + + +\paragraph{Accuracy Analysis} As shown in Table~\ref{table:accuracy_compare_with_training_time_methods}, SMART consistently preserves the classification accuracy of all base models across all training-time calibration methods. This is a critical property of post-hoc calibration methods, as improving confidence estimates should not come at the cost of predictive performance. The perfect accuracy preservation is by design, as SMART's temperature scaling mechanism operates solely on the scaling of logits without altering their relative ordering, thus maintaining the same class predictions. This contrasts with some training-time methods that may involve trade-offs between accuracy and calibration quality during the model optimization process. The preservation of accuracy across diverse architectures and datasets further validates SMART's practical utility as a calibration method that can be safely applied in real-world scenarios where maintaining predictive performance is essential. + +\subsection{AdaECE Performance} + +\begin{table*}[ht] +\centering +\caption{\textbf{Comparison of Train-time Calibration Methods Using AdaECE($\downarrow$, \%, 15bins) Across Various Datasets and Models.} The best-performing method for each dataset-model combination is in bold, and our method (SMART) is highlighted. Results are from the best run of 5 seeds.} +\label{table:adaece_compare_with_training_time_methods} +\scriptsize +\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}cccccccccccccc} +\toprule +\textbf{Dataset} & \textbf{Model} &\multicolumn{2}{c}{\textbf{NLL}} &\multicolumn{2}{c}{\textbf{Brier Loss}} &\multicolumn{2}{c}{\textbf{MMCE}} &\multicolumn{2}{c}{\textbf{LS-0.05}} &\multicolumn{2}{c}{\textbf{FLSD-53}} &\multicolumn{2}{c}{\textbf{FL-3}} \\ + & & base & ours & base & ours & base & ours & base & ours & base & ours & base & ours \\ +\midrule +\multirow{4}{*}{CIFAR10} + & ResNet-50 & 4.33 & \cellgray{\textbf{0.80}} & 1.74 & \cellgray{\textbf{1.01}} & 4.55 & \cellgray{\textbf{0.67}} & 3.88 & \cellgray{\textbf{2.18}} & 1.56 & \cellgray{\textbf{0.45}} & 1.95 & \cellgray{\textbf{0.48}} \\ + & ResNet-110 & 4.40 & \cellgray{\textbf{1.22}} & 2.61 & \cellgray{\textbf{0.56}} & 5.07 & \cellgray{\textbf{0.93}} & 4.46 & \cellgray{\textbf{3.66}} & 2.07 & \cellgray{\textbf{0.40}} & 1.64 & \cellgray{\textbf{0.52}} \\ + & DenseNet-121 & 4.49 & \cellgray{\textbf{0.61}} & 2.01 & \cellgray{\textbf{0.51}} & 5.10 & \cellgray{\textbf{0.96}} & 4.40 & \cellgray{\textbf{2.95}} & 1.38 & \cellgray{\textbf{0.62}} & 1.23 & \cellgray{\textbf{0.83}} \\ + & Wide-ResNet & 3.24 & \cellgray{\textbf{0.44}} & 1.70 & \cellgray{\textbf{0.44}} & 3.29 & \cellgray{\textbf{0.53}} & 4.27 & \cellgray{\textbf{0.97}} & 1.52 & \cellgray{\textbf{0.44}} & 1.84 & \cellgray{\textbf{0.59}} \\ +\midrule +\multirow{4}{*}{CIFAR100} + & ResNet-50 & 17.53 & \cellgray{\textbf{1.00}} & 6.54 & \cellgray{\textbf{1.41}} & 15.31 & \cellgray{\textbf{1.08}} & 7.63 & \cellgray{\textbf{1.75}} & 4.40 & \cellgray{\textbf{1.35}} & 5.08 & \cellgray{\textbf{0.95}} \\ + & ResNet-110 & 19.06 & \cellgray{\textbf{1.67}} & 7.73 & \cellgray{\textbf{0.93}} & 19.13 & \cellgray{\textbf{1.98}} & 11.07 & \cellgray{\textbf{2.72}} & 8.54 & \cellgray{\textbf{0.93}} & 8.65 & \cellgray{\textbf{1.22}} \\ + & DenseNet-121 & 20.99 & \cellgray{\textbf{2.23}} & 5.04 & \cellgray{\textbf{1.02}} & 19.10 & \cellgray{\textbf{1.73}} & 12.83 & \cellgray{\textbf{1.96}} & 3.54 & \cellgray{\textbf{0.93}} & 4.14 & \cellgray{\textbf{0.97}} \\ + & Wide-ResNet & 15.34 & \cellgray{\textbf{1.55}} & 4.28 & \cellgray{\textbf{0.97}} & 13.16 & \cellgray{\textbf{1.12}} & 5.13 & \cellgray{\textbf{2.11}} & 2.77 & \cellgray{\textbf{0.75}} & 2.07 & \cellgray{\textbf{1.15}} \\ +\bottomrule +\end{tabular*} +\vspace{-0.1in} +\end{table*} + +\paragraph{AdaECE Analysis} The adaptive ECE results in Table~\ref{table:adaece_compare_with_training_time_methods} provide further validation of SMART's effectiveness when combined with various training-time calibration methods. AdaECE, which uses adaptive binning to ensure equal sample counts in each bin, offers a more robust calibration measure than standard ECE by eliminating potential biases from uneven confidence distributions. SMART consistently improves AdaECE across all training methods, with particularly dramatic improvements for models trained with NLL and MMCE, where we observe reductions of up to 18$\times$ (17.53\% → 1.00\% for CIFAR-100 ResNet-50). + +The most substantial AdaECE improvements occur on CIFAR-100, which has ten times more classes than CIFAR-10 and thus represents a more challenging calibration scenario. This suggests that SMART's effectiveness scales favorably with task complexity. Even for models already trained with calibration-oriented objectives like Focal Loss or FLSD, SMART provides further substantial improvements, indicating that its margin-based temperature adjustment captures complementary information to these training-time approaches. Notably, the combination of SMART with FLSD-53 achieves some of the lowest overall AdaECE values (e.g., 0.40\% on CIFAR-10 ResNet-110), suggesting a particularly effective synergy between these methods. + +\subsection{Classwise ECE Performance} + +\begin{table*}[ht] +\centering +\caption{\textbf{Comparison of Train-time Calibration Methods Using Classwise ECE($\downarrow$, \%, 15bins) Across Various Datasets and Models.} The best-performing method for each dataset-model combination is in bold, and our method (SMART) is highlighted. Results are from the best run of 5 seeds.} +\label{table:cece_compare_with_training_time_methods} +\scriptsize +\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}cccccccccccccc} +\toprule +\textbf{Dataset} & \textbf{Model} &\multicolumn{2}{c}{\textbf{NLL}} &\multicolumn{2}{c}{\textbf{Brier Loss}} &\multicolumn{2}{c}{\textbf{MMCE}} &\multicolumn{2}{c}{\textbf{LS-0.05}} &\multicolumn{2}{c}{\textbf{FLSD-53}} &\multicolumn{2}{c}{\textbf{FL-3}} \\ + & & base & ours & base & ours & base & ours & base & ours & base & ours & base & ours \\ +\midrule +\multirow{4}{*}{CIFAR10} + & ResNet-50 & 0.91 & \cellgray{\textbf{0.43}} & 0.46 & \cellgray{\textbf{0.40}} & 0.94 & \cellgray{\textbf{0.51}} & 0.71 & \cellgray{\textbf{0.51}} & 0.42 & \cellgray{\textbf{0.37}} & 0.43 & \cellgray{\textbf{0.38}} \\ + & ResNet-110 & 0.92 & \cellgray{\textbf{0.49}} & 0.59 & \cellgray{\textbf{0.45}} & 1.04 & \cellgray{\textbf{0.54}} & 0.66 & \cellgray{\textbf{0.54}} & 0.47 & \cellgray{\textbf{0.41}} & 0.44 & \cellgray{\textbf{0.38}} \\ + & DenseNet-121 & 0.92 & \cellgray{\textbf{0.45}} & 0.46 & \cellgray{\textbf{0.41}} & 1.04 & \cellgray{\textbf{0.59}} & 0.60 & \cellgray{\textbf{0.50}} & 0.41 & \cellgray{\textbf{0.38}} & 0.42 & \cellgray{\textbf{0.35}} \\ + & Wide-ResNet & 0.68 & \cellgray{\textbf{0.37}} & 0.44 & \cellgray{\textbf{0.39}} & 0.70 & \cellgray{\textbf{0.38}} & 0.79 & \cellgray{\textbf{0.40}} & 0.41 & \cellgray{\textbf{0.29}} & 0.44 & \cellgray{\textbf{0.34}} \\ +\midrule +\multirow{4}{*}{CIFAR100} + & ResNet-50 & 0.38 & \cellgray{\textbf{0.21}} & 0.22 & \cellgray{\textbf{0.20}} & 0.34 & \cellgray{\textbf{0.20}} & 0.23 & \cellgray{\textbf{0.21}} & 0.20 & \cellgray{\textbf{0.20}} & 0.20 & \cellgray{\textbf{0.20}} \\ + & ResNet-110 & 0.41 & \cellgray{\textbf{0.20}} & 0.24 & \cellgray{\textbf{0.21}} & 0.42 & \cellgray{\textbf{0.21}} & 0.26 & \cellgray{\textbf{0.20}} & 0.24 & \cellgray{\textbf{0.20}} & 0.24 & \cellgray{\textbf{0.21}} \\ + & DenseNet-121 & 0.45 & \cellgray{\textbf{0.23}} & 0.20 & \cellgray{\textbf{0.20}} & 0.42 & \cellgray{\textbf{0.23}} & 0.29 & \cellgray{\textbf{0.21}} & \textbf{0.19} & \cellgray{0.20} & 0.20 & \cellgray{\textbf{0.20}} \\ + & Wide-ResNet & 0.34 & \cellgray{\textbf{0.19}} & 0.19 & \cellgray{\textbf{0.19}} & 0.30 & \cellgray{\textbf{0.19}} & 0.21 & \cellgray{\textbf{0.20}} & 0.18 & \cellgray{\textbf{0.18}} & 0.18 & \cellgray{\textbf{0.18}} \\ +\bottomrule +\end{tabular*} +\vspace{-0.1in} +\end{table*} + +\paragraph{CECE Analysis} Classwise ECE (CECE) provides insights into calibration performance at the individual class level rather than aggregated across all classes. The formula for classwise ECE is: + +\begin{equation} +\text{Classwise-ECE} = \frac{1}{\mathcal{K}} \sum_{i=1}^{B} \sum_{j=1}^{\mathcal{K}} \frac{|B_{i,j}|}{N} |I_{i,j} - C_{i,j}| +\end{equation} + +where the calibration error is computed separately for each class $j$ across all bins $i$, then averaged across all $\mathcal{K}$ classes. This metric is particularly valuable for understanding whether calibration improvements are uniformly distributed across classes or concentrated in specific categories. + +Table~\ref{table:cece_compare_with_training_time_methods} demonstrates SMART's ability to improve per-class calibration across almost all training methods and architectures. The improvements are particularly prominent for models trained with NLL and MMCE, where CECE values are typically reduced by 50\% or more after applying SMART (e.g., from 0.91\% to 0.43\% for CIFAR-10 ResNet-50). This substantial improvement suggests that SMART's margin-based temperature scaling effectively addresses class-specific miscalibration patterns that may arise during training with these standard objectives. + +Interestingly, CECE values are consistently lower on CIFAR-100 compared to CIFAR-10 despite the higher class count, which contrasts with the pattern observed for ECE and AdaECE. This phenomenon occurs because CECE averages calibration errors across classes, and with 100 classes, individual class miscalibrations tend to average out more effectively than with only 10 classes. Additionally, the higher granularity of class divisions in CIFAR-100 may lead to more balanced per-class confidence distributions, making the averaging effect more pronounced. + +For models already trained with calibration-oriented losses like FLSD-53 and FL-3, SMART provides more modest improvements in CECE, and in a few cases maintains the same level of performance. This suggests that these training-time methods are already effective at addressing per-class calibration issues through their specialized loss formulations that inherently consider class-wise balance. However, SMART can still provide complementary benefits in most scenarios, particularly for classes that may remain poorly calibrated even after specialized training procedures. + +\subsection{Negative Log-Likelihood Performance} + +\begin{table*}[ht] +\centering +\caption{\textbf{Comparison of Train-time Calibration Methods Using NLL($\downarrow$, \%) Across Various Datasets and Models.} The best-performing method for each dataset-model combination is in bold, and our method (SMART) is highlighted. Results are from the best run of 5 seeds.} +\label{table:nll_compare_with_training_time_methods} +\scriptsize +\setlength{\tabcolsep}{5pt} +\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}cccccccccccccc} +\toprule +\textbf{Dataset} & \textbf{Model} & +\multicolumn{2}{c}{\textbf{NLL}} & +\multicolumn{2}{c}{\textbf{Brier Loss}} & +\multicolumn{2}{c}{\textbf{MMCE}} & +\multicolumn{2}{c}{\textbf{LS-0.05}} & +\multicolumn{2}{c}{\textbf{FLSD-53}} & +\multicolumn{2}{c}{\textbf{FL-3}} \\ + & & \textbf{Base} & \textbf{Ours} & + \textbf{Base} & \textbf{Ours} & + \textbf{Base} & \textbf{Ours} & + \textbf{Base} & \textbf{Ours} & + \textbf{Base} & \textbf{Ours} & + \textbf{Base} & \textbf{Ours} \\ +\midrule +\multirow{4}{*}{CIFAR-10} + & ResNet-50 & 41.2 & \cellgray \textbf{19.7} & 18.7 & \cellgray \textbf{18.4} & 44.8 & \cellgray \textbf{21.0} & 27.7 & \cellgray \textbf{27.7} & 17.6 & \cellgray \textbf{17.1} & 18.4 & \cellgray \textbf{17.9} \\ + & ResNet-110 & 47.5 & \cellgray \textbf{22.5} & 20.4 & \cellgray \textbf{19.4} & 55.7 & \cellgray \textbf{23.6} & 29.9 & \cellgray \textbf{29.4} & 18.5 & \cellgray \textbf{17.9} & 17.8 & \cellgray \textbf{17.3} \\ + & DenseNet-121 & 42.9 & \cellgray \textbf{20.8} & 19.1 & \cellgray \textbf{18.6} & 52.1 & \cellgray \textbf{24.1} & 28.7 & \cellgray \textbf{28.7} & 18.4 & \cellgray \textbf{18.1} & 18.0 & \cellgray \textbf{17.9} \\ + & Wide-ResNet & 26.8 & \cellgray \textbf{14.9} & 15.9 & \cellgray \textbf{15.4} & 28.5 & \cellgray \textbf{15.9} & 21.7 & \cellgray \textbf{19.9} & 14.6 & \cellgray \textbf{13.7} & 15.2 & \cellgray \textbf{14.9} \\ +\midrule +\multirow{4}{*}{CIFAR-100} + & ResNet-50 & 153.7 & \cellgray \textbf{105.3} & 99.6 & \cellgray \textbf{99.5} & 125.3 & \cellgray \textbf{100.7} & 121.0 & \cellgray \textbf{120.1} & 88.0 & \cellgray 88.4 & 87.5 & \cellgray 88.1 \\ + & ResNet-110 & 179.2 & \cellgray \textbf{104.0} & 110.7 & \cellgray \textbf{110.0} & 180.6 & \cellgray \textbf{106.1} & 133.1 & \cellgray \textbf{128.8} & 89.9 & \cellgray \textbf{88.3} & 90.9 & \cellgray \textbf{90.0} \\ + & DenseNet-121 & 205.6 & \cellgray \textbf{119.1} & 98.3 & \cellgray 98.9 & 166.6 & \cellgray \textbf{112.6} & 142.0 & \cellgray \textbf{134.3} & 85.5 & \cellgray 86.5 & 87.1 & \cellgray 87.3 \\ + & Wide-ResNet & 140.1 & \cellgray \textbf{95.2} & 84.6 & \cellgray 84.9 & 119.6 & \cellgray \textbf{94.1} & 108.1 & \cellgray \textbf{106.5} & 76.9 & \cellgray 77.4 & 74.7 & \cellgray 75.8 \\ +\bottomrule +\end{tabular*} +\end{table*} + + + +\paragraph{NLL Analysis} NLL is a probabilistic metric that measures both calibration quality and discriminative power. Table~\ref{table:nll_compare_with_training_time_methods} shows that SMART improves NLL for most models, with the most significant gains observed for NLL, MMCE, and LS-0.05 trained models. The improvements are particularly striking for CIFAR-10, where NLL is reduced by up to 60\% after applying SMART (e.g., 41.22 → 19.70 for ResNet-50 with NLL). + +However, a different pattern emerges for models trained with specialized losses like FLSD-53 and FL-3 on CIFAR-100, where SMART sometimes leads to slight increases in NLL despite improvements in calibration metrics like ECE and AdaECE. This suggests that these specialized training losses optimize directly for NLL-like objectives, creating a scenario where SMART's temperature scaling might slightly disturb the carefully optimized probability distributions. Nevertheless, the overall trend across metrics indicates that SMART maintains or improves model performance in the vast majority of cases. + +% \subsection{Comparative Analysis Across Metrics} + +% Examining the results across all metrics provides a comprehensive view of SMART's effectiveness when combined with training-time calibration methods. Several key insights emerge: + +% \begin{enumerate} +% \item \textbf{Complementary Benefits:} SMART provides substantial improvements across almost all combinations of architectures, datasets, and training methods, indicating that its margin-based temperature scaling captures calibration signals complementary to those targeted by training-time approaches. + +% \item \textbf{Most Effective Combinations:} The combinations of SMART with FLSD-53 and FL-3 often achieve the best overall calibration performance, suggesting a particularly effective synergy between these focal-loss based training methods and SMART's post-hoc adjustments. + +% \item \textbf{Architectural Differences:} Wide-ResNet consistently shows the best calibration performance across most metrics and training methods when combined with SMART, possibly due to its architectural properties that produce more informative margins. + +% \item \textbf{Scale with Task Complexity:} The relative improvements from applying SMART are generally larger on CIFAR-100 than CIFAR-10, suggesting that SMART's benefits scale favorably with task complexity and class count. +% \end{enumerate} + +% These findings further validate SMART as a versatile, effective calibration method that can be applied as a complementary step to various training-time calibration approaches, providing additional performance improvements with minimal computational overhead and without compromising accuracy. + + + +\section{Calibration Performance Under Specific Corruption Types} +\label{appendix:corruption_analysis} + +To provide deeper insights into SMART's robustness across different corruption scenarios, we examine the calibration error reduction achieved by various methods on individual corruption types in ImageNet-C. We analyze performance across two architectures (ResNet-50 and ViT-B/16) and two metrics (ECE and AdaECE), providing a comprehensive view of how different calibration approaches respond to specific distribution shifts. This granular analysis helps understand which corruption types pose the greatest calibration challenges and how architectural differences influence calibration robustness. + +\begin{figure}[ht] +\centering +\includegraphics[width=0.95\textwidth]{figs/resnet50_ece_reduced_error.pdf} +\caption{\textbf{ECE reduction($\uparrow$, \%, 15bins) across corruption types for ResNet-50.} SMART consistently achieves superior calibration improvements across diverse corruption scenarios, demonstrating exceptional robustness to distribution shifts.} +\label{fig:resnet50_ece_corruption_analysis} +\end{figure} + +\paragraph{ResNet-50 ECE Analysis} The corruption-specific analysis reveals that SMART demonstrates remarkable consistency, achieving the highest ECE reduction across most corruption categories with improvements often exceeding 20\%. The inclusion of Spline calibration exposes a critical limitation of non-parametric methods: extreme brittleness under distribution shifts. While Spline achieves competitive results on certain corruptions like Snow, it completely fails on others such as Brightness and Contrast, highlighting how non-parametric approaches overfit to validation characteristics and break down when faced with novel corruptions. + +This contrasts sharply with SMART's robust performance across all corruption types. The key insight is that SMART's margin indicator captures decision boundary information that remains meaningful regardless of input degradation type—whether geometric distortions, noise, or digital artifacts. Temperature Scaling and other global methods show predictable limitations on uniform corruptions, while parametric methods like PTS exhibit moderate consistency but still significant variability. SMART's sample-specific adaptation based on decision boundary information provides the most reliable calibration improvements, making it uniquely suitable for real-world scenarios where corruption characteristics are unpredictable. + +\begin{figure}[ht] +\centering +\includegraphics[width=0.95\textwidth]{figs/resnet50_adaece_reduced_error.pdf} +\caption{\textbf{AdaECE reduction($\uparrow$, \%, 15bins) across corruption types for ResNet-50.} SMART maintains consistent superiority across corruption types under adaptive binning, confirming robust calibration improvements independent of evaluation methodology.} +\label{fig:resnet50_adaece_corruption_analysis} +\end{figure} + +\paragraph{ResNet-50 AdaECE Analysis} The AdaECE results closely mirror the ECE patterns, confirming that SMART's calibration improvements are fundamental rather than evaluation artifacts. SMART achieves the highest reduction rates across most corruptions, with particularly strong performance on geometric distortions approaching 25\% improvement. Spline's brittleness persists under adaptive binning—performing reasonably on weather corruptions but failing on uniform transforms, confirming that its limitations stem from overfitting rather than evaluation methodology. The near-identical performance rankings across both metrics demonstrate that SMART's margin approach captures robust calibration signals regardless of how calibration quality is measured. + +\begin{figure}[ht] +\centering +\includegraphics[width=0.95\textwidth]{figs/vit_b_16_ece_reduced_error.pdf} +\caption{\textbf{ECE reduction($\uparrow$, \%, 15bins) across corruption types for ViT-B/16.} Transformer architectures exhibit distinct calibration challenges under corruption, with global methods often failing while SMART maintains consistent improvements.} +\label{fig:vit_ece_corruption_analysis} +\end{figure} + +\paragraph{ViT-B/16 ECE Analysis} The transformer results reveal striking architectural differences in calibration behavior under corruption. Most notably, Temperature Scaling frequently worsens calibration, showing negative improvements on multiple corruption types including Shot Noise, Speckle Noise, Snow, Brightness, Pixelate, Jpeg Compression, Saturate and Spatter. This demonstrates that transformers' attention mechanisms and different inductive biases make them fundamentally incompatible with global temperature adjustments under distribution shifts. + +SMART maintains consistent positive improvements across all corruption types, though generally more modest than with ResNet-50. This architectural difference suggests that while transformers are inherently better calibrated, they also present unique challenges that require more sophisticated approaches than global scaling. The convergence of all methods on Fog corruption (around 25\% improvement) indicates that certain atmospheric corruptions create calibration conditions where architectural differences become less relevant. + +A key insight emerges: the margin's decision boundary information remains meaningful across architectures, while global statistics become unreliable for transformers under corruption. PTS and CTS show more consistent improvements than TS, but SMART's sample-specific adaptation consistently outperforms all alternatives, confirming its architectural robustness. + +\begin{figure}[ht] +\centering +\includegraphics[width=0.95\textwidth]{figs/vit_b_16_adaece_reduced_error.pdf} +\caption{\textbf{AdaECE reduction($\uparrow$, \%, 15bins) across corruption types for ViT-B/16.} Transformer calibration patterns remain consistent under adaptive binning, confirming architectural-specific calibration challenges and SMART's robustness.} +\label{fig:vit_adaece_corruption_analysis} +\end{figure} + +\paragraph{ViT-B/16 AdaECE Analysis} The AdaECE results closely replicate the ECE patterns, confirming that transformer calibration behaviors are fundamental architectural characteristics rather than evaluation artifacts. Temperature Scaling's negative performance persists under adaptive binning, while SMART maintains consistent positive improvements across all corruption types. This metric independence demonstrates that SMART's margin approach captures robust decision boundary information that remains effective regardless of how calibration quality is measured. + +\section{Margin Perspective on Calibration} +\label{sec:logit_gap_perspective} + + +\begin{figure*}[ht] +\centering +\begin{subfigure}[b]{0.32\textwidth} + \includegraphics[width=\textwidth]{figs/overall_reliability_imagenet_vit_b_16.pdf} + \caption{Overall calibration for ImgNet} + \label{fig:overall_imagenet_vit} +\end{subfigure} +\hfill +\begin{subfigure}[b]{0.32\textwidth} + \includegraphics[width=\textwidth]{figs/group_3_reliability_imagenet_vit_b_16.pdf} + \caption{High margin samples} + \label{fig:high_gap_imagenet_vit} +\end{subfigure} +\hfill +\begin{subfigure}[b]{0.32\textwidth} + \includegraphics[width=\textwidth]{figs/group_1_reliability_imagenet_vit_b_16.pdf} + \caption{Low margin samples} + \label{fig:low_gap_imagenet_vit} +\end{subfigure} + +\begin{subfigure}[b]{0.32\textwidth} + \includegraphics[width=\textwidth]{figs/overall_reliability_ciafr100_resnet50.pdf} + \caption{Overall calibration for Cifar-100} + \label{fig:overall_cifar_resnet} +\end{subfigure} +\hfill +\begin{subfigure}[b]{0.32\textwidth} + \includegraphics[width=\textwidth]{figs/group_3_reliability_ciafr100_resnet50.pdf} + \caption{High margin samples Cifar-100} + \label{fig:high_gap_cifar_resnet} +\end{subfigure} +\hfill +\begin{subfigure}[b]{0.32\textwidth} + \includegraphics[width=\textwidth]{figs/logitsgap_vs_confidence_change.pdf} + \caption{Margin vs. confidence change} + \label{fig:gap_vs_change} +\end{subfigure} + +\caption{\textbf{Margin reveals hidden calibration patterns across the confidence spectrum.} ImageNet ViT-B/16 shows near-perfect overall calibration (a) but reveals systematic under-confidence in high margin samples (b) and well-calibrated low margin samples (c). CIFAR-100 ResNet-50 demonstrates that even with overall over-confidence (d), high margin samples remain under-confident (e). Panel (f) shows SMART provides targeted adjustments whilst TS and PTS show suboptimal patterns.} +\label{fig:logit_gap_analysis} +\end{figure*} + +Traditional calibration analysis evaluates models from an overall perspective, potentially masking important sample-specific miscalibration patterns. By examining calibration behaviour across margin values, we uncover fundamental insights about how neural networks distribute confidence and validate our method visually. + +Figure~\ref{fig:logit_gap_analysis} demonstrates heterogeneity across margin groups. For ImageNet with ViT-B/16, whilst overall calibration appears near-perfect (Figure~\ref{fig:overall_imagenet_vit}), decomposing by margin reveals distinct patterns: low margin samples achieve good calibration (Figure~\ref{fig:low_gap_imagenet_vit}), whilst high margin samples show systematic under-confidence (Figure~\ref{fig:high_gap_imagenet_vit}). This pattern persists across different conditions, as shown in CIFAR-100 with ResNet-50 (Figures~\ref{fig:overall_cifar_resnet} and~\ref{fig:high_gap_cifar_resnet}), indicating that margin-based groupings reveal fundamental calibration characteristics transcending dataset-specific or architecture-specific behaviors. + +\paragraph{The Under-Confidence Paradox in High Margin Samples} Perhaps the most counterintuitive finding emerges from examining high margin samples. Despite representing easy classifications with substantial separation between top predictions, these samples consistently exhibit under-confidence rather than expected over-confidence. High margin samples from ImageNet ViT-B/16 show systematic under-confidence, with predicted confidence consistently lower than empirical accuracy (Figure~\ref{fig:high_gap_imagenet_vit}). This pattern persists even when overall model behaviour differs dramatically, as CIFAR-100 ResNet-50 maintains under-confidence in high margin samples despite overall over-confidence (Figure~\ref{fig:high_gap_cifar_resnet}). + +\paragraph{Method-Specific Failures from the Margin Perspective} The confidence adjustment patterns in Figure~\ref{fig:gap_vs_change} expose fundamental limitations in existing approaches. Temperature Scaling's uniform adjustment completely ignores heterogeneous calibration needs across margin groups, applying identical modifications regardless of sample characteristics. More critically, PTS makes substantial adjustments to low margin samples that already achieve good calibration and require minimal intervention. This unnecessary manipulation exemplifies how increased dimensionality introduces noise for precise temperature parameterisation. In contrast, SMART provides minimal adjustments to low margin samples that are already well-calibrated, whilst delivering targeted confidence increases to high margin samples suffering from under-confidence. This adaptive behavior emerges naturally from our lightweight margin-to-temperature mapping, demonstrating how principled architectural choices translate into appropriate calibration strategies. + + +% \section{Additional Visualization of SmoothSoftECE} +% \label{app:hubersoeftece-viz} + +% \begin{figure}[t] +% \centering +% \begin{minipage}{0.48\linewidth} +% \centering +% \includegraphics[width=\linewidth]{figs/test_loss_convergence_vit_b_16.pdf} +% \end{minipage}\hfill +% \begin{minipage}{0.48\linewidth} +% \centering +% \includegraphics[width=\linewidth]{figs/gradient_norm_evolution_vit_b_16.pdf} +% \end{minipage} +% \caption{Post-hoc calibration on ImageNet with ViT-B/16: test loss (left) and gradient norm (right) over training epochs for NLL, SoftECE, and SmoothSoftECE.} +% \label{fig:vitb16-hubersoeftece} +% \end{figure} + +% Figure~\ref{fig:vitb16-hubersoeftece} illustrates how the three objectives behave during optimization for a ViT-B/16 calibrator on ImageNet. The left panel shows that NLL rapidly plateaus at a comparatively high value, while SoftECE continues to decrease but remains consistently above SmoothSoftECE. SmoothSoftECE converges faster and reaches the lowest final loss. The right panel plots the corresponding gradient norms: SoftECE exhibits large and persistent oscillations, suggesting noisy and unstable gradients; NLL’s gradient norm decays smoothly but, consistent with the loss curve, does not lead to substantial improvement in the calibration objective. In contrast, SmoothSoftECE shows a more gradual and stable decay of the gradient norm, indicating a better-conditioned optimization process. These curves are representative of typical runs and are meant to provide intuition rather than a separate benchmark. + + + + + +\subsection{Sensitivity to Hyperparameters $\lambda$ and $\delta$} +\label{subsec:hyperparameter_sensitivity} + +We examine the sensitivity of SMART's performance to the bandwidth parameter $\lambda$ and Charbonnier smoothing parameter $\delta$ in Equation~\eqref{eq:charbonnier_softece}. Tables~\ref{tab:hyperparam_resnet50} and~\ref{tab:hyperparam_vitb16} report ECE (15 bins) on ImageNet for ResNet-50 and ViT-B/16 across different $(\lambda, \delta)$ combinations. + +\begin{table}[h] +\centering +\caption{ECE (\%, $\downarrow$, 15 bins) on ImageNet ResNet-50 for different $(\lambda, \delta)$ combinations.} +\label{tab:hyperparam_resnet50} +\small +\begin{tabular}{@{}lcccc@{}} +\toprule +$\lambda \backslash \delta$ & 0.001 & 0.010 & 0.100 & 1.000 \\ +\midrule +0.01 & 0.66 & 0.83 & 1.02 & 0.67 \\ +\cellgray 0.05 & \cellgray 0.61 & \cellgray 0.66 & \cellgray 0.67 & \cellgray 0.66 \\ +0.10 & 0.66 & 3.11 & 0.71 & 0.72 \\ +0.50 & 0.85 & 0.95 & 1.29 & 1.26 \\ +1.00 & 0.79 & 1.15 & 2.49 & 2.51 \\ +\bottomrule +\end{tabular} +\end{table} + +\begin{table}[h] +\centering +\caption{ECE (\%, $\downarrow$, 15 bins) on ImageNet ViT-B/16 for different $(\lambda, \delta)$ combinations.} +\label{tab:hyperparam_vitb16} +\small +\begin{tabular}{@{}lcccc@{}} +\toprule +$\lambda \backslash \delta$ & 0.001 & 0.010 & 0.100 & 1.000 \\ +\midrule +0.01 & 1.32 & 0.84 & 0.78 & 0.85 \\ +\cellgray 0.05 & \cellgray 0.84 & \cellgray 0.80 & \cellgray 0.89 & \cellgray 0.86 \\ +0.10 & 0.99 & 0.97 & 0.81 & 2.26 \\ +0.50 & 2.09 & 2.02 & 2.06 & 2.05 \\ +1.00 & 2.04 & 2.48 & 2.56 & 2.56 \\ +\bottomrule +\end{tabular} +\end{table} + +The results demonstrate that performance remains stable within a reasonable range of values. For $\lambda \in \{0.01, 0.05, 0.10\}$, ECE varies by less than 0.2\% across different $\delta$ choices, indicating robustness to the Charbonnier smoothing parameter. Larger values ($\lambda \geq 0.50$) lead to degraded performance due to over-localization of kernel weights, creating high variance in calibration estimates. Our choice of $\lambda=0.05$ and $\delta=0.001$ (highlighted rows) provides consistent performance across both CNN and transformer architectures, though the method is not particularly sensitive to $\delta$ within the range $[0.001, 0.100]$ when $\lambda$ is appropriately chosen. + +\section{Additional Analysis of the Margin--Temperature Relationship} +\label{app:margin_temperature} + +\begin{figure}[t] + \centering + \begin{minipage}{0.48\linewidth} + \centering + \includegraphics[width=\linewidth]{figs/logitsgap_temperature_density_map_resnet50.pdf} + \end{minipage}\hfill + \begin{minipage}{0.48\linewidth} + \centering + \includegraphics[width=\linewidth]{figs/logitsgap_temperature_density_map_vit_b_16.pdf} + \end{minipage} + \caption{Empirical margin--temperature relationship learned by Left: ImageNet with a ResNet-50 backbone, where the mapping is approximately linear and monotone increasing (Pearson $r=0.94$). Right: ImageNet with a ViT-B/16 backbone, where the mapping becomes non-monotonic with a pronounced U-shaped pattern (Pearson $r=-0.63$ for the best linear fit).} + \label{fig:margin_temperature_appendix} +\end{figure} + +Figure~\ref{fig:margin_temperature_appendix} illustrates that the learned margin--temperature mapping is not constrained to be monotonic. For ImageNet ResNet-50, the mapping closely follows an increasing linear trend: samples with larger logit margins receive higher temperatures (softer probabilities), while low-margin samples are assigned temperatures closer to one. In contrast, on ImageNet with ViT-B/16 the mapping is clearly non-monotonic, with an approximately U-shaped dependence on the margin. This behavior indicates that the relationship between margin and miscalibration is architecture- and dataset-dependent; SMART adapts to these differences rather than enforcing a fixed monotone form, and understanding the underlying theoretical reasons is left for future work. diff --git a/icml2026/example_paper.pdf b/icml2026/example_paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..08e4c6c7bce4de6c6e57ee02d606a89ad428415a --- /dev/null +++ b/icml2026/example_paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3e8fe0e952de8702ca4697dba09ca52e83bd649767b92a8340d34a591caa4d1b +size 193509 diff --git a/icml2026/fancyhdr.sty b/icml2026/fancyhdr.sty new file mode 100644 index 0000000000000000000000000000000000000000..b3d811f9028c2f60b4437f9c834a5dfff1d02a9d --- /dev/null +++ b/icml2026/fancyhdr.sty @@ -0,0 +1,864 @@ +%% +%% This is file `fancyhdr.sty', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% fancyhdr.dtx (with options: `fancyhdr') +%% +%% This is a generated file. +%% +%% This file may be distributed and/or modified under the conditions of +%% the LaTeX Project Public License, either version 1.3 of this license +%% or (at your option) any later version. The latest version of this +%% license is in: +%% +%% http://www.latex-project.org/lppl.txt +%% +%% and version 1.3 or later is part of all distributions of LaTeX version +%% 2005/12/01 or later. +%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\NeedsTeXFormat{LaTeX2e}[2018-04-01] +\ProvidesPackage{fancyhdr}% + [2025/02/07 v5.2 + Extensive control of page headers and footers]% +% Copyright (C) 1994-2025 by Pieter van Oostrum +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\ifdefined\NewDocumentCommand\else\RequirePackage{xparse}\fi +\newif\iff@nch@check +\f@nch@checktrue +\DeclareOption{nocheck}{% + \f@nch@checkfalse +} +\let\f@nch@gbl\relax +\newif\iff@nch@compatViii +\DeclareOption{compatV3}{% + \PackageWarningNoLine{fancyhdr}{The `compatV3' option is deprecated.\MessageBreak + It will disappear in one of the following releases.\MessageBreak + Please change your document to work\MessageBreak + without this option} + \let\f@nch@gbl\global + \f@nch@compatViiitrue +} +\newif\iff@nch@twoside +\f@nch@twosidefalse +\DeclareOption{twoside}{% + \if@twoside\else\f@nch@twosidetrue\fi +} +\newcommand\f@nch@def[2]{% + \def\temp@a{#2}\ifx\temp@a\@empty\f@nch@gbl\def#1{}% + \else\f@nch@gbl\def#1{#2\strut}\fi} +\DeclareOption{myheadings}{% + \@ifundefined{chapter}{% + \def\ps@myheadings{\ps@f@nch@fancyproto \let\@mkboth\@gobbletwo + \fancyhf{} + \fancyhead[LE,RO]{\thepage}% + \fancyhead[RE]{\slshape\leftmark}% + \fancyhead[LO]{\slshape\rightmark}% + \let\sectionmark\@gobble + \let\subsectionmark\@gobble + }% + }% + {\def\ps@myheadings{\ps@f@nch@fancyproto \let\@mkboth\@gobbletwo + \fancyhf{} + \fancyhead[LE,RO]{\thepage}% + \fancyhead[RE]{\slshape\leftmark}% + \fancyhead[LO]{\slshape\rightmark}% + \let\chaptermark\@gobble + \let\sectionmark\@gobble + }% + }% +} +\DeclareOption{headings}{% + \@ifundefined{chapter}{% + \if@twoside + \def\ps@headings{\ps@f@nch@fancyproto \def\@mkboth{\protect\markboth} + \fancyhf{} + \fancyhead[LE,RO]{\thepage}% + \fancyhead[RE]{\slshape\leftmark}% + \fancyhead[LO]{\slshape\rightmark}% + \def\sectionmark##1{% + \markboth{\MakeUppercase{% + \ifnum \c@secnumdepth >\z@ \thesection\quad \fi##1}}{}}% + \def\subsectionmark##1{% + \markright{% + \ifnum \c@secnumdepth >\@ne \thesubsection\quad \fi##1}}% + }% + \else + \def\ps@headings{\ps@f@nch@fancyproto \def\@mkboth{\protect\markboth} + \fancyhf{} + \fancyhead[LE,RO]{\thepage}% + \fancyhead[RE]{\slshape\leftmark}% + \fancyhead[LO]{\slshape\rightmark}% + \def\sectionmark##1{% + \markright {\MakeUppercase{% + \ifnum \c@secnumdepth >\z@ \thesection\quad \fi##1}}}% + \let\subsectionmark\@gobble % Not needed but inserted for safety + }% + \fi + }{\if@twoside + \def\ps@headings{\ps@f@nch@fancyproto \def\@mkboth{\protect\markboth} + \fancyhf{} + \fancyhead[LE,RO]{\thepage}% + \fancyhead[RE]{\slshape\leftmark}% + \fancyhead[LO]{\slshape\rightmark}% + \def\chaptermark##1{% + \markboth{\MakeUppercase{% + \ifnum \c@secnumdepth >\m@ne \if@mainmatter + \@chapapp\ \thechapter. \ \fi\fi##1}}{}}% + \def\sectionmark##1{% + \markright {\MakeUppercase{% + \ifnum \c@secnumdepth >\z@ \thesection. \ \fi##1}}}% + }% + \else + \def\ps@headings{\ps@f@nch@fancyproto \def\@mkboth{\protect\markboth} + \fancyhf{} + \fancyhead[LE,RO]{\thepage}% + \fancyhead[RE]{\slshape\leftmark}% + \fancyhead[LO]{\slshape\rightmark}% + \def\chaptermark##1{% + \markright{\MakeUppercase{% + \ifnum \c@secnumdepth >\m@ne \if@mainmatter + \@chapapp\ \thechapter. \ \fi\fi##1}}}% + \let\sectionmark\@gobble % Not needed but inserted for safety + }% + \fi + }% +} +\ProcessOptions* +\newcommand{\f@nch@forc}[3]{\expandafter\f@nchf@rc\expandafter#1\expandafter{#2}{#3}} +\newcommand{\f@nchf@rc}[3]{\def\temp@ty{#2}\ifx\@empty\temp@ty\else + \f@nch@rc#1#2\f@nch@rc{#3}\fi} +\long\def\f@nch@rc#1#2#3\f@nch@rc#4{\def#1{#2}#4\f@nchf@rc#1{#3}{#4}} +\newcommand{\f@nch@for}[3]{\edef\@fortmp{#2}% + \expandafter\@forloop#2,\@nil,\@nil\@@#1{#3}} +\newcommand\f@nch@default[3]{% + \edef\temp@a{\lowercase{\edef\noexpand\temp@a{#3}}}\temp@a \def#1{}% + \f@nch@forc\tmpf@ra{#2}% + {\expandafter\f@nch@ifin\tmpf@ra\temp@a{\edef#1{#1\tmpf@ra}}{}}% + \ifx\@empty#1\def#1{#2}\fi} +\newcommand{\f@nch@ifin}[4]{% + \edef\temp@a{#2}\def\temp@b##1#1##2\temp@b{\def\temp@b{##1}}% + \expandafter\temp@b#2#1\temp@b\ifx\temp@a\temp@b #4\else #3\fi} +\newcommand{\fancyhead}[2][]{\f@nch@fancyhf\fancyhead h[#1]{#2}}% +\newcommand{\fancyfoot}[2][]{\f@nch@fancyhf\fancyfoot f[#1]{#2}}% +\newcommand{\fancyhf}[2][]{\f@nch@fancyhf\fancyhf {}[#1]{#2}}% +\newcommand{\fancyheadoffset}[2][]{\f@nch@fancyhfoffs\fancyheadoffset h[#1]{#2}}% +\newcommand{\fancyfootoffset}[2][]{\f@nch@fancyhfoffs\fancyfootoffset f[#1]{#2}}% +\newcommand{\fancyhfoffset}[2][]{\f@nch@fancyhfoffs\fancyhfoffset {}[#1]{#2}}% +\def\f@nch@fancyhf@Echeck#1{% + \if@twoside\else + \iff@nch@twoside\else + \if\f@nch@@eo e% + \PackageWarning{fancyhdr} {\string#1's `E' option without twoside option is useless.\MessageBreak + Please consider using the `twoside' option}% + \fi\fi\fi +} +\long\def\f@nch@fancyhf#1#2[#3]#4{% + \def\temp@c{}% + \f@nch@forc\tmpf@ra{#3}% + {\expandafter\f@nch@ifin\tmpf@ra{eolcrhf,EOLCRHF}% + {}{\edef\temp@c{\temp@c\tmpf@ra}}}% + \ifx\@empty\temp@c\else \PackageError{fancyhdr}{Illegal char `\temp@c' in + \string#1 argument: [#3]}{}% + \fi \f@nch@for\temp@c{#3}% + {\f@nch@default\f@nch@@eo{eo}\temp@c + \f@nch@fancyhf@Echeck{#1}% + \f@nch@default\f@nch@@lcr{lcr}\temp@c + \f@nch@default\f@nch@@hf{hf}{#2\temp@c}% + \f@nch@forc\f@nch@eo\f@nch@@eo + {\f@nch@forc\f@nch@lcr\f@nch@@lcr + {\f@nch@forc\f@nch@hf\f@nch@@hf + {\expandafter\f@nch@def\csname + f@nch@\f@nch@eo\f@nch@lcr\f@nch@hf\endcsname {#4}}}}}} +\def\f@nch@fancyhfoffs#1#2[#3]#4{% + \def\temp@c{}% + \f@nch@forc\tmpf@ra{#3}% + {\expandafter\f@nch@ifin\tmpf@ra{eolrhf,EOLRHF}% + {}{\edef\temp@c{\temp@c\tmpf@ra}}}% + \ifx\@empty\temp@c\else \PackageError{fancyhdr}{Illegal char `\temp@c' in + \string#1 argument: [#3]}{}% + \fi \f@nch@for\temp@c{#3}% + {\f@nch@default\f@nch@@eo{eo}\temp@c + \f@nch@fancyhf@Echeck{#1}% + \f@nch@default\f@nch@@lcr{lr}\temp@c + \f@nch@default\f@nch@@hf{hf}{#2\temp@c}% + \f@nch@forc\f@nch@eo\f@nch@@eo + {\f@nch@forc\f@nch@lcr\f@nch@@lcr + {\f@nch@forc\f@nch@hf\f@nch@@hf + {\expandafter\setlength\csname + f@nch@offset@\f@nch@eo\f@nch@lcr\f@nch@hf\endcsname {#4}}}}}% + \f@nch@setoffs} +\NewDocumentCommand {\fancyheadwidth}{ s O{} O{} m } + {\f@nch@fancyhfwidth{#1}\fancyheadwidth h[#2][#3]{#4}}% +\NewDocumentCommand {\fancyfootwidth}{ s O{} O{} m } + {\f@nch@fancyhfwidth{#1}\fancyfootwidth f[#2][#3]{#4}}% +\NewDocumentCommand {\fancyhfwidth} { s O{} O{} m } + {\f@nch@fancyhfwidth{#1}\fancyhfwidth {}[#2][#3]{#4}}% +\def\f@nch@fancyhfwidth#1#2#3[#4][#5]#6{% + \setlength\@tempdima{#6}% + \def\temp@c{}% + \f@nch@forc\tmpf@ra{#4}% + {\expandafter\f@nch@ifin\tmpf@ra{eolcrhf,EOLCRHF}% + {}{\edef\temp@c{\temp@c\tmpf@ra}}}% + \ifx\@empty\temp@c\else \PackageError{fancyhdr}{Illegal char `\temp@c' in + \string#2 argument: [#4]}{}% + \fi + \f@nch@for\temp@c{#4}% + {\f@nch@default\f@nch@@eo{eo}\temp@c + \f@nch@fancyhf@Echeck{#2}% + \f@nch@default\f@nch@@lcr{lcr}\temp@c + \f@nch@default\f@nch@@hf{hf}{#3\temp@c}% + \f@nch@forc\f@nch@eo\f@nch@@eo + {\f@nch@forc\f@nch@lcr\f@nch@@lcr + {\f@nch@forc\f@nch@hf\f@nch@@hf + {% + \IfBooleanTF{#1}{% + \expandafter\edef\csname + f@nch@width@\f@nch@eo\f@nch@lcr\f@nch@hf\endcsname{\the\@tempdima}% + }% + {% + \expandafter\def\csname + f@nch@width@\f@nch@eo\f@nch@lcr\f@nch@hf\endcsname{#6}% + }% + \csname f@nchdrwdt@align@v@\f@nch@hf\endcsname + \edef\f@nch@align@@h{\f@nch@lcr}% + \def\temp@a{#5}% + \ifx\temp@a\@empty \else \f@nchdrwdt@align#5\@nil{#2}\fi + \expandafter\edef\csname + f@nch@align@\f@nch@eo\f@nch@lcr\f@nch@hf\endcsname + {\f@nch@align@@v\f@nch@align@@h}}}}}} +\def\f@nch@width@elh{\headwidth} +\def\f@nch@width@ech{\headwidth} +\def\f@nch@width@erh{\headwidth} +\def\f@nch@width@olh{\headwidth} +\def\f@nch@width@och{\headwidth} +\def\f@nch@width@orh{\headwidth} +\def\f@nch@width@elf{\headwidth} +\def\f@nch@width@ecf{\headwidth} +\def\f@nch@width@erf{\headwidth} +\def\f@nch@width@olf{\headwidth} +\def\f@nch@width@ocf{\headwidth} +\def\f@nch@width@orf{\headwidth} +\def\f@nch@align@elh{bl} +\def\f@nch@align@ech{bc} +\def\f@nch@align@erh{br} +\def\f@nch@align@olh{bl} +\def\f@nch@align@och{bc} +\def\f@nch@align@orh{br} +\def\f@nch@align@elf{tl} +\def\f@nch@align@ecf{tc} +\def\f@nch@align@erf{tr} +\def\f@nch@align@olf{tl} +\def\f@nch@align@ocf{tc} +\def\f@nch@align@orf{tr} +\def\f@nchdrwdt@align@v@h{\def\f@nch@align@@v{b}}% +\def\f@nchdrwdt@align@v@f{\def\f@nch@align@@v{t}}% +\long\def\f@nchdrwdt@align#1#2\@nil#3{% + \f@nch@ifin{#1}{TtcbB-}{% + \f@nch@ifin{#1}{-}{}{\def\f@nch@align@@v{#1}}% + \def\@tempa{#2}% + \ifx\@tempa\@empty \else \def\f@nch@align@@h{#2}\fi + }% + {\def\f@nch@align@@h{#1}}% + \expandafter\f@nch@ifin\expandafter{\f@nch@align@@h}{lcrj}{}% + {\PackageError{fancyhdr} + {\string#3: Illegal char `\f@nch@align@@h'\MessageBreak + in alignment argument}{}}% +} +\newcommand{\lhead}[2][\f@nch@olh]% + {\f@nch@def\f@nch@olh{#2}\f@nch@def\f@nch@elh{#1}} +\newcommand{\chead}[2][\f@nch@och]% + {\f@nch@def\f@nch@och{#2}\f@nch@def\f@nch@ech{#1}} +\newcommand{\rhead}[2][\f@nch@orh]% + {\f@nch@def\f@nch@orh{#2}\f@nch@def\f@nch@erh{#1}} +\newcommand{\lfoot}[2][\f@nch@olf]% + {\f@nch@def\f@nch@olf{#2}\f@nch@def\f@nch@elf{#1}} +\newcommand{\cfoot}[2][\f@nch@ocf]% + {\f@nch@def\f@nch@ocf{#2}\f@nch@def\f@nch@ecf{#1}} +\newcommand{\rfoot}[2][\f@nch@orf]% + {\f@nch@def\f@nch@orf{#2}\f@nch@def\f@nch@erf{#1}} +\newlength{\f@nch@headwidth} \let\headwidth\f@nch@headwidth +\newlength{\f@nch@offset@elh} +\newlength{\f@nch@offset@erh} +\newlength{\f@nch@offset@olh} +\newlength{\f@nch@offset@orh} +\newlength{\f@nch@offset@elf} +\newlength{\f@nch@offset@erf} +\newlength{\f@nch@offset@olf} +\newlength{\f@nch@offset@orf} +\newcommand{\headrulewidth}{0.4pt} +\newcommand{\footrulewidth}{0pt} +\@ifundefined{headruleskip}% + {\newcommand{\headruleskip}{0pt}}{} +\@ifundefined{footruleskip}% + {\newcommand{\footruleskip}{.3\normalbaselineskip}}{} +\newcommand{\plainheadrulewidth}{0pt} +\newcommand{\plainfootrulewidth}{0pt} +\newif\if@fancyplain \@fancyplainfalse +\def\fancyplain#1#2{\if@fancyplain#1\else#2\fi} +\headwidth=-123456789sp +\let\f@nch@raggedleft\raggedleft +\let\f@nch@raggedright\raggedright +\let\f@nch@centering\centering +\let\f@nch@everypar\everypar +\ifdefined\ExplSyntaxOn + \ExplSyntaxOn + \providecommand\IfFormatAtLeastTF{\@ifl@t@r\fmtversion} + \IfFormatAtLeastTF{2021-06-01}{ + \def\f@nch@saveclr@parhook #1{ + \expandafter\let\csname f@nch@__hook~#1\expandafter\endcsname + \csname __hook~#1\endcsname + \expandafter\let\csname f@nch@__hook_toplevel~#1\expandafter\endcsname + \csname __hook_toplevel~#1\endcsname + \expandafter\let\csname f@nch@__hook_next~#1\expandafter\endcsname + \csname __hook_next~#1\endcsname + \expandafter\let\csname f@nch@g__hook_#1_code_prop\expandafter\endcsname + \csname g__hook_#1_code_prop\endcsname + \RemoveFromHook{#1}[*] + \ClearHookNext{#1} + } + \def\f@nch@restore@parhook #1{ + \global\expandafter\let\csname __hook~#1\expandafter\endcsname + \csname f@nch@__hook~#1\endcsname + \global\expandafter\let\csname __hook_toplevel~#1\expandafter\endcsname + \csname f@nch@__hook_toplevel~#1\endcsname + \global\expandafter\let\csname __hook_next~#1\expandafter\endcsname + \csname f@nch@__hook_next~#1\endcsname + \global\expandafter\let\csname g__hook_#1_code_prop\expandafter\endcsname + \csname f@nch@g__hook_#1_code_prop\endcsname + } + \def\f@nch@resetpar{ + \f@nch@everypar{} + \f@nch@saveclr@parhook{para/before} + \f@nch@saveclr@parhook{para/begin} + \f@nch@saveclr@parhook{para/end} + \f@nch@saveclr@parhook{para/after} + } + \def\f@nch@restorepar{ + \f@nch@restore@parhook{para/before} + \f@nch@restore@parhook{para/begin} + \f@nch@restore@parhook{para/end} + \f@nch@restore@parhook{para/after} + } + }{ + \def\f@nch@resetpar{ + \f@nch@everypar{} + } + \def\f@nch@restorepar{} + } + \ExplSyntaxOff +\else + \def\f@nch@resetpar{% + \f@nch@everypar{}% + } + \def\f@nch@restorepar{} +\fi +\newcommand\f@nch@noUppercase[2][]{#2} +\def\f@nch@reset{\f@nch@resetpar\restorecr\endlinechar=13 + \catcode`\\=0\catcode`\{=1\catcode`\}=2\catcode`\$=3\catcode`\&=4 + \catcode`\#=6\catcode`\^=7\catcode`\_=8\catcode`\ =10\catcode`\@=11 + \catcode`\:=11\catcode`\~=13\catcode`\%=14 + \catcode0=15 %NULL + \catcode9=10 %TAB + \let\\\@normalcr \let\raggedleft\f@nch@raggedleft + \let\raggedright\f@nch@raggedright \let\centering\f@nch@centering + \def\baselinestretch{1}% + \hsize=\headwidth + \def\nouppercase##1{{% + \let\uppercase\relax\let\MakeUppercase\f@nch@noUppercase + \expandafter\let\csname MakeUppercase \endcsname\relax + \expandafter\def\csname MakeUppercase\space\space\space\endcsname + [####1]####2{####2}% + ##1}}% + \@ifundefined{@normalsize} {\normalsize} % for ucthesis.cls + {\@normalsize}% + } +\newcommand*{\fancycenter}[1][1em]{% + \@ifnextchar[{\f@nch@center{#1}}{\f@nch@center{#1}[3]}% +} +\def\f@nch@center#1[#2]#3#4#5{% + \def\@tempa{#4}\ifx\@tempa\@empty + \hbox to\linewidth{\color@begingroup{#3}\hfil {#5}\color@endgroup}% + \else + \setlength\@tempdima{#1}% + \setlength{\@tempdimb}{#2\@tempdima}% + \@tempdimc \@tempdimb \advance\@tempdimc -\@tempdima + \setlength\@tempskipa{\@tempdimb \@plus 1fil \@minus \@tempdimc}% + \@tempskipb\@tempskipa + \def\@tempa{#3}\ifx\@tempa\@empty + \addtolength\@tempskipa{\z@ \@minus \@tempdima}% + \fi + \def\@tempa{#5}\ifx\@tempa\@empty % empty right + \addtolength\@tempskipb{\z@ \@minus \@tempdima}% + \fi + \settowidth{\@tempdimb}{#3}% + \settowidth{\@tempdimc}{#5}% + \ifdim\@tempdimb>\@tempdimc + \advance\@tempdimb -\@tempdimc + \addtolength\@tempskipb{\@tempdimb \@minus \@tempdimb}% + \else + \advance\@tempdimc -\@tempdimb + \addtolength\@tempskipa{\@tempdimc \@minus \@tempdimc}% + \fi + \hbox to\linewidth{\color@begingroup{#3}\hskip \@tempskipa + {#4}\hskip \@tempskipb {#5}\color@endgroup}% + \fi +} +\newcommand{\f@nch@headinit}{} +\newcommand{\fancyheadinit}[1]{% + \def\f@nch@headinit{#1}% +} +\newcommand{\f@nch@footinit}{} +\newcommand{\fancyfootinit}[1]{% + \def\f@nch@footinit{#1}% +} +\newcommand{\fancyhfinit}[1]{% + \def\f@nch@headinit{#1}% + \def\f@nch@footinit{#1}% +} +\ifdefined\NewMirroredHookPair + \NewMirroredHookPair{fancyhdr/before}{fancyhdr/after} + \NewMirroredHookPair{fancyhdr/head/begin}{fancyhdr/head/end} + \NewMirroredHookPair{fancyhdr/foot/begin}{fancyhdr/foot/end} +\fi +\newlength\f@nch@height +\newlength\f@nch@footalignment +\newif\iff@nch@footalign\f@nch@footalignfalse +\newcommand{\fancyfootalign}[1]{% + \def\temp@a{#1}% + \ifx\temp@a\@empty + \f@nch@footalignfalse + \else + \f@nch@footaligntrue + \setlength\f@nch@footalignment{#1}% + \fi +} +\newcommand\fancyhdrsettoheight[2]{% + \expandafter\ifx\csname f@nch@#2\endcsname\fancyhdrsettoheight + \else\PackageError{fancyhdr}{Unknown parameter #2 in \string\fancyhdrsettoheight}{}\fi + \setbox\@tempboxa\hbox{{\f@nch@checkfalse\csname @#2\endcsname}}% + \setlength{#1}\f@nch@height + \setbox\@tempboxa\box\voidb@x +} +\let\f@nch@oddhead\fancyhdrsettoheight +\let\f@nch@evenhead\fancyhdrsettoheight +\let\f@nch@oddfoot\fancyhdrsettoheight +\let\f@nch@evenfoot\fancyhdrsettoheight +\newcommand\f@nch@vbox[2]{% + \setbox0\vbox{#2}% + \global\f@nch@height=\ht0 + \ifdim\ht0>#1\relax + \iff@nch@check + \dimen0=#1\advance\dimen0-\ht0 + \PackageWarning{fancyhdr}{% + \string#1 is too small (\the#1): \MessageBreak + Make it at least \the\ht0, for example:\MessageBreak + \string\setlength{\string#1}{\the\ht0}% + \iff@nch@compatViii .\MessageBreak + We now make it that large for the rest of the document.\MessageBreak + This may cause the page layout to be inconsistent, however + \fi + \ifx#1\headheight .\MessageBreak + You might also make \topmargin smaller:\MessageBreak + \string\addtolength{\string\topmargin}{\the\dimen0}% + \fi + \@gobble + }% + \iff@nch@compatViii + \dimen0=#1\relax + \global#1=\ht0\relax + \ht0=\dimen0 % + \else + \ht0=#1\relax + \fi + \else + \ht0=#1\relax + \fi + \fi + \box0} +\newcommand\f@nch@head[6]{% + \f@nch@reset + \ifdefined\UseHook\UseHook{fancyhdr/before}\UseHook{fancyhdr/head/begin}\fi + \f@nch@headinit\relax + #1% + \hbox to\headwidth{% + \f@nch@vbox\headheight{% + \f@nch@hfbox{#2}{#3}{#4}{#6}{h}% + \vskip\headruleskip\relax + \headrule + }% + }% + #5% + \ifdefined\UseHook\UseHook{fancyhdr/head/end}\UseHook{fancyhdr/after}\fi + \f@nch@restorepar +} +\newcommand\f@nch@foot[6]{% + \f@nch@reset + \ifdefined\UseHook\UseHook{fancyhdr/before}\UseHook{fancyhdr/foot/begin}\fi + \f@nch@footinit\relax + #1% + \hbox to\headwidth{% + \f@nch@vbox\footskip{% + \setbox0=\vbox{\footrule}\unvbox0 + \vskip\footruleskip + \f@nch@hfbox{#2}{#3}{#4}{#6}{f}% + \iff@nch@footalign \vskip\f@nch@footalignment \fi + }% + }% + #5% + \ifdefined\UseHook\UseHook{fancyhdr/foot/end}\UseHook{fancyhdr/after}\fi + \f@nch@restorepar +} +\newlength\f@nch@widthL +\newlength\f@nch@widthC +\newlength\f@nch@widthR +\newcommand\f@nch@hfbox[5]{% + \setlength\f@nch@widthL{\csname f@nch@width@#4l#5\endcsname}% + \setlength\f@nch@widthC{\csname f@nch@width@#4c#5\endcsname}% + \setlength\f@nch@widthR{\csname f@nch@width@#4r#5\endcsname}% + \let\@tempa\f@nch@hfbox@center + \ifdim \dimexpr \f@nch@widthL+\f@nch@widthC+\f@nch@widthR>\headwidth + \else + \ifdim \dimexpr \f@nch@widthL+0.5\f@nch@widthC>0.5\headwidth + \let \@tempa\f@nch@hfbox@fit + \fi + \ifdim \dimexpr \f@nch@widthR+0.5\f@nch@widthC>0.5\headwidth + \let \@tempa\f@nch@hfbox@fit + \fi + \fi + \@tempa{#1}{#2}{#3}#4#5% +} +\newcommand\f@nch@hfbox@center[5]{% + \hbox to \headwidth{% + \rlap{\f@nch@parbox{#1}\f@nch@widthL{#4}l{#5}}% + \hfill + \f@nch@parbox{#2}\f@nch@widthC{#4}c{#5}% + \hfill + \llap{\f@nch@parbox{#3}\f@nch@widthR{#4}r{#5}}% + }% +} +\newcommand\f@nch@hfbox@fit[5]{% + \hbox to \headwidth{% + \f@nch@parbox{#1}\f@nch@widthL{#4}l{#5}% + \hfill + \f@nch@parbox{#2}\f@nch@widthC{#4}c{#5}% + \hfill + \f@nch@parbox{#3}\f@nch@widthR{#4}r{#5}% + }% +}% +\newcommand\f@nch@parbox[5]{% + \expandafter\expandafter\expandafter\f@nch@parbox@align + \csname f@nch@align@#3#4#5\endcsname + \parbox[\f@nch@align@@v]{#2}% + {% + \f@nch@align@@pre + \f@nch@align@@h\leavevmode\ignorespaces#1% + \f@nch@align@@post + }% +} +\newcommand\f@nch@parbox@align[2]{% + \def\f@nch@align@@pre{}% + \def\f@nch@align@@post{}% + \csname f@nch@parbox@align@v#1\endcsname + \csname f@nch@parbox@align@h#2\endcsname +} +\def\f@nch@parbox@align@vT{\def\f@nch@align@@v{t}\def\f@nch@align@@pre{\vspace{0pt}}} +\def\f@nch@parbox@align@vt{\def\f@nch@align@@v{t}} +\def\f@nch@parbox@align@vc{\def\f@nch@align@@v{c}} +\def\f@nch@parbox@align@vb{\def\f@nch@align@@v{b}} +\def\f@nch@parbox@align@vB{\def\f@nch@align@@v{b}\def\f@nch@align@@post{\vspace{0pt}}} +\def\f@nch@parbox@align@hl{\def\f@nch@align@@h{\raggedright}} +\def\f@nch@parbox@align@hc{\def\f@nch@align@@h{\centering}} +\def\f@nch@parbox@align@hr{\def\f@nch@align@@h{\raggedleft}} +\def\f@nch@parbox@align@hj{\def\f@nch@align@@h{}} +\@ifundefined{@chapapp}{\let\@chapapp\chaptername}{}% +\def\f@nch@initialise{% + \@ifundefined{chapter}% + {\def\sectionmark##1{\markboth{\MakeUppercase{\ifnum \c@secnumdepth>\z@ + \thesection\hskip 1em\relax + \fi ##1}}{}}% + \def\subsectionmark##1{\markright {\ifnum \c@secnumdepth >\@ne + \thesubsection\hskip 1em\relax \fi ##1}}}% + {\def\chaptermark##1{\markboth {\MakeUppercase{\ifnum + \c@secnumdepth>\m@ne \@chapapp\ \thechapter. \ \fi ##1}}{}}% + \def\sectionmark##1{\markright{\MakeUppercase{\ifnum \c@secnumdepth >\z@ + \thesection. \ \fi ##1}}}% + }% + \def\headrule{{\if@fancyplain\let\headrulewidth\plainheadrulewidth\fi + \hrule\@height\headrulewidth\@width\headwidth + \vskip-\headrulewidth}}% + \def\footrule{{\if@fancyplain\let\footrulewidth\plainfootrulewidth\fi + \hrule\@width\headwidth\@height\footrulewidth}}% + \def\headrulewidth{0.4pt}% + \def\footrulewidth{0pt}% + \def\headruleskip{0pt}% + \def\footruleskip{0.3\normalbaselineskip}% + \fancyhf{}% + \if@twoside + \fancyhead[el,or]{\fancyplain{}{\slshape\rightmark}}% + \fancyhead[er,ol]{\fancyplain{}{\slshape\leftmark}}% + \else + \fancyhead[l]{\fancyplain{}{\slshape\rightmark}}% + \fancyhead[r]{\fancyplain{}{\slshape\leftmark}}% + \fi + \fancyfoot[c]{\rmfamily\thepage}% page number +} +\f@nch@initialise +\def\ps@f@nch@fancyproto{% + \ifdim\headwidth<0sp + \global\advance\headwidth123456789sp\global\advance\headwidth\textwidth + \fi + \gdef\ps@f@nch@fancyproto{\@fancyplainfalse\ps@f@nch@fancycore}% + \@fancyplainfalse\ps@f@nch@fancycore +}% +\@namedef{f@nch@ps@f@nch@fancyproto-is-fancyhdr}{} +\def\ps@fancy{\ps@f@nch@fancyproto} +\@namedef{f@nch@ps@fancy-is-fancyhdr}{} +\def\ps@fancyplain{\ps@f@nch@fancyproto \let\ps@plain\ps@plain@fancy} +\def\ps@plain@fancy{\@fancyplaintrue\ps@f@nch@fancycore} +\let\f@nch@ps@empty\ps@empty +\def\ps@f@nch@fancycore{% + \f@nch@ps@empty + \def\@mkboth{\protect\markboth}% + \def\f@nch@oddhead{\f@nch@head\f@nch@Oolh\f@nch@olh\f@nch@och\f@nch@orh\f@nch@Oorh{o}}% + \def\@oddhead{% + \iff@nch@twoside + \ifodd\c@page + \f@nch@oddhead + \else + \@evenhead + \fi + \else + \f@nch@oddhead + \fi + } + \def\f@nch@oddfoot{\f@nch@foot\f@nch@Oolf\f@nch@olf\f@nch@ocf\f@nch@orf\f@nch@Oorf{o}}% + \def\@oddfoot{% + \iff@nch@twoside + \ifodd\c@page + \f@nch@oddfoot + \else + \@evenfoot + \fi + \else + \f@nch@oddfoot + \fi + } + \def\@evenhead{\f@nch@head\f@nch@Oelh\f@nch@elh\f@nch@ech\f@nch@erh\f@nch@Oerh{e}}% + \def\@evenfoot{\f@nch@foot\f@nch@Oelf\f@nch@elf\f@nch@ecf\f@nch@erf\f@nch@Oerf{e}}% +} +\def\f@nch@Oolh{\if@reversemargin\hss\else\relax\fi} +\def\f@nch@Oorh{\if@reversemargin\relax\else\hss\fi} +\let\f@nch@Oelh\f@nch@Oorh +\let\f@nch@Oerh\f@nch@Oolh +\let\f@nch@Oolf\f@nch@Oolh +\let\f@nch@Oorf\f@nch@Oorh +\let\f@nch@Oelf\f@nch@Oelh +\let\f@nch@Oerf\f@nch@Oerh +\def\f@nch@offsolh{\headwidth=\textwidth\advance\headwidth\f@nch@offset@olh + \advance\headwidth\f@nch@offset@orh\hskip-\f@nch@offset@olh} +\def\f@nch@offselh{\headwidth=\textwidth\advance\headwidth\f@nch@offset@elh + \advance\headwidth\f@nch@offset@erh\hskip-\f@nch@offset@elh} +\def\f@nch@offsolf{\headwidth=\textwidth\advance\headwidth\f@nch@offset@olf + \advance\headwidth\f@nch@offset@orf\hskip-\f@nch@offset@olf} +\def\f@nch@offself{\headwidth=\textwidth\advance\headwidth\f@nch@offset@elf + \advance\headwidth\f@nch@offset@erf\hskip-\f@nch@offset@elf} +\def\f@nch@setoffs{% + \f@nch@gbl\let\headwidth\f@nch@headwidth + \f@nch@gbl\def\f@nch@Oolh{\f@nch@offsolh}% + \f@nch@gbl\def\f@nch@Oelh{\f@nch@offselh}% + \f@nch@gbl\def\f@nch@Oorh{\hss}% + \f@nch@gbl\def\f@nch@Oerh{\hss}% + \f@nch@gbl\def\f@nch@Oolf{\f@nch@offsolf}% + \f@nch@gbl\def\f@nch@Oelf{\f@nch@offself}% + \f@nch@gbl\def\f@nch@Oorf{\hss}% + \f@nch@gbl\def\f@nch@Oerf{\hss}% +} +\newif\iff@nch@footnote +\AtBeginDocument{% + \let\latex@makecol\@makecol + \def\@makecol{\ifvoid\footins\f@nch@footnotefalse\else\f@nch@footnotetrue\fi + \let\f@nch@topfloat\@toplist\let\f@nch@botfloat\@botlist\latex@makecol}% +} +\newcommand\iftopfloat[2]{\ifx\f@nch@topfloat\@empty #2\else #1\fi}% +\newcommand\ifbotfloat[2]{\ifx\f@nch@botfloat\@empty #2\else #1\fi}% +\newcommand\iffloatpage[2]{\if@fcolmade #1\else #2\fi}% +\newcommand\iffootnote[2]{\iff@nch@footnote #1\else #2\fi}% +\ifx\@temptokenb\undefined \csname newtoks\endcsname\@temptokenb\fi +\newif\iff@nch@pagestyle@star +\newcommand\fancypagestyle{% + \@ifstar{\f@nch@pagestyle@startrue\f@nch@pagestyle}% + {\f@nch@pagestyle@starfalse\f@nch@pagestyle}% +} +\newcommand\f@nch@pagestyle[1]{% + \@ifnextchar[{\f@nch@@pagestyle{#1}}{\f@nch@@pagestyle{#1}[f@nch@fancyproto]}% +} +\long\def\f@nch@@pagestyle#1[#2]#3{% + \@ifundefined{ps@#2}{% + \PackageError{fancyhdr}{\string\fancypagestyle: Unknown base page style `#2'}{}% + }{% + \@ifundefined{f@nch@ps@#2-is-fancyhdr}{% + \PackageError{fancyhdr}{\string\fancypagestyle: Base page style `#2' is not fancyhdr-based}{}% + }% + {% + \f@nch@pagestyle@setup + \def\temp@b{\@namedef{ps@#1}}% + \expandafter\temp@b\expandafter{\the\@temptokenb + \let\f@nch@gbl\relax\@nameuse{ps@#2}#3\relax}% + \@namedef{f@nch@ps@#1-is-fancyhdr}{}% + }% + }% +} +\newcommand\f@nch@pagestyle@setup{% + \iff@nch@pagestyle@star + \iff@nch@check\@temptokenb={\f@nch@checktrue}\else\@temptokenb={\f@nch@checkfalse}\fi + \@tfor\temp@a:= + \f@nch@olh\f@nch@och\f@nch@orh\f@nch@elh\f@nch@ech\f@nch@erh + \f@nch@olf\f@nch@ocf\f@nch@orf\f@nch@elf\f@nch@ecf\f@nch@erf + \f@nch@width@elh\f@nch@width@ech\f@nch@width@erh\f@nch@width@olh + \f@nch@width@och\f@nch@width@orh\f@nch@width@elf\f@nch@width@ecf + \f@nch@width@erf\f@nch@width@olf\f@nch@width@ocf\f@nch@width@orf + \f@nch@align@elh\f@nch@align@ech\f@nch@align@erh\f@nch@align@olh + \f@nch@align@och\f@nch@align@orh\f@nch@align@elf\f@nch@align@ecf + \f@nch@align@erf\f@nch@align@olf\f@nch@align@ocf\f@nch@align@orf + \f@nch@Oolh\f@nch@Oorh\f@nch@Oelh\f@nch@Oerh + \f@nch@Oolf\f@nch@Oorf\f@nch@Oelf\f@nch@Oerf + \f@nch@headinit\f@nch@footinit + \headrule\headrulewidth\footrule\footrulewidth + \do {% + \toks@=\expandafter\expandafter\expandafter{\temp@a}% + \toks@=\expandafter\expandafter\expandafter{% + \expandafter\expandafter\expandafter\def + \expandafter\expandafter\temp@a\expandafter{\the\toks@}}% + \edef\temp@b{\@temptokenb={\the\@temptokenb\the\toks@}}% + \temp@b + }% + \@tfor\temp@a:= + \f@nch@offset@olh\f@nch@offset@orh\f@nch@offset@elh\f@nch@offset@erh + \f@nch@offset@olf\f@nch@offset@orf\f@nch@offset@elf\f@nch@offset@erf + \do {% + \toks@=\expandafter\expandafter\expandafter{\expandafter\the\temp@a}% + \toks@=\expandafter\expandafter\expandafter{% + \expandafter\expandafter\expandafter\setlength + \expandafter\expandafter\temp@a\expandafter{\the\toks@}}% + \edef\temp@b{\@temptokenb={\the\@temptokenb\the\toks@}}% + \temp@b + }% + \else + \@temptokenb={}% + \fi +} +\newcommand\fancypagestyleassign[2]{% + \@ifundefined{ps@#2}{% + \PackageError{fancyhdr}{\string\fancypagestyleassign: Unknown page style `#2'}{}% + }{% + \expandafter\let + \csname ps@#1\expandafter\endcsname + \csname ps@#2\endcsname + \@ifundefined{f@nch@ps@#2-is-fancyhdr}{% + \expandafter\let\csname f@nch@ps@#1-is-fancyhdr\endcsname\@undefined + }{% + \@namedef{f@nch@ps@#1-is-fancyhdr}{}% + }% + }% +} +\fancypagestyle*{fancydefault}{\f@nch@initialise} +\def\f@nchdrbox@topstrut{\vrule height\ht\strutbox width\z@} +\def\f@nchdrbox@botstrut{\vrule depth\dp\strutbox width\z@} +\def\f@nchdrbox@nostrut{\noalign{\vspace{0pt}}\let\f@nchdrbox@@crstrut\f@nchdrbox@botstrut} +\NewDocumentCommand{\fancyhdrbox}{ O{cl} o m }{% +\begingroup + \let\f@nchdrbox@@pre\f@nchdrbox@topstrut + \let\f@nchdrbox@@postx\f@nchdrbox@botstrut + \let\f@nchdrbox@@posty\relax + \let\f@nchdrbox@@crstrut\strut + \IfNoValueTF{#2}% + {\let\f@nchdrbox@@halignto\@empty}% + {\setlength\@tempdima{#2}% + \def\f@nchdrbox@@halignto{to\@tempdima}}% + \def\@tempa{#1}% + \ifx\@tempa\@empty + \f@nchdrbox@align cl\@nil{#3}% + \else + \f@nchdrbox@align #1\@nil{#3}% + \fi +\endgroup +} +\protected\def\f@nchdrbox@cr{% + {\ifnum0=`}\fi\@ifstar\@f@nchdrbox@xcr\@f@nchdrbox@xcr} + +\def\@f@nchdrbox@xcr{% + \unskip\f@nchdrbox@@crstrut + \@ifnextchar[\@f@nchdrbox@argc{\ifnum0=`{\fi}\cr}% +} + +\def\@f@nchdrbox@argc[#1]{% + \ifnum0=`{\fi}% + \ifdim #1>\z@ + \unskip\@f@nchdrbox@xargc{#1}% + \else + \@f@nchdrbox@yargc{#1}% + \fi} + +\def\@f@nchdrbox@xargc#1{\@tempdima #1\advance\@tempdima \dp \strutbox + \vrule \@height\z@ \@depth\@tempdima \@width\z@ \cr} + +\def\@f@nchdrbox@yargc#1{\cr\noalign{\setlength\@tempdima{#1}\vskip\@tempdima}} +\def\f@nchdrbox@T{\let\f@nchdrbox@@pre\f@nchdrbox@nostrut + \f@nchdrbox@t} +\def\f@nchdrbox@t{\def\f@nchdrbox@@v{t}\def\f@nchdrbox@@h{l}} +\def\f@nchdrbox@c{\def\f@nchdrbox@@v{c}\def\f@nchdrbox@@h{c}} +\def\f@nchdrbox@b{\def\f@nchdrbox@@v{b}\def\f@nchdrbox@@h{l}} +\def\f@nchdrbox@B{\let\f@nchdrbox@@postx\relax + \def\f@nchdrbox@@posty{\vspace{0pt}}% + \f@nchdrbox@b} +\long\def\f@nchdrbox@align#1#2\@nil#3{% + \f@nch@ifin{#1}{TtcbB}{% + \@nameuse{f@nchdrbox@#1}% + \def\@tempa{#2}% + \ifx\@tempa\@empty\else \def\f@nchdrbox@@h{#2}\fi + }% + {\def\f@nchdrbox@@v{c}\def\f@nchdrbox@@h{#1}}% + \expandafter\f@nch@ifin\expandafter{\f@nchdrbox@@h}{lcr}{}% + {\PackageError{fancyhdr}{\string\fancyhdrbox: Illegal char `\f@nchdrbox@@h'\MessageBreak + in alignment argument}{}}% + \let\\\f@nchdrbox@cr + \setbox0=\if \f@nchdrbox@@v t\vtop + \else \vbox + \fi + {% + \ialign \f@nchdrbox@@halignto + \bgroup \relax + {\if \f@nchdrbox@@h l\hskip 1sp\else \hfil \fi + \ignorespaces ##\unskip + \if\f@nchdrbox@@h r\else \hfil \fi + }% + \tabskip\z@skip \cr + \f@nchdrbox@@pre + #3\unskip \f@nchdrbox@@postx + \crcr + \egroup + \f@nchdrbox@@posty + }% + \if\f@nchdrbox@@v c\@tempdima=\ht0\advance\@tempdima\dp0% + \ht0=0.5\@tempdima\dp0=0.5\@tempdima\fi + \leavevmode \box0 +} +\@ifclassloaded{newlfm} +{ + \let\ps@@empty\f@nch@ps@empty + \AtBeginDocument{% + \renewcommand{\@zfancyhead}[5]{\relax\hbox to\headwidth{\f@nch@reset + \@zfancyvbox\headheight{\hbox + {\rlap{\parbox[b]{\headwidth}{\raggedright\f@nch@olh}}\hfill + \parbox[b]{\headwidth}{\centering\f@nch@olh}\hfill + \llap{\parbox[b]{\headwidth}{\raggedleft\f@nch@orh}}}% + \zheadrule}}\relax}% + } +} +{} +\endinput +%% +%% End of file `fancyhdr.sty'. diff --git a/icml2026/figs/adaece_cifar_figure.pdf b/icml2026/figs/adaece_cifar_figure.pdf new file mode 100644 index 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/dev/null and b/icml2026/figs/vit_b_16_ece_reduced_error.pdf differ diff --git a/icml2026/icml2026.bib b/icml2026/icml2026.bib new file mode 100644 index 0000000000000000000000000000000000000000..5b7254aa228593fdea3d67f58ae7fa6bcc45de46 --- /dev/null +++ b/icml2026/icml2026.bib @@ -0,0 +1,1223 @@ +@misc{zwaan2019bridging, + title={Bridging the gap between uncertainty, confidence and diagnostic accuracy: calibration is key}, + author={Zwaan, Laura and Hautz, Wolf E}, + journal={BMJ quality \& safety}, + volume={28}, + number={5}, + pages={352--355}, + year={2019}, + publisher={BMJ Publishing Group Ltd} +} + + + + + +@inproceedings{luo2025beyond, + added-at = {2025-12-05T00:00:00.000+0100}, + author = {Luo, Haoyang and Tao, Linwei and Dong, Minjing and Xu, Chang}, + biburl = {https://www.bibsonomy.org/bibtex/2ff6f9571f60a72e2d6b71dcae9c0d0f9/dblp}, + booktitle = {ICML}, + ee = {https://openreview.net/forum?id=zDZkyvZqFR}, + interhash = {58681d87ddf6d793599ce25fe4bdffed}, + intrahash = {ff6f9571f60a72e2d6b71dcae9c0d0f9}, + keywords = {dblp}, + publisher = {OpenReview.net}, + timestamp = {2025-12-08T09:10:01.000+0100}, + title = {Beyond One-Hot Labels: Semantic Mixing for Model Calibration.}, + url = {http://dblp.uni-trier.de/db/conf/icml/icml2025.html#LuoTD025}, + year = 2025 +} + + +@inproceedings{tao2023dual, + title={Dual focal loss for calibration}, + author={Tao, Linwei and Dong, Minjing and Xu, Chang}, + booktitle={International Conference on Machine Learning}, + pages={33833--33849}, + year={2023}, + organization={PMLR} +} + +@inproceedings{gupta2021calibration, + title={Calibration of neural networks using splines}, + author={Gupta, Kartik and Rahimi, Amir and Ajanthan, Thalaiyasingam and Mensink, Thomas and Sminchisescu, Cristian and Hartley, Richard}, + booktitle={International Conference on Learning Representations}, + year={2021} +} + +@article{yang2023beyond, + title={Beyond probability partitions: Calibrating neural networks with semantic aware grouping}, + author={Yang, Jia-Qi and Zhan, De-Chuan and Gan, Le}, + journal={Advances in Neural Information Processing Systems}, + volume={36}, + pages={58448--58460}, + year={2023} +} +@inproceedings{hendrycks2019benchmarking, + title={Benchmarking Neural Network Robustness to Common Corruptions and Perturbations}, + author={Hendrycks, Dan and Dietterich, Thomas}, + booktitle={International Conference on Learning Representations (ICLR)}, + year={2019} +} +@inproceedings{kakade2004deterministic, + title={Deterministic calibration and Nash equilibrium}, + author={Kakade, Sham M and Foster, Dean P}, + booktitle={International Conference on Computational Learning Theory}, + pages={33--48}, + year={2004}, + organization={Springer} +} +@inproceedings{dosovitskiy2021an, + title={An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale}, + author={Dosovitskiy, Alexey and Beyer, Lucas and Kolesnikov, Alexander and Weissenborn, Dirk and Zhai, Xiaohua and Unterthiner, Thomas and Dehghani, Mostafa and Minderer, Matthias and Heigold, Georg and Gelly, Sylvain and Uszkoreit, Jakob and Houlsby, Neil}, + booktitle={International Conference on Learning Representations}, + year={2021} +} +@article{blasiok2023does, + title={When does optimizing a proper loss yield calibration?}, + author={Blasiok, Jaroslaw and Gopalan, Parikshit and Hu, Lunjia and Nakkiran, Preetum}, + journal={Advances in Neural Information Processing Systems}, + volume={36}, + pages={72071--72095}, + year={2023} +} +@inproceedings{liu2019large, + title={Large-scale long-tailed recognition in an open world}, + author={Liu, Ziwei and Miao, Zhongqi and Zhan, Xiaohang and Wang, Jiayun and Gong, Boqing and Yu, Stella X}, + booktitle={Proceedings of the IEEE/CVF conference on computer vision and pattern recognition}, + pages={2537--2546}, + year={2019} +} +@inproceedings{liu2021swin, + title={Swin transformer: Hierarchical vision transformer using shifted windows}, + author={Liu, Ze and Lin, Yutong and Cao, Yue and Hu, Han and Wei, Yixuan and Zhang, Zheng and Lin, Stephen and Guo, Baining}, + booktitle={Proceedings of the IEEE/CVF international conference on computer vision}, + pages={10012--10022}, + year={2021} +} +@article{paszke2019pytorch, + title={Pytorch: An imperative style, high-performance deep learning library}, + author={Paszke, Adam and Gross, Sam and Massa, Francisco and Lerer, Adam and Bradbury, James and Chanan, Gregory and Killeen, Trevor and Lin, Zeming and Gimelshein, Natalia and Antiga, Luca and others}, + journal={Advances in neural information processing systems}, + volume={32}, + year={2019} +} +@article{wang2019learning, + title={Learning robust global representations by penalizing local predictive power}, + author={Wang, Haohan and Ge, Songwei and Lipton, Zachary and Xing, Eric P}, + journal={Advances in neural information processing systems}, + volume={32}, + year={2019} +} +@article{xiong2024proximity, + title={Proximity-informed calibration for deep neural networks}, + author={Xiong, Miao and Deng, Ailin and Koh, Pang Wei W and Wu, Jiaying and Li, Shen and Xu, Jianqing and Hooi, Bryan}, + journal={Advances in Neural Information Processing Systems}, + volume={36}, + pages={68511--68538}, + year={2024} +} +@inproceedings{tao2025feature, + title={Feature clipping for uncertainty calibration}, + author={Tao, Linwei and Dong, Minjing and Xu, Chang}, + booktitle={Proceedings of the AAAI Conference on Artificial Intelligence}, + volume={39}, + number={19}, + pages={20841--20849}, + year={2025} +} + +@inproceedings{frenkel2021network, + title={Network calibration by class-based temperature scaling}, + author={Frenkel, Lior and Goldberger, Jacob}, + booktitle={2021 29th European Signal Processing Conference (EUSIPCO)}, + pages={1486--1490}, + year={2021}, + organization={IEEE} +} +@inproceedings{tomani2022parameterized, + title={Parameterized temperature scaling for boosting the expressive power in post-hoc uncertainty calibration}, + author={Tomani, Christian and Cremers, Daniel and Buettner, Florian}, + booktitle={European conference on computer vision}, + pages={555--569}, + year={2022}, + organization={Springer} +} +@article{goodfellow2014explaining, + title={Explaining and harnessing adversarial examples}, + author={Goodfellow, Ian J and Shlens, Jonathon and Szegedy, Christian}, + journal={arXiv preprint arXiv:1412.6572}, + year={2014} +} +@inproceedings{ding2021local, + title={Local Temperature Scaling for Probability Calibration}, + author={Ding, Zhipeng and Han, Xu and Liu, Peirong and Niethammer, Marc}, + booktitle={International Conference on Computer Vision}, + year={2021} +} +@inproceedings{hendrycksaugmix, + title={AugMix: A Simple Data Processing Method to Improve Robustness and Uncertainty}, + author={Hendrycks, Dan and Mu, Norman and Cubuk, Ekin Dogus and Zoph, Barret and Gilmer, Justin and Lakshminarayanan, Balaji}, + booktitle={International Conference on Learning Representations}, + year={2019} +} +@inproceedings{wei2022mitigating, + title={Mitigating neural network overconfidence with logit normalization}, + author={Wei, Hongxin and Xie, Renchunzi and Cheng, Hao and Feng, Lei and An, Bo and Li, Yixuan}, + booktitle={International conference on machine learning}, + pages={23631--23644}, + year={2022}, + organization={PMLR} +} + +@article{feng2019trust, + title={Can We Trust You? On Calibration of a Probabilistic Object Detector for Autonomous Driving}, + author={Di Feng and Lars Rosenbaum and Claudius Glaeser and Fabian Timm and Klaus C. J. Dietmayer}, + journal={ArXiv}, + year={2019}, + volume={abs/1909.12358}, + url={https://api.semanticscholar.org/CorpusID:203591605} +} + + +@inproceedings{caruana2015intelligible, + title={Intelligible models for healthcare: Predicting pneumonia risk and hospital 30-day readmission}, + author={Caruana, Rich and Lou, Yin and Gehrke, Johannes and Koch, Paul and Sturm, Marc and Elhadad, Noemie}, + booktitle={Proceedings of the 21th ACM SIGKDD international conference on knowledge discovery and data mining}, + pages={1721--1730}, + year={2015} +} +@article{cortes1995support, + title={Support-vector networks}, + author={Cortes, Corinna and Vapnik, Vladimir}, + journal={Machine learning}, + volume={20}, + number={3}, + pages={273--297}, + year={1995}, + publisher={Springer} +} +@article{hadsell2006dimensionality, + title={Dimensionality reduction by learning an invariant mapping}, + author={Hadsell, Raia and Chopra, Sumit and LeCun, Yann}, + journal={IEEE transactions on pattern analysis and machine intelligence}, + volume={28}, + number={11}, + pages={185--190}, + year={2006}, + publisher={IEEE} +} +@inproceedings{schroff2015facenet, + title={FaceNet: A unified embedding for face recognition and clustering}, + author={Schroff, Florian and Kalenichenko, Dmitry and Philbin, James}, + booktitle={Proceedings of the IEEE conference on computer vision and pattern recognition}, + pages={815--823}, + year={2015} +} +@article{weston2011wsabie, + title={Wsabie: Scaling up to large vocabulary image annotation}, + author={Weston, Jason and Bengio, Samy}, + journal={Journal of machine learning research}, + volume={12}, + number={Feb}, + pages={2177--2204}, + year={2011} +} +@article{gruber2022better, + title={Better uncertainty calibration via proper scores for classification and beyond}, + author={Gruber, Sebastian and Buettner, Florian}, + journal={Advances in Neural Information Processing Systems}, + volume={35}, + pages={8618--8632}, + year={2022} +} +@article{popordanoska2022consistent, + title={A consistent and differentiable lp canonical calibration error estimator}, + author={Popordanoska, Teodora and Sayer, Raphael and Blaschko, Matthew}, + journal={Advances in Neural Information Processing Systems}, + volume={35}, + pages={7933--7946}, + year={2022} +} +@article{lin2013network, + title={Network in network}, + author={Lin, Min and Chen, Qiang and Yan, Shuicheng}, + journal={arXiv preprint arXiv:1312.4400}, + year={2013} +} +@techreport{joachims1996probabilistic, + title={A Probabilistic Analysis of the Rocchio Algorithm with TFIDF for Text Categorization.}, + author={Joachims, Thorsten}, + year={1996}, + institution={Carnegie-mellon univ pittsburgh pa dept of computer science} +} +@misc{ +ghosh2022adafocal, +title={AdaFocal: Calibration-aware Adaptive Focal Loss}, +author={Arindam Ghosh and Thomas Schaaf and Matthew R. Gormley}, +year={2022}, +url={https://openreview.net/forum?id=CoMOKHYWf2} +} +@incollection{prechelt1998early, + title={Early stopping-but when?}, + author={Prechelt, Lutz}, + booktitle={Neural Networks: Tricks of the trade}, + pages={55--69}, + year={1998}, + publisher={Springer} +} + +@inproceedings{deng2009imagenet, + title={Imagenet: A large-scale hierarchical image database}, + author={Deng, Jia and Dong, Wei and Socher, Richard and Li, Li-Jia and Li, Kai and Fei-Fei, Li}, + booktitle={2009 IEEE conference on computer vision and pattern recognition}, + pages={248--255}, + year={2009}, + organization={Ieee} +} +@inproceedings{huang2017densely, + title={Densely connected convolutional networks}, + author={Huang, Gao and Liu, Zhuang and Van Der Maaten, Laurens and Weinberger, Kilian Q}, + booktitle={Proceedings of the IEEE conference on computer vision and pattern recognition}, + pages={4700--4708}, + year={2017} +} +@article{muller2019does, + title={When does label smoothing help?}, + author={M{\"u}ller, Rafael and Kornblith, Simon and Hinton, Geoffrey E}, + journal={Advances in neural information processing systems}, + volume={32}, + year={2019} +} + + +@article{brier1950verification, + title={VERIFICATION OF FORECASTS EXPRESSED IN TERMS OF PROBABILITY}, + author={Glenn W. Brier}, + journal={Monthly Weather Review}, + year={1950}, + volume={78}, + pages={1-3}, + url={https://api.semanticscholar.org/CorpusID:122906757} +} + + +@InProceedings{kumar2018trainable, + title = {Trainable Calibration Measures for Neural Networks from Kernel Mean Embeddings}, + author = {Kumar, Aviral and Sarawagi, Sunita and Jain, Ujjwal}, + booktitle = {Proceedings of the 35th International Conference on Machine Learning}, + pages = {2805--2814}, + year = {2018}, + editor = {Dy, Jennifer and Krause, Andreas}, + volume = {80}, + series = {Proceedings of Machine Learning Research}, + month = {10--15 Jul}, + publisher = {PMLR}, + pdf = {http://proceedings.mlr.press/v80/kumar18a/kumar18a.pdf}, + url = {https://proceedings.mlr.press/v80/kumar18a.html}, + 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.} +} + +@article{zagoruyko2016wide, + title={Wide residual networks}, + author={Zagoruyko, Sergey and Komodakis, Nikos}, + journal={arXiv preprint arXiv:1605.07146}, + year={2016} +} +@inproceedings{he2016deep, + title={Deep residual learning for image recognition}, + author={He, Kaiming and Zhang, Xiangyu and Ren, Shaoqing and Sun, Jian}, + booktitle={Proceedings of the IEEE conference on computer vision and pattern recognition}, + pages={770--778}, + year={2016} +} +@article{krizhevsky2009learning, + title={Learning multiple layers of features from tiny images}, + author={Krizhevsky, Alex and Hinton, Geoffrey and others}, + year={2009}, + publisher={Citeseer} +} +@inproceedings{szegedy2016rethinking, + title={Rethinking the inception architecture for computer vision}, + author={Szegedy, Christian and Vanhoucke, Vincent and Ioffe, Sergey and Shlens, Jon and Wojna, Zbigniew}, + booktitle={Proceedings of the IEEE conference on computer vision and pattern recognition}, + pages={2818--2826}, + year={2016} +} + +@article{jang2016categorical, + title={Categorical reparameterization with gumbel-softmax}, + author={Jang, Eric and Gu, Shixiang and Poole, Ben}, + journal={arXiv preprint arXiv:1611.01144}, + year={2016} +} +@incollection{miikkulainen2019evolving, + title={Evolving deep neural networks}, + author={Miikkulainen, Risto and Liang, Jason and Meyerson, Elliot and Rawal, Aditya and Fink, Daniel and Francon, Olivier and Raju, Bala and Shahrzad, Hormoz and Navruzyan, Arshak and Duffy, Nigel and others}, + booktitle={Artificial intelligence in the age of neural networks and brain computing}, + pages={293--312}, + year={2019}, + publisher={Elsevier} +} +@inproceedings{naeini2015obtaining, + title={Obtaining well calibrated probabilities using bayesian binning}, + author={Naeini, Mahdi Pakdaman and Cooper, Gregory and Hauskrecht, Milos}, + booktitle={Twenty-Ninth AAAI Conference on Artificial Intelligence}, + year={2015} +} +@InProceedings{pmlr-v119-li20c, + title = {Neural Architecture Search in A Proxy Validation Loss Landscape}, + author = {Li, Yanxi and Dong, Minjing and Wang, Yunhe and Xu, Chang}, + booktitle = {Proceedings of the 37th International Conference on Machine Learning}, + pages = {5853--5862}, + year = {2020}, + editor = {III, Hal Daumé and Singh, Aarti}, + volume = {119}, + series = {Proceedings of Machine Learning Research}, + month = {13--18 Jul}, + publisher = {PMLR}, + pdf = {http://proceedings.mlr.press/v119/li20c/li20c.pdf}, + url = {https://proceedings.mlr.press/v119/li20c.html}, + abstract = {This paper searches for the optimal neural architecture by minimizing a proxy of validation loss. Existing neural architecture search (NAS) methods used to discover the optimal neural architecture that best fits the validation examples given the up-to-date network weights. However, back propagation with a number of validation examples could be time consuming, especially when it needs to be repeated many times in NAS. Though these intermediate validation results are invaluable, they would be wasted if we cannot use them to predict the future from the past. In this paper, we propose to approximate the validation loss landscape by learning a mapping from neural architectures to their corresponding validate losses. The optimal neural architecture thus can be easily identified as the minimum of this proxy validation loss landscape. A novel sampling strategy is further developed for an efficient approximation of the loss landscape. Theoretical analysis indicates that the validation loss estimator learnt with our sampling strategy can reach a lower error rate and a lower label complexity compared with a uniform sampling. Experimental results on benchmarks demonstrate that the architecture searched by the proposed algorithm can achieve a satisfactory accuracy with less time cost.} +} + +@inproceedings{guo2017calibration, + title={On calibration of modern neural networks}, + author={Guo, Chuan and Pleiss, Geoff and Sun, Yu and Weinberger, Kilian Q}, + booktitle={International conference on machine learning}, + pages={1321--1330}, + year={2017}, + organization={PMLR} +} +@article{minderer2021revisiting, + title={Revisiting the calibration of modern neural networks}, + author={Minderer, Matthias and Djolonga, Josip and Romijnders, Rob and Hubis, Frances and Zhai, Xiaohua and Houlsby, Neil and Tran, Dustin and Lucic, Mario}, + journal={Advances in Neural Information Processing Systems}, + volume={34}, + pages={15682--15694}, + year={2021} +} +@article{bojarski2016end, + title={End to end learning for self-driving cars}, + author={Bojarski, Mariusz and Del Testa, Davide and Dworakowski, Daniel and Firner, Bernhard and Flepp, Beat and Goyal, Prasoon and Jackel, Lawrence D and Monfort, Mathew and Muller, Urs and Zhang, Jiakai and others}, + journal={arXiv preprint arXiv:1604.07316}, + year={2016} +} +@article{esteva2017dermatologist, + title={Dermatologist-level classification of skin cancer with deep neural networks}, + author={Esteva, Andre and Kuprel, Brett and Novoa, Roberto A and Ko, Justin and Swetter, Susan M and Blau, Helen M and Thrun, Sebastian}, + journal={nature}, + volume={542}, + number={7639}, + pages={115--118}, + year={2017}, + publisher={Nature Publishing Group} +} +@article{sonderby2020metnet, + title={Metnet: A neural weather model for precipitation forecasting}, + author={S{\o}nderby, Casper Kaae and Espeholt, Lasse and Heek, Jonathan and Dehghani, Mostafa and Oliver, Avital and Salimans, Tim and Agrawal, Shreya and Hickey, Jason and Kalchbrenner, Nal}, + journal={arXiv preprint arXiv:2003.12140}, + year={2020} +} +@article{degroot1983comparison, + title={The comparison and evaluation of forecasters}, + author={DeGroot, Morris H and Fienberg, Stephen E}, + journal={Journal of the Royal Statistical Society: Series D (The Statistician)}, + volume={32}, + number={1-2}, + pages={12--22}, + year={1983}, + publisher={Wiley Online Library} +} + +@article{goodfellow2013multi, + title={Multi-digit number recognition from street view imagery using deep convolutional neural networks}, + author={Goodfellow, Ian J and Bulatov, Yaroslav and Ibarz, Julian and Arnoud, Sacha and Shet, Vinay}, + journal={arXiv preprint arXiv:1312.6082}, + year={2013} +} +@inproceedings{ding2020revisiting, + title={Revisiting the evaluation of uncertainty estimation and its application to explore model complexity-uncertainty trade-off}, + author={Ding, Yukun and Liu, Jinglan and Xiong, Jinjun and Shi, Yiyu}, + booktitle={Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops}, + pages={4--5}, + year={2020} +} +@article{kumar2019verified, + title={Verified uncertainty calibration}, + author={Kumar, Ananya and Liang, Percy S and Ma, Tengyu}, + journal={Advances in Neural Information Processing Systems}, + volume={32}, + year={2019} +} +@inproceedings{nixon2019measuring, + title={Measuring Calibration in Deep Learning.}, + author={Nixon, Jeremy and Dusenberry, Michael W and Zhang, Linchuan and Jerfel, Ghassen and Tran, Dustin}, + booktitle={CVPR Workshops}, + volume={2}, + year={2019} +} +@article{hendrycks2018benchmarking, + title={Benchmarking neural network robustness to common corruptions and surface variations}, + author={Hendrycks, Dan and Dietterich, Thomas G}, + journal={arXiv preprint arXiv:1807.01697}, + year={2018} +} +@article{ovadia2019can, + title={Can you trust your model's uncertainty? evaluating predictive uncertainty under dataset shift}, + author={Ovadia, Yaniv and Fertig, Emily and Ren, Jie and Nado, Zachary and Sculley, David and Nowozin, Sebastian and Dillon, Joshua and Lakshminarayanan, Balaji and Snoek, Jasper}, + journal={Advances in neural information processing systems}, + volume={32}, + year={2019} +} +@inproceedings{wenger2020non, + title={Non-parametric calibration for classification}, + author={Wenger, Jonathan and Kjellstr{\"o}m, Hedvig and Triebel, Rudolph}, + booktitle={International Conference on Artificial Intelligence and Statistics}, + pages={178--190}, + year={2020}, + organization={PMLR} +} +@inproceedings{kull2017beta, + title={Beta calibration: a well-founded and easily implemented improvement on logistic calibration for binary classifiers}, + author={Kull, Meelis and Silva Filho, Telmo and Flach, Peter}, + booktitle={Artificial Intelligence and Statistics}, + pages={623--631}, + year={2017}, + organization={PMLR} +} + +@article{karandikar2021soft, + title={Soft calibration objectives for neural networks}, + author={Karandikar, Archit and Cain, Nicholas and Tran, Dustin and Lakshminarayanan, Balaji and Shlens, Jonathon and Mozer, Michael C and Roelofs, Becca}, + journal={Advances in Neural Information Processing Systems}, + volume={34}, + pages={29768--29779}, + year={2021} +} +@article{hui2020evaluation, + title={Evaluation of neural architectures trained with square loss vs cross-entropy in classification tasks}, + author={Hui, Like and Belkin, Mikhail}, + journal={arXiv preprint arXiv:2006.07322}, + year={2020} +} +@inproceedings{platt1999probabilistic, + abstract = {The output of a classifier should be a calibrated posterior probability to enable post-processing. Standard SVMs do not provide such probabilities. One method to create probabilities is to directly train a kernel classifier with a logit link function and a regularized maximum likelihood score. However, training with a maximum likelihood score will produce non-sparse kernel machines. Instead, we train an SVM, then train the parameters of an additional sigmoid function to map the SVM outputs into probabilities. This chapter compares classification error rate and likelihood scores for an SVM plus sigmoid versus a kernel method trained with a regularized likelihood error function. These methods are tested on three data-mining-style data sets. The SVM+sigmoid yields probabilities of comparable quality to the regularized maximum likelihood kernel method, while still retaining the sparseness of the SVM.}, + added-at = {2012-07-13T11:59:27.000+0200}, + author = {Platt, John C.}, + biburl = {https://www.bibsonomy.org/bibtex/21a15c25b220dac6b6f222c18fc7e91d2/jabreftest}, + booktitle = {Advances in Large Margin Classifiers}, + citeseerurl = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.41.1639}, + file = {Platt1999.pdf:1999/Platt1999.pdf:PDF}, + groups = {public}, + interhash = {c5df9f9137085cad9cafce3c347b2508}, + intrahash = {1a15c25b220dac6b6f222c18fc7e91d2}, + keywords = {}, + pages = {61--74}, + publisher = {MIT Press}, + timestamp = {2012-07-13T11:59:27.000+0200}, + title = {Probabilistic Outputs for Support Vector Machines and Comparisons to Regularized Likelihood Methods}, + username = {jabreftest}, + year = 1999 +} + + +@inproceedings{chen2020simple, + title={A simple framework for contrastive learning of visual representations}, + author={Chen, Ting and Kornblith, Simon and Norouzi, Mohammad and Hinton, Geoffrey}, + booktitle={International conference on machine learning}, + pages={1597--1607}, + year={2020}, + organization={PMLR} +} +@article{hinton2015distilling, + title={Distilling the knowledge in a neural network}, + author={Hinton, Geoffrey and Vinyals, Oriol and Dean, Jeff and others}, + journal={arXiv preprint arXiv:1503.02531}, + volume={2}, + number={7}, + year={2015} +} + +@article{zadrozny2001obtaining, + title={Obtaining calibrated probability estimates from decision trees and naive bayesian classifiers}, + author={Zadrozny, Bianca and Elkan, Charles}, + journal={18th International Conferrence on Machine Learning}, + year={2001} +} + +@inproceedings{zadrozny2002transforming, + title={Transforming classifier scores into accurate multiclass probability estimates}, + author={Zadrozny, Bianca and Elkan, Charles}, + booktitle={Proceedings of the eighth ACM SIGKDD international conference on Knowledge discovery and data mining}, + pages={694--699}, + year={2002} +} +@article{nguyen2015posterior, + title={Posterior calibration and exploratory analysis for natural language processing models}, + author={Nguyen, Khanh and O'Connor, Brendan}, + journal={arXiv preprint arXiv:1508.05154}, + year={2015} +} +@article{kull2019beyond, + title={Beyond temperature scaling: Obtaining well-calibrated multi-class probabilities with dirichlet calibration}, + author={Kull, Meelis and Perello Nieto, Miquel and K{\"a}ngsepp, Markus and Silva Filho, Telmo and Song, Hao and Flach, Peter}, + journal={Advances in neural information processing systems}, + volume={32}, + year={2019} +} +@inproceedings{vaicenavicius2019evaluating, + title={Evaluating model calibration in classification}, + author={Vaicenavicius, Juozas and Widmann, David and Andersson, Carl and Lindsten, Fredrik and Roll, Jacob and Sch{\"o}n, Thomas}, + booktitle={The 22nd International Conference on Artificial Intelligence and Statistics}, + pages={3459--3467}, + year={2019}, + organization={PMLR} +} +@article{widmann2019calibration, + title={Calibration tests in multi-class classification: A unifying framework}, + author={Widmann, David and Lindsten, Fredrik and Zachariah, Dave}, + journal={Advances in Neural Information Processing Systems}, + volume={32}, + year={2019} +} + +@inproceedings{roelofs2022mitigating, + title={Mitigating bias in calibration error estimation}, + author={Roelofs, Rebecca and Cain, Nicholas and Shlens, Jonathon and Mozer, Michael C}, + booktitle={International Conference on Artificial Intelligence and Statistics}, + pages={4036--4054}, + year={2022}, + organization={PMLR} +} +@article{wang2022yolov7, + title={YOLOv7: Trainable bag-of-freebies sets new state-of-the-art for real-time object detectors}, + author={Wang, Chien-Yao and Bochkovskiy, Alexey and Liao, Hong-Yuan Mark}, + journal={arXiv preprint arXiv:2207.02696}, + year={2022} +} +@inproceedings{wortsman2022model, + title={Model soups: averaging weights of multiple fine-tuned models improves accuracy without increasing inference time}, + author={Wortsman, Mitchell and Ilharco, Gabriel and Gadre, Samir Ya and Roelofs, Rebecca and Gontijo-Lopes, Raphael and Morcos, Ari S and Namkoong, Hongseok and Farhadi, Ali and Carmon, Yair and Kornblith, Simon and others}, + booktitle={International Conference on Machine Learning}, + pages={23965--23998}, + year={2022}, + organization={PMLR} +} +@article{yu2022coca, + title={Coca: Contrastive captioners are image-text foundation models}, + author={Yu, Jiahui and Wang, Zirui and Vasudevan, Vijay and Yeung, Legg and Seyedhosseini, Mojtaba and Wu, Yonghui}, + journal={arXiv preprint arXiv:2205.01917}, + year={2022} +} +@inproceedings{brock2021high, + title={High-performance large-scale image recognition without normalization}, + author={Brock, Andy and De, Soham and Smith, Samuel L and Simonyan, Karen}, + booktitle={International Conference on Machine Learning}, + pages={1059--1071}, + year={2021}, + organization={PMLR} +} +@inproceedings{pham2021meta, + title={Meta pseudo labels}, + author={Pham, Hieu and Dai, Zihang and Xie, Qizhe and Le, Quoc V}, + booktitle={Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition}, + pages={11557--11568}, + year={2021} +} +@article{wang2021you, + title={You only learn one representation: Unified network for multiple tasks}, + author={Wang, Chien-Yao and Yeh, I-Hau and Liao, Hong-Yuan Mark}, + journal={arXiv preprint arXiv:2105.04206}, + year={2021} +} +@inproceedings{qiao2021detectors, + title={Detectors: Detecting objects with recursive feature pyramid and switchable atrous convolution}, + author={Qiao, Siyuan and Chen, Liang-Chieh and Yuille, Alan}, + booktitle={Proceedings of the IEEE/CVF conference on computer vision and pattern recognition}, + pages={10213--10224}, + year={2021} +} +@article{zhou2021probabilistic, + title={Probabilistic two-stage detection}, + author={Zhou, Xingyi and Koltun, Vladlen and Kr{\"a}henb{\"u}hl, Philipp}, + journal={arXiv preprint arXiv:2103.07461}, + year={2021} +} +@article{gupta2020calibration, + title={Calibration of neural networks using splines}, + author={Gupta, Kartik and Rahimi, Amir and Ajanthan, Thalaiyasingam and Mensink, Thomas and Sminchisescu, Cristian and Hartley, Richard}, + journal={arXiv preprint arXiv:2006.12800}, + year={2020} +} +@article{krishnan2020improving, + title={Improving model calibration with accuracy versus uncertainty optimization}, + author={Krishnan, Ranganath and Tickoo, Omesh}, + journal={Advances in Neural Information Processing Systems}, + volume={33}, + pages={18237--18248}, + year={2020} +} +@article{mukhoti2020calibrating, + title={Calibrating deep neural networks using focal loss}, + author={Mukhoti, Jishnu and Kulharia, Viveka and Sanyal, Amartya and Golodetz, Stuart and Torr, Philip and Dokania, Puneet}, + journal={Advances in Neural Information Processing Systems}, + volume={33}, + pages={15288--15299}, + year={2020} +} + +@inproceedings{lin2017focal, + title={Focal loss for dense object detection}, + author={Lin, Tsung-Yi and Goyal, Priya and Girshick, Ross and He, Kaiming and Doll{\'a}r, Piotr}, + booktitle={Proceedings of the IEEE international conference on computer vision}, + pages={2980--2988}, + year={2017} +} +@article{thulasidasan2019mixup, + title={On mixup training: Improved calibration and predictive uncertainty for deep neural networks}, + author={Thulasidasan, Sunil and Chennupati, Gopinath and Bilmes, Jeff A and Bhattacharya, Tanmoy and Michalak, Sarah}, + journal={Advances in Neural Information Processing Systems}, + volume={32}, + year={2019} +} +@article{hendrycks2019augmix, + title={Augmix: A simple data processing method to improve robustness and uncertainty}, + author={Hendrycks, Dan and Mu, Norman and Cubuk, Ekin D and Zoph, Barret and Gilmer, Justin and Lakshminarayanan, Balaji}, + journal={arXiv preprint arXiv:1912.02781}, + year={2019} +} +@article{lakshminarayanan2017simple, + title={Simple and scalable predictive uncertainty estimation using deep ensembles}, + author={Lakshminarayanan, Balaji and Pritzel, Alexander and Blundell, Charles}, + journal={Advances in neural information processing systems}, + volume={30}, + year={2017} +} +@article{wen2020batchensemble, + title={Batchensemble: an alternative approach to efficient ensemble and lifelong learning}, + author={Wen, Yeming and Tran, Dustin and Ba, Jimmy}, + journal={arXiv preprint arXiv:2002.06715}, + year={2020} +} +@inproceedings{dusenberry2020efficient, + title={Efficient and scalable bayesian neural nets with rank-1 factors}, + author={Dusenberry, Michael and Jerfel, Ghassen and Wen, Yeming and Ma, Yian and Snoek, Jasper and Heller, Katherine and Lakshminarayanan, Balaji and Tran, Dustin}, + booktitle={International conference on machine learning}, + pages={2782--2792}, + year={2020}, + organization={PMLR} +} +@article{bai2021understanding, + title={Understanding and improving early stopping for learning with noisy labels}, + author={Bai, Yingbin and Yang, Erkun and Han, Bo and Yang, Yanhua and Li, Jiatong and Mao, Yinian and Niu, Gang and Liu, Tongliang}, + journal={Advances in Neural Information Processing Systems}, + volume={34}, + pages={24392--24403}, + year={2021} +} + + + + + +@inproceedings{lopez2017gradient, + title={Gradient episodic memory for continual learning}, + author={Lopez-Paz, David and Ranzato, Marc'Aurelio}, + booktitle={Advances in neural information processing systems}, + year={2017} +} + +@inproceedings{AGEM, + title={Efficient Lifelong Learning with A-GEM}, + author={Chaudhry, Arslan and Ranzato, Marc’Aurelio and Rohrbach, Marcus and Elhoseiny, Mohamed}, + booktitle={International Conference on Learning Representations}, + year={2018} +} + +@article{guo2019learning, + title={Learning with Long-term Remembering: Following the Lead of Mixed Stochastic Gradient}, + author={Guo, Yunhui and Liu, Mingrui and Yang, Tianbao and Rosing, Tajana}, + journal={arXiv preprint arXiv:1909.11763}, + year={2019} +} + +@article{riemer2018learning, + title={Learning to learn without forgetting by maximizing transfer and minimizing interference}, + author={Riemer, Matthew and Cases, Ignacio and Ajemian, Robert and Liu, Miao and Rish, Irina and Tu, Yuhai and Tesauro, Gerald}, + journal={arXiv preprint arXiv:1810.11910}, + year={2018} +} + +@article{kirkpatrick2017overcoming, + title={Overcoming catastrophic forgetting in neural networks}, + author={Kirkpatrick, James and Pascanu, Razvan and Rabinowitz, Neil and Veness, Joel and Desjardins, Guillaume and Rusu, Andrei A and Milan, Kieran and Quan, John and Ramalho, Tiago and Grabska-Barwinska, Agnieszka and others}, + journal={Proceedings of the national academy of sciences}, + year={2017}, +} + +@article{zenke2017continual, + title={Continual learning through synaptic intelligence}, + author={Zenke, Friedemann and Poole, Ben and Ganguli, Surya}, + journal={Proceedings of machine learning research}, + year={2017}, +} + +@article{wah2011caltech, + title={The caltech-ucsd birds-200-2011 dataset}, + author={Wah, Catherine and Branson, Steve and Welinder, Peter and Perona, Pietro and Belongie, Serge}, + year={2011}, + publisher={California Institute of Technology} +} + +@inproceedings{lampert2009learning, + title={Learning to detect unseen object classes by between-class attribute transfer}, + author={Lampert, Christoph H and Nickisch, Hannes and Harmeling, Stefan}, + booktitle={2009 IEEE Conference on Computer Vision and Pattern Recognition}, + year={2009}, +} + +@article{french1999catastrophic, + title={Catastrophic forgetting in connectionist networks}, + author={French, Robert M}, + journal={Trends in cognitive sciences}, + year={1999}, +} + +@inproceedings{ioffe2015batch, + title={Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift}, + author={Ioffe, Sergey and Szegedy, Christian}, + booktitle={International Conference on Machine Learning}, + year={2015} +} + +@inproceedings{chaudhry2018riemannian, + title={Riemannian walk for incremental learning: Understanding forgetting and intransigence}, + author={Chaudhry, Arslan and Dokania, Puneet K and Ajanthan, Thalaiyasingam and Torr, Philip HS}, + booktitle={Proceedings of the European Conference on Computer Vision (ECCV)}, + year={2018} +} + +@article{de2019continual, + title={Continual learning: A comparative study on how to defy forgetting in classification tasks}, + author={De Lange, Matthias and Aljundi, Rahaf and Masana, Marc and Parisot, Sarah and Jia, Xu and Leonardis, Ales and Slabaugh, Gregory and Tuytelaars, Tinne}, + journal={arXiv preprint arXiv:1909.08383}, + year={2019} +} +@inproceedings{li2016learning, + author = {Li, Zhizhong and Hoiem, Derek}, + title = {Learning Without Forgetting}, + booktitle = {European Conference on Computer Vision}, + year = {2016}, +} + +@inproceedings{Chaudhry2018, + author = {Chaudhry, Arslan and Dokania, Puneet K and Ajanthan, Thalaiyasingam and Torr, Philip HS}, + title = {Riemannian walk for incremental learning: Understanding forgetting and intransigence}, + booktitle = {Proceedings of the European Conference on Computer Vision (ECCV)}, + year = {2018}, +} + +@inproceedings{dhar2019learning, + author = {Dhar, Prithviraj and Singh, Rajat Vikram and Peng, Kuan-Chuan and Wu, Ziyan and Chellappa, Rama}, + title = {Learning without memorizing}, + booktitle = {Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition}, + year = {2019}, +} + +@inproceedings{mallya2018piggyback, + author = {Mallya, Arun and Davis, Dillon and Lazebnik, Svetlana}, + title = {Piggyback: Adapting a single network to multiple tasks by learning to mask weights}, + booktitle = {Proceedings of the European Conference on Computer Vision (ECCV)}, + year = {2018}, +} + +@article{yoon2017lifelong, + author = {Yoon, Jaehong and Yang, Eunho and Lee, Jeongtae and Hwang, Sung Ju}, + title = {Lifelong learning with dynamically expandable networks}, + journal = {arXiv preprint arXiv:1708.01547}, + year = {2017}, +} + +@article{lin2019multi, + author = {Lin, Jiabin and Liu, Hai-Lin and Xue, Bing and Zhang, Mengjie and Gu, Fangqing}, + title = {Multi-objective Multi-tasking Optimization Based on Incremental Learning}, + journal = {IEEE Transactions on Evolutionary Computation}, + year = {2019}, +} + +@inproceedings{lin2019pareto, + author = {Lin, Xi and Zhen, Hui-Ling and Li, Zhenhua and Zhang, Qing-Fu and Kwong, Sam}, + title = {Pareto Multi-Task Learning}, + booktitle = {Advances in Neural Information Processing Systems}, + year = {2019}, +} + +@inproceedings{sener2018multi, + author = {Sener, Ozan and Koltun, Vladlen}, + title = {Multi-task learning as multi-objective optimization}, + booktitle = {Advances in Neural Information Processing Systems}, + year = {2018}, +} + +@article{ring1997child, + author = {Ring, Mark B}, + title = {CHILD: A first step towards continual learning}, + journal = {Machine Learning}, + year = {1997}, +} + +@inproceedings{thrun1998lifelong, + author = {Thrun, Sebastian}, + title = {Lifelong learning algorithms}, + booktitle = {Learning to learn}, + publisher = {Springer}, + year = {1998}, +} + +@inproceedings{aljundi2018memory, + title={Memory aware synapses: Learning what (not) to forget}, + author={Aljundi, Rahaf and Babiloni, Francesca and Elhoseiny, Mohamed and Rohrbach, Marcus and Tuytelaars, Tinne}, + booktitle={Proceedings of the European Conference on Computer Vision (ECCV)}, + year={2018} +} + +@article{farajtabar2019orthogonal, + author = {Farajtabar, Mehrdad and Azizan, Navid and Mott, Alex and Li, Ang}, + title = {Orthogonal Gradient Descent for Continual Learning}, + journal = {arXiv preprint arXiv:1910.07104}, + year = {2019}, +} + +@article{tetko1995neural, + author = {Tetko, Igor V and Livingstone, David J and Luik, Alexander I}, + title = {Neural network studies. 1. Comparison of overfitting and overtraining}, + journal = {Journal of chemical information and computer sciences}, + year = {1995}, +} + +@article{dietterich1995overfitting, + author = {Dietterich, Tom}, + title = {Overfitting and undercomputing in machine learning}, + journal = {ACM computing surveys (CSUR)}, + year = {1995}, +} + +@inproceedings{castro2018end, + author = {Castro, Francisco M and Mar{\'\i}n-Jim{\'e}nez, Manuel J and Guil, Nicol{\'a}s and Schmid, Cordelia and Alahari, Karteek}, + title = {End-to-end incremental learning}, + booktitle = {Proceedings of the European Conference on Computer Vision (ECCV)}, + year = {2018}, +} + +@InProceedings{belouadah2019il2m, + author = {Belouadah, Eden and Popescu, Adrian}, + title = {Il2m: Class incremental learning with dual memory}, + booktitle = {Proceedings of the IEEE International Conference on Computer Vision}, + year = {2019}, +} + +@InProceedings{hou2018lifelong, + author = {Hou, Saihui and Pan, Xinyu and Change Loy, Chen and Wang, Zilei and Lin, Dahua}, + title = {Lifelong learning via progressive distillation and retrospection}, + booktitle = {Proceedings of the European Conference on Computer Vision (ECCV)}, + year = {2018}, +} + +@Article{maaten2008visualizing, + author = {Maaten, Laurens van der and Hinton, Geoffrey}, + title = {Visualizing data using t-SNE}, + journal = {Journal of machine learning research}, + year = {2008}, +} + +@Article{chaudhry2019on, + title={On tiny episodic memories in continual learning}, + author={Chaudhry, Arslan and Rohrbach, Marcus and Elhoseiny, Mohamed and Ajanthan, Thalaiyasingam and Dokania, Puneet K and Torr, Philip HS and Ranzato, Marc'Aurelio}, + journal={arXiv preprint arXiv:1902.10486}, + year={2019} +} + +@inproceedings{rolnick2018experience, + title={Experience replay for continual learning}, + author={Rolnick, David and Ahuja, Arun and Schwarz, Jonathan and Lillicrap, Timothy and Wayne, Gregory}, + booktitle={Advances in Neural Information Processing Systems}, + year={2019} +} + +@inproceedings{deng2018arcface, + title={Arcface: Additive angular margin loss for deep face recognition}, + author={Deng, Jiankang and Guo, Jia and Xue, Niannan and Zafeiriou, Stefanos}, + booktitle={Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition}, + year={2019} +} + +@inproceedings{liu2016large-margin, + title={Large-margin softmax loss for convolutional neural networks.}, + author={Liu, Weiyang and Wen, Yandong and Yu, Zhiding and Yang, Meng}, + booktitle={ICML}, + year={2016} +} + +@InProceedings{Yang2014, + author = {Yongxin Yang and Timothy M. Hospedales}, + booktitle = {International Conference on Learning Representations}, + title = {A Unified Perspective on Multi-Domain and Multi-Task Learning}, + year = {2014}, +} + +@InProceedings{DBLP:conf/icml/LiuWYY16, + author = {Weiyang Liu and Yandong Wen and Zhiding Yu and Meng Yang}, + booktitle = {Proceedings of the 33nd International Conference on Machine Learning, {ICML} 2016, New York City, NY, USA, June 19-24, 2016}, + title = {Large-Margin Softmax Loss for Convolutional Neural Networks}, + year = {2016}, +} + +@inproceedings{nam2015, + title={Learning multi-domain convolutional neural networks for visual tracking}, + author={Nam, Hyeonseob and Han, Bohyung}, + booktitle={Proceedings of the IEEE conference on computer vision and pattern recognition}, + year={2016} +} + +@InProceedings{Tang2020, + author = {Hui Tang and Kui Jia}, + booktitle = {The Thirty-Fourth {AAAI} Conference on Artificial Intelligence, {AAAI} 2020, The Thirty-Second Innovative Applications of Artificial Intelligence Conference, {IAAI} 2020, The Tenth {AAAI} Symposium on Educational Advances in Artificial Intelligence, {EAAI} 2020, New York, NY, USA, February 7-12, 2020}, + title = {Discriminative Adversarial Domain Adaptation}, + year = {2020}, +} + +@Article{Rebuffi2016, + author = {Sylvestre{-}Alvise Rebuffi and Alexander Kolesnikov and Christoph H. Lampert}, + journal = {CoRR}, + title = {iCaRL: Incremental Classifier and Representation Learning}, + year = {2016}, +} + +@Article{Pellegrini2019, + title={Latent replay for real-time continual learning}, + author={Pellegrini, Lorenzo and Graffieti, Gabrile and Lomonaco, Vincenzo and Maltoni, Davide}, + journal={arXiv preprint arXiv:1912.01100}, + year={2019} +} + +@Article{Ven2018, + title={Generative replay with feedback connections as a general strategy for continual learning}, + author={van de Ven, Gido M and Tolias, Andreas S}, + journal={arXiv preprint arXiv:1809.10635}, + year={2018} +} + +@Article{Shen2020, + title={Generative Feature Replay with Orthogonal Weight Modification for Continual Learning}, + author={Shen, Gehui and Zhang, Song and Chen, Xiang and Deng, Zhi-Hong}, + journal={arXiv preprint arXiv:2005.03490}, + year={2020} +} + +@inproceedings{Lesort2018, + title={Marginal replay vs conditional replay for continual learning}, + author={Lesort, Timoth{\'e}e and Gepperth, Alexander and Stoian, Andrei and Filliat, David}, + booktitle={International Conference on Artificial Neural Networks}, + year={2019}, +} + +@InProceedings{DBLP:conf/cvpr/LiuWYLRS17, + author = {Weiyang Liu and Yandong Wen and Zhiding Yu and Ming Li and Bhiksha Raj and Le Song}, + booktitle = {2017 {IEEE} Conference on Computer Vision and Pattern Recognition, {CVPR} 2017, Honolulu, HI, USA, July 21-26, 2017}, + title = {SphereFace: Deep Hypersphere Embedding for Face Recognition}, + year = {2017}, +} + +@InProceedings{DBLP:conf/cvpr/WangWZJGZL018, + author = {Hao Wang and Yitong Wang and Zheng Zhou and Xing Ji and Dihong Gong and Jingchao Zhou and Zhifeng Li and Wei Liu}, + booktitle = {2018 {IEEE} Conference on Computer Vision and Pattern Recognition, {CVPR} 2018, Salt Lake City, UT, USA, June 18-22, 2018}, + title = {CosFace: Large Margin Cosine Loss for Deep Face Recognition}, + year = {2018}, +} + +@Article{DBLP:journals/spl/WangCLL18, + author = {Feng Wang and Jian Cheng and Weiyang Liu and Haijun Liu}, + journal = {{IEEE} Signal Process. Lett.}, + title = {Additive Margin Softmax for Face Verification}, + year = {2018}, +} + +@Article{Hinton2015, + author = {Geoffrey E. Hinton and Oriol Vinyals and Jeffrey Dean}, + journal = {CoRR}, + title = {Distilling the Knowledge in a Neural Network}, + year = {2015}, +} + +@Article{Chaudhry2019, + author = {Arslan Chaudhry and Marcus Rohrbach and Mohamed Elhoseiny and Thalaiyasingam Ajanthan and Puneet K. Dokania and Philip H. S. Torr and Marc'Aurelio Ranzato}, + title = {On Tiny Episodic Memories in Continual Learning}, + journal={arXiv preprint arXiv:1902.10486}, + year={2019} +} + +@Article{Peng2016, + author = {Nanyun Peng and Mark Dredze}, + journal = {CoRR}, + title = {Multi-task Multi-domain Representation Learning for Sequence Tagging}, + year = {2016}, +} + +@Article{Fourure2017, + author = {Damien Fourure and R{\'{e}}mi Emonet and {\'{E}}lisa Fromont and Damien Muselet and Natalia Neverova and Alain Tr{\'{e}}meau and Christian Wolf}, + journal = {Neurocomputing}, + title = {Multi-task, multi-domain learning: Application to semantic segmentation and pose regression}, + year = {2017}, +} + +@InProceedings{deng2018visual, + author = {Deng, Chaorui and Wu, Qi and Wu, Qingyao and Hu, Fuyuan and Lyu, Fan and Tan, Mingkui}, + booktitle = {Proceedings of the IEEE conference on computer vision and pattern recognition}, + title = {Visual grounding via accumulated attention}, + year = {2018}, + pages = {7746--7755}, +} + +@Article{lyu2019attend, + author = {Lyu, Fan and Wu, Qi and Hu, Fuyuan and Wu, Qingyao and Tan, Mingkui}, + journal = {IEEE Transactions on Multimedia}, + title = {Attend and imagine: Multi-label image classification with visual attention and recurrent neural networks}, + year = {2019}, + pages = {1971--1981}, + publisher = {IEEE}, +} + +@Article{lyu2020vtgraphnet, + author = {Lyu, Fan and Feng, Wei and Wang, Song}, + journal = {Neurocomputing}, + title = {vtGraphNet: Learning weakly-supervised scene graph for complex visual grounding}, + year = {2020}, + pages = {51--60}, + publisher = {Elsevier}, +} +@article{bohdal2021meta, + title={Meta-calibration: Meta-learning of model calibration using differentiable expected calibration error}, + author={Bohdal, Ondrej and Yang, Yongxin and Hospedales, Timothy}, + journal={arXiv preprint arXiv:2106.09613}, + year={2021} +} +@article{wang2020uncertainty, + title={Uncertainty Estimation and Calibration with Finite-State Probabilistic RNNs}, + author={Wang, Cheng and Lawrence, Carolin and Niepert, Mathias}, + journal={arXiv preprint arXiv:2011.12010}, + year={2020} +} + +@article{bartlett2006convexity, + title={Convexity, classification, and risk bounds}, + author={Bartlett, Peter L and Jordan, Michael I and McAuliffe, Jon D}, + journal={Journal of the American Statistical Association}, + volume={101}, + number={473}, + pages={138--156}, + year={2006}, + publisher={Taylor \& Francis} +} + +@article{tewari2007consistency, + title={On the Consistency of Multiclass Classification Methods.}, + author={Tewari, Ambuj and Bartlett, Peter L}, + journal={Journal of Machine Learning Research}, + volume={8}, + number={5}, + year={2007} +} + +@inproceedings{charoenphakdee2021focal, + title={On focal loss for class-posterior probability estimation: A theoretical perspective}, + author={Charoenphakdee, Nontawat and Vongkulbhisal, Jayakorn and Chairatanakul, Nuttapong and Sugiyama, Masashi}, + booktitle={Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition}, + pages={5202--5211}, + year={2021} +} + +@article{wang2021rethinking, + title={Rethinking calibration of deep neural networks: Do not be afraid of overconfidence}, + author={Wang, Deng-Bao and Feng, Lei and Zhang, Min-Ling}, + journal={Advances in Neural Information Processing Systems}, + volume={34}, + pages={11809--11820}, + year={2021} +} + +@inproceedings{tian2019fcos, + title={Fcos: Fully convolutional one-stage object detection}, + author={Tian, Zhi and Shen, Chunhua and Chen, Hao and He, Tong}, + booktitle={Proceedings of the IEEE/CVF international conference on computer vision}, + pages={9627--9636}, + year={2019} +} + +@inproceedings{he2017mask, + title={Mask r-cnn}, + author={He, Kaiming and Gkioxari, Georgia and Doll{\'a}r, Piotr and Girshick, Ross}, + booktitle={Proceedings of the IEEE international conference on computer vision}, + pages={2961--2969}, + year={2017} +} + +@inproceedings{long2015fully, + title={Fully convolutional networks for semantic segmentation}, + author={Long, Jonathan and Shelhamer, Evan and Darrell, Trevor}, + booktitle={Proceedings of the IEEE conference on computer vision and pattern recognition}, + pages={3431--3440}, + year={2015} +} + +@article{cheng2021per, + title={Per-pixel classification is not all you need for semantic segmentation}, + author={Cheng, Bowen and Schwing, Alex and Kirillov, Alexander}, + journal={Advances in Neural Information Processing Systems}, + volume={34}, + pages={17864--17875}, + year={2021} +} + +@article{joo2020revisiting, + title={Revisiting explicit regularization in neural networks for well-calibrated predictive uncertainty}, + author={Joo, Taejong and Chung, Uijung}, + journal={arXiv preprint arXiv:2006.06399}, + year={2020} +} + +@Comment{jabref-meta: databaseType:bibtex;} +@article{rahaman2021uncertainty, + title={Uncertainty quantification and deep ensembles}, + author={Rahaman, Rahul and others}, + journal={Advances in Neural Information Processing Systems}, + volume={34}, + pages={20063--20075}, + year={2021} +} +@inproceedings{zhang2020mix, + title={Mix-n-match: Ensemble and compositional methods for uncertainty calibration in deep learning}, + author={Zhang, Jize and Kailkhura, Bhavya and Han, T Yong-Jin}, + booktitle={International conference on machine learning}, + pages={11117--11128}, + year={2020}, + organization={PMLR} +} +@inproceedings{niculescu2005predicting, + title={Predicting good probabilities with supervised learning}, + author={Niculescu-Mizil, Alexandru and Caruana, Rich}, + booktitle={Proceedings of the 22nd international conference on Machine learning}, + pages={625--632}, + year={2005} +} + +@article{zhang2004statistical, + title={Statistical analysis of some multi-category large margin classification methods}, + author={Zhang, Tong}, + journal={Journal of Machine Learning Research}, + volume={5}, + number={Oct}, + pages={1225--1251}, + year={2004} +} \ No newline at end of file diff --git a/icml2026/icml2026.bst b/icml2026/icml2026.bst new file mode 100644 index 0000000000000000000000000000000000000000..f1a50e87853c5a25a1f072fecc8a99398413257f --- /dev/null +++ b/icml2026/icml2026.bst @@ -0,0 +1,1443 @@ +%% File: `icml2025.bst' +%% A modification of `plainnl.bst' for use with natbib package +%% +%% Copyright 2010 Hal Daum\'e III +%% Modified by J. Fürnkranz +%% - Changed labels from (X and Y, 2000) to (X & Y, 2000) +%% - Changed References to last name first and abbreviated first names. +%% Modified by Iain Murray 2018 (who suggests adopting a standard .bst in future...) +%% - Made it actually use abbreviated first names +%% +%% Copyright 1993-2007 Patrick W Daly +%% Max-Planck-Institut f\"ur Sonnensystemforschung +%% Max-Planck-Str. 2 +%% D-37191 Katlenburg-Lindau +%% Germany +%% E-mail: daly@mps.mpg.de +%% +%% This program can be redistributed and/or modified under the terms +%% of the LaTeX Project Public License Distributed from CTAN +%% archives in directory macros/latex/base/lppl.txt; either +%% version 1 of the License, or any later version. +%% + % Version and source file information: + % \ProvidesFile{icml2010.mbs}[2007/11/26 1.93 (PWD)] + % + % BibTeX `plainnat' family + % version 0.99b for BibTeX versions 0.99a or later, + % for LaTeX versions 2.09 and 2e. + % + % For use with the `natbib.sty' package; emulates the corresponding + % member of the `plain' family, but with author-year citations. + % + % With version 6.0 of `natbib.sty', it may also be used for numerical + % citations, while retaining the commands \citeauthor, \citefullauthor, + % and \citeyear to print the corresponding information. + % + % For version 7.0 of `natbib.sty', the KEY field replaces missing + % authors/editors, and the date is left blank in \bibitem. + % + % Includes field EID for the sequence/citation number of electronic journals + % which is used instead of page numbers. + % + % Includes fields ISBN and ISSN. + % + % Includes field URL for Internet addresses. + % + % Includes field DOI for Digital Object Idenfifiers. + % + % Works best with the url.sty package of Donald Arseneau. + % + % Works with identical authors and year are further sorted by + % citation key, to preserve any natural sequence. + % +ENTRY + { address + author + booktitle + chapter + doi + eid + edition + editor + howpublished + institution + isbn + issn + journal + key + month + note + number + organization + pages + publisher + school + series + title + type + url + volume + year + } + {} + { label extra.label sort.label short.list } + +INTEGERS { output.state before.all mid.sentence after.sentence after.block } + +FUNCTION {init.state.consts} +{ #0 'before.all := + #1 'mid.sentence := + #2 'after.sentence := + #3 'after.block := +} + +STRINGS { s t } + +FUNCTION {output.nonnull} +{ 's := + output.state mid.sentence = + { ", " * write$ } + { output.state after.block = + { add.period$ write$ + newline$ + "\newblock " write$ + } + { output.state before.all = + 'write$ + { add.period$ " " * write$ } + if$ + } + if$ + mid.sentence 'output.state := + } + if$ + s +} + +FUNCTION {output} +{ duplicate$ empty$ + 'pop$ + 'output.nonnull + if$ +} + +FUNCTION {output.check} +{ 't := + duplicate$ empty$ + { pop$ "empty " t * " in " * cite$ * warning$ } + 'output.nonnull + if$ +} + +FUNCTION {fin.entry} +{ add.period$ + write$ + newline$ +} + +FUNCTION {new.block} +{ output.state before.all = + 'skip$ + { after.block 'output.state := } + if$ +} + +FUNCTION {new.sentence} +{ output.state after.block = + 'skip$ + { output.state before.all = + 'skip$ + { after.sentence 'output.state := } + if$ + } + if$ +} + +FUNCTION {not} +{ { #0 } + { #1 } + if$ +} + +FUNCTION {and} +{ 'skip$ + { pop$ #0 } + if$ +} + +FUNCTION {or} +{ { pop$ #1 } + 'skip$ + if$ +} + +FUNCTION {new.block.checka} +{ empty$ + 'skip$ + 'new.block + if$ +} + +FUNCTION {new.block.checkb} +{ empty$ + swap$ empty$ + and + 'skip$ + 'new.block + if$ +} + +FUNCTION {new.sentence.checka} +{ empty$ + 'skip$ + 'new.sentence + if$ +} + +FUNCTION {new.sentence.checkb} +{ empty$ + swap$ empty$ + and + 'skip$ + 'new.sentence + if$ +} + +FUNCTION {field.or.null} +{ duplicate$ empty$ + { pop$ "" } + 'skip$ + if$ +} + +FUNCTION {emphasize} +{ duplicate$ empty$ + { pop$ "" } + { "\emph{" swap$ * "}" * } + if$ +} + +INTEGERS { nameptr namesleft numnames } + +FUNCTION {format.names} +{ 's := + #1 'nameptr := + s num.names$ 'numnames := + numnames 'namesleft := + { namesleft #0 > } + { s nameptr "{vv~}{ll}{, jj}{, f.}" format.name$ 't := + nameptr #1 > + { namesleft #1 > + { ", " * t * } + { numnames #2 > + { "," * } + 'skip$ + if$ + t "others" = + { " et~al." * } + { " and " * t * } + if$ + } + if$ + } + 't + if$ + nameptr #1 + 'nameptr := + namesleft #1 - 'namesleft := + } + while$ +} + +FUNCTION {format.key} +{ empty$ + { key field.or.null } + { "" } + if$ +} + +FUNCTION {format.authors} +{ author empty$ + { "" } + { author format.names } + if$ +} + +FUNCTION {format.editors} +{ editor empty$ + { "" } + { editor format.names + editor num.names$ #1 > + { " (eds.)" * } + { " (ed.)" * } + if$ + } + if$ +} + +FUNCTION {format.isbn} +{ isbn empty$ + { "" } + { new.block "ISBN " isbn * } + if$ +} + +FUNCTION {format.issn} +{ issn empty$ + { "" } + { new.block "ISSN " issn * } + if$ +} + +FUNCTION {format.url} +{ url empty$ + { "" } + { new.block "URL \url{" url * "}" * } + if$ +} + +FUNCTION {format.doi} +{ doi empty$ + { "" } + { new.block "\doi{" doi * "}" * } + if$ +} + +FUNCTION {format.title} +{ title empty$ + { "" } + { title "t" change.case$ } + if$ +} + +FUNCTION {format.full.names} +{'s := + #1 'nameptr := + s num.names$ 'numnames := + numnames 'namesleft := + { namesleft #0 > } + { s nameptr + "{vv~}{ll}" format.name$ 't := + nameptr #1 > + { + namesleft #1 > + { ", " * t * } + { + numnames #2 > + { "," * } + 'skip$ + if$ + t "others" = + { " et~al." * } + { " and " * t * } + if$ + } + if$ + } + 't + if$ + nameptr #1 + 'nameptr := + namesleft #1 - 'namesleft := + } + while$ +} + +FUNCTION {author.editor.full} +{ author empty$ + { editor empty$ + { "" } + { editor format.full.names } + if$ + } + { author format.full.names } + if$ +} + +FUNCTION {author.full} +{ author empty$ + { "" } + { author format.full.names } + if$ +} + +FUNCTION {editor.full} +{ editor empty$ + { "" } + { editor format.full.names } + if$ +} + +FUNCTION {make.full.names} +{ type$ "book" = + type$ "inbook" = + or + 'author.editor.full + { type$ "proceedings" = + 'editor.full + 'author.full + if$ + } + if$ +} + +FUNCTION {output.bibitem} +{ newline$ + "\bibitem[" write$ + label write$ + ")" make.full.names duplicate$ short.list = + { pop$ } + { * } + if$ + "]{" * write$ + cite$ write$ + "}" write$ + newline$ + "" + before.all 'output.state := +} + +FUNCTION {n.dashify} +{ 't := + "" + { t empty$ not } + { t #1 #1 substring$ "-" = + { t #1 #2 substring$ "--" = not + { "--" * + t #2 global.max$ substring$ 't := + } + { { t #1 #1 substring$ "-" = } + { "-" * + t #2 global.max$ substring$ 't := + } + while$ + } + if$ + } + { t #1 #1 substring$ * + t #2 global.max$ substring$ 't := + } + if$ + } + while$ +} + +FUNCTION {format.date} +{ year duplicate$ empty$ + { "empty year in " cite$ * warning$ + pop$ "" } + 'skip$ + if$ + month empty$ + 'skip$ + { month + " " * swap$ * + } + if$ + extra.label * +} + +FUNCTION {format.btitle} +{ title emphasize +} + +FUNCTION {tie.or.space.connect} +{ duplicate$ text.length$ #3 < + { "~" } + { " " } + if$ + swap$ * * +} + +FUNCTION {either.or.check} +{ empty$ + 'pop$ + { "can't use both " swap$ * " fields in " * cite$ * warning$ } + if$ +} + +FUNCTION {format.bvolume} +{ volume empty$ + { "" } + { "volume" volume tie.or.space.connect + series empty$ + 'skip$ + { " of " * series emphasize * } + if$ + "volume and number" number either.or.check + } + if$ +} + +FUNCTION {format.number.series} +{ volume empty$ + { number empty$ + { series field.or.null } + { output.state mid.sentence = + { "number" } + { "Number" } + if$ + number tie.or.space.connect + series empty$ + { "there's a number but no series in " cite$ * warning$ } + { " in " * series * } + if$ + } + if$ + } + { "" } + if$ +} + +FUNCTION {format.edition} +{ edition empty$ + { "" } + { output.state mid.sentence = + { edition "l" change.case$ " edition" * } + { edition "t" change.case$ " edition" * } + if$ + } + if$ +} + +INTEGERS { multiresult } + +FUNCTION {multi.page.check} +{ 't := + #0 'multiresult := + { multiresult not + t empty$ not + and + } + { t #1 #1 substring$ + duplicate$ "-" = + swap$ duplicate$ "," = + swap$ "+" = + or or + { #1 'multiresult := } + { t #2 global.max$ substring$ 't := } + if$ + } + while$ + multiresult +} + +FUNCTION {format.pages} +{ pages empty$ + { "" } + { pages multi.page.check + { "pp.\ " pages n.dashify tie.or.space.connect } + { "pp.\ " pages tie.or.space.connect } + if$ + } + if$ +} + +FUNCTION {format.eid} +{ eid empty$ + { "" } + { "art." eid tie.or.space.connect } + if$ +} + +FUNCTION {format.vol.num.pages} +{ volume field.or.null + number empty$ + 'skip$ + { "\penalty0 (" number * ")" * * + volume empty$ + { "there's a number but no volume in " cite$ * warning$ } + 'skip$ + if$ + } + if$ + pages empty$ + 'skip$ + { duplicate$ empty$ + { pop$ format.pages } + { ":\penalty0 " * pages n.dashify * } + if$ + } + if$ +} + +FUNCTION {format.vol.num.eid} +{ volume field.or.null + number empty$ + 'skip$ + { "\penalty0 (" number * ")" * * + volume empty$ + { "there's a number but no volume in " cite$ * warning$ } + 'skip$ + if$ + } + if$ + eid empty$ + 'skip$ + { duplicate$ empty$ + { pop$ format.eid } + { ":\penalty0 " * eid * } + if$ + } + if$ +} + +FUNCTION {format.chapter.pages} +{ chapter empty$ + 'format.pages + { type empty$ + { "chapter" } + { type "l" change.case$ } + if$ + chapter tie.or.space.connect + pages empty$ + 'skip$ + { ", " * format.pages * } + if$ + } + if$ +} + +FUNCTION {format.in.ed.booktitle} +{ booktitle empty$ + { "" } + { editor empty$ + { "In " booktitle emphasize * } + { "In " format.editors * ", " * booktitle emphasize * } + if$ + } + if$ +} + +FUNCTION {empty.misc.check} +{ author empty$ title empty$ howpublished empty$ + month empty$ year empty$ note empty$ + and and and and and + key empty$ not and + { "all relevant fields are empty in " cite$ * warning$ } + 'skip$ + if$ +} + +FUNCTION {format.thesis.type} +{ type empty$ + 'skip$ + { pop$ + type "t" change.case$ + } + if$ +} + +FUNCTION {format.tr.number} +{ type empty$ + { "Technical Report" } + 'type + if$ + number empty$ + { "t" change.case$ } + { number tie.or.space.connect } + if$ +} + +FUNCTION {format.article.crossref} +{ key empty$ + { journal empty$ + { "need key or journal for " cite$ * " to crossref " * crossref * + warning$ + "" + } + { "In \emph{" journal * "}" * } + if$ + } + { "In " } + if$ + " \citet{" * crossref * "}" * +} + +FUNCTION {format.book.crossref} +{ volume empty$ + { "empty volume in " cite$ * "'s crossref of " * crossref * warning$ + "In " + } + { "Volume" volume tie.or.space.connect + " of " * + } + if$ + editor empty$ + editor field.or.null author field.or.null = + or + { key empty$ + { series empty$ + { "need editor, key, or series for " cite$ * " to crossref " * + crossref * warning$ + "" * + } + { "\emph{" * series * "}" * } + if$ + } + 'skip$ + if$ + } + 'skip$ + if$ + " \citet{" * crossref * "}" * +} + +FUNCTION {format.incoll.inproc.crossref} +{ editor empty$ + editor field.or.null author field.or.null = + or + { key empty$ + { booktitle empty$ + { "need editor, key, or booktitle for " cite$ * " to crossref " * + crossref * warning$ + "" + } + { "In \emph{" booktitle * "}" * } + if$ + } + { "In " } + if$ + } + { "In " } + if$ + " \citet{" * crossref * "}" * +} + +FUNCTION {article} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + crossref missing$ + { journal emphasize "journal" output.check + eid empty$ + { format.vol.num.pages output } + { format.vol.num.eid output } + if$ + format.date "year" output.check + } + { format.article.crossref output.nonnull + eid empty$ + { format.pages output } + { format.eid output } + if$ + } + if$ + format.issn output + format.doi output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {book} +{ output.bibitem + author empty$ + { format.editors "author and editor" output.check + editor format.key output + } + { format.authors output.nonnull + crossref missing$ + { "author and editor" editor either.or.check } + 'skip$ + if$ + } + if$ + new.block + format.btitle "title" output.check + crossref missing$ + { format.bvolume output + new.block + format.number.series output + new.sentence + publisher "publisher" output.check + address output + } + { new.block + format.book.crossref output.nonnull + } + if$ + format.edition output + format.date "year" output.check + format.isbn output + format.doi output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {booklet} +{ output.bibitem + format.authors output + author format.key output + new.block + format.title "title" output.check + howpublished address new.block.checkb + howpublished output + address output + format.date output + format.isbn output + format.doi output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {inbook} +{ output.bibitem + author empty$ + { format.editors "author and editor" output.check + editor format.key output + } + { format.authors output.nonnull + crossref missing$ + { "author and editor" editor either.or.check } + 'skip$ + if$ + } + if$ + new.block + format.btitle "title" output.check + crossref missing$ + { format.bvolume output + format.chapter.pages "chapter and pages" output.check + new.block + format.number.series output + new.sentence + publisher "publisher" output.check + address output + } + { format.chapter.pages "chapter and pages" output.check + new.block + format.book.crossref output.nonnull + } + if$ + format.edition output + format.date "year" output.check + format.isbn output + format.doi output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {incollection} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + crossref missing$ + { format.in.ed.booktitle "booktitle" output.check + format.bvolume output + format.number.series output + format.chapter.pages output + new.sentence + publisher "publisher" output.check + address output + format.edition output + format.date "year" output.check + } + { format.incoll.inproc.crossref output.nonnull + format.chapter.pages output + } + if$ + format.isbn output + format.doi output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {inproceedings} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + crossref missing$ + { format.in.ed.booktitle "booktitle" output.check + format.bvolume output + format.number.series output + format.pages output + address empty$ + { organization publisher new.sentence.checkb + organization output + publisher output + format.date "year" output.check + } + { address output.nonnull + format.date "year" output.check + new.sentence + organization output + publisher output + } + if$ + } + { format.incoll.inproc.crossref output.nonnull + format.pages output + } + if$ + format.isbn output + format.doi output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {conference} { inproceedings } + +FUNCTION {manual} +{ output.bibitem + format.authors output + author format.key output + new.block + format.btitle "title" output.check + organization address new.block.checkb + organization output + address output + format.edition output + format.date output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {mastersthesis} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + "Master's thesis" format.thesis.type output.nonnull + school "school" output.check + address output + format.date "year" output.check + format.url output + new.block + note output + fin.entry +} + +FUNCTION {misc} +{ output.bibitem + format.authors output + author format.key output + title howpublished new.block.checkb + format.title output + howpublished new.block.checka + howpublished output + format.date output + format.issn output + format.url output + new.block + note output + fin.entry + empty.misc.check +} + +FUNCTION {phdthesis} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.btitle "title" output.check + new.block + "PhD thesis" format.thesis.type output.nonnull + school "school" output.check + address output + format.date "year" output.check + format.url output + new.block + note output + fin.entry +} + +FUNCTION {proceedings} +{ output.bibitem + format.editors output + editor format.key output + new.block + format.btitle "title" output.check + format.bvolume output + format.number.series output + address output + format.date "year" output.check + new.sentence + organization output + publisher output + format.isbn output + format.doi output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {techreport} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + format.tr.number output.nonnull + institution "institution" output.check + address output + format.date "year" output.check + format.url output + new.block + note output + fin.entry +} + +FUNCTION {unpublished} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + note "note" output.check + format.date output + format.url output + fin.entry +} + +FUNCTION {default.type} { misc } + + +MACRO {jan} {"January"} + +MACRO {feb} {"February"} + +MACRO {mar} {"March"} + +MACRO {apr} {"April"} + +MACRO {may} {"May"} + +MACRO {jun} {"June"} + +MACRO {jul} {"July"} + +MACRO {aug} {"August"} + +MACRO {sep} {"September"} + +MACRO {oct} {"October"} + +MACRO {nov} {"November"} + +MACRO {dec} {"December"} + + + +MACRO {acmcs} {"ACM Computing Surveys"} + +MACRO {acta} {"Acta Informatica"} + +MACRO {cacm} {"Communications of the ACM"} + +MACRO {ibmjrd} {"IBM Journal of Research and Development"} + +MACRO {ibmsj} {"IBM Systems Journal"} + +MACRO {ieeese} {"IEEE Transactions on Software Engineering"} + +MACRO {ieeetc} {"IEEE Transactions on Computers"} + +MACRO {ieeetcad} + {"IEEE Transactions on Computer-Aided Design of Integrated Circuits"} + +MACRO {ipl} {"Information Processing Letters"} + +MACRO {jacm} {"Journal of the ACM"} + +MACRO {jcss} {"Journal of Computer and System Sciences"} + +MACRO {scp} {"Science of Computer Programming"} + +MACRO {sicomp} {"SIAM Journal on Computing"} + +MACRO {tocs} {"ACM Transactions on Computer Systems"} + +MACRO {tods} {"ACM Transactions on Database Systems"} + +MACRO {tog} {"ACM Transactions on Graphics"} + +MACRO {toms} {"ACM Transactions on Mathematical Software"} + +MACRO {toois} {"ACM Transactions on Office Information Systems"} + +MACRO {toplas} {"ACM Transactions on Programming Languages and Systems"} + +MACRO {tcs} {"Theoretical Computer Science"} + + +READ + +FUNCTION {sortify} +{ purify$ + "l" change.case$ +} + +INTEGERS { len } + +FUNCTION {chop.word} +{ 's := + 'len := + s #1 len substring$ = + { s len #1 + global.max$ substring$ } + 's + if$ +} + +FUNCTION {format.lab.names} +{ 's := + s #1 "{vv~}{ll}" format.name$ + s num.names$ duplicate$ + #2 > + { pop$ " et~al." * } + { #2 < + 'skip$ + { s #2 "{ff }{vv }{ll}{ jj}" format.name$ "others" = + { " et~al." * } + { " \& " * s #2 "{vv~}{ll}" format.name$ * } + if$ + } + if$ + } + if$ +} + +FUNCTION {author.key.label} +{ author empty$ + { key empty$ + { cite$ #1 #3 substring$ } + 'key + if$ + } + { author format.lab.names } + if$ +} + +FUNCTION {author.editor.key.label} +{ author empty$ + { editor empty$ + { key empty$ + { cite$ #1 #3 substring$ } + 'key + if$ + } + { editor format.lab.names } + if$ + } + { author format.lab.names } + if$ +} + +FUNCTION {author.key.organization.label} +{ author empty$ + { key empty$ + { organization empty$ + { cite$ #1 #3 substring$ } + { "The " #4 organization chop.word #3 text.prefix$ } + if$ + } + 'key + if$ + } + { author format.lab.names } + if$ +} + +FUNCTION {editor.key.organization.label} +{ editor empty$ + { key empty$ + { organization empty$ + { cite$ #1 #3 substring$ } + { "The " #4 organization chop.word #3 text.prefix$ } + if$ + } + 'key + if$ + } + { editor format.lab.names } + if$ +} + +FUNCTION {calc.short.authors} +{ type$ "book" = + type$ "inbook" = + or + 'author.editor.key.label + { type$ "proceedings" = + 'editor.key.organization.label + { type$ "manual" = + 'author.key.organization.label + 'author.key.label + if$ + } + if$ + } + if$ + 'short.list := +} + +FUNCTION {calc.label} +{ calc.short.authors + short.list + "(" + * + year duplicate$ empty$ + short.list key field.or.null = or + { pop$ "" } + 'skip$ + if$ + * + 'label := +} + +FUNCTION {sort.format.names} +{ 's := + #1 'nameptr := + "" + s num.names$ 'numnames := + numnames 'namesleft := + { namesleft #0 > } + { + s nameptr "{vv{ } }{ll{ }}{ f{ }}{ jj{ }}" format.name$ 't := + nameptr #1 > + { + " " * + namesleft #1 = t "others" = and + { "zzzzz" * } + { numnames #2 > nameptr #2 = and + { "zz" * year field.or.null * " " * } + 'skip$ + if$ + t sortify * + } + if$ + } + { t sortify * } + if$ + nameptr #1 + 'nameptr := + namesleft #1 - 'namesleft := + } + while$ +} + +FUNCTION {sort.format.title} +{ 't := + "A " #2 + "An " #3 + "The " #4 t chop.word + chop.word + chop.word + sortify + #1 global.max$ substring$ +} + +FUNCTION {author.sort} +{ author empty$ + { key empty$ + { "to sort, need author or key in " cite$ * warning$ + "" + } + { key sortify } + if$ + } + { author sort.format.names } + if$ +} + +FUNCTION {author.editor.sort} +{ author empty$ + { editor empty$ + { key empty$ + { "to sort, need author, editor, or key in " cite$ * warning$ + "" + } + { key sortify } + if$ + } + { editor sort.format.names } + if$ + } + { author sort.format.names } + if$ +} + +FUNCTION {author.organization.sort} +{ author empty$ + { organization empty$ + { key empty$ + { "to sort, need author, organization, or key in " cite$ * warning$ + "" + } + { key sortify } + if$ + } + { "The " #4 organization chop.word sortify } + if$ + } + { author sort.format.names } + if$ +} + +FUNCTION {editor.organization.sort} +{ editor empty$ + { organization empty$ + { key empty$ + { "to sort, need editor, organization, or key in " cite$ * warning$ + "" + } + { key sortify } + if$ + } + { "The " #4 organization chop.word sortify } + if$ + } + { editor sort.format.names } + if$ +} + + +FUNCTION {presort} +{ calc.label + label sortify + " " + * + type$ "book" = + type$ "inbook" = + or + 'author.editor.sort + { type$ "proceedings" = + 'editor.organization.sort + { type$ "manual" = + 'author.organization.sort + 'author.sort + if$ + } + if$ + } + if$ + " " + * + year field.or.null sortify + * + " " + * + cite$ + * + #1 entry.max$ substring$ + 'sort.label := + sort.label * + #1 entry.max$ substring$ + 'sort.key$ := +} + +ITERATE {presort} + +SORT + +STRINGS { longest.label last.label next.extra } + +INTEGERS { longest.label.width last.extra.num number.label } + +FUNCTION {initialize.longest.label} +{ "" 'longest.label := + #0 int.to.chr$ 'last.label := + "" 'next.extra := + #0 'longest.label.width := + #0 'last.extra.num := + #0 'number.label := +} + +FUNCTION {forward.pass} +{ last.label label = + { last.extra.num #1 + 'last.extra.num := + last.extra.num int.to.chr$ 'extra.label := + } + { "a" chr.to.int$ 'last.extra.num := + "" 'extra.label := + label 'last.label := + } + if$ + number.label #1 + 'number.label := +} + +FUNCTION {reverse.pass} +{ next.extra "b" = + { "a" 'extra.label := } + 'skip$ + if$ + extra.label 'next.extra := + extra.label + duplicate$ empty$ + 'skip$ + { "{\natexlab{" swap$ * "}}" * } + if$ + 'extra.label := + label extra.label * 'label := +} + +EXECUTE {initialize.longest.label} + +ITERATE {forward.pass} + +REVERSE {reverse.pass} + +FUNCTION {bib.sort.order} +{ sort.label 'sort.key$ := +} + +ITERATE {bib.sort.order} + +SORT + +FUNCTION {begin.bib} +{ preamble$ empty$ + 'skip$ + { preamble$ write$ newline$ } + if$ + "\begin{thebibliography}{" number.label int.to.str$ * "}" * + write$ newline$ + "\providecommand{\natexlab}[1]{#1}" + write$ newline$ + "\providecommand{\url}[1]{\texttt{#1}}" + write$ newline$ + "\expandafter\ifx\csname urlstyle\endcsname\relax" + write$ newline$ + " \providecommand{\doi}[1]{doi: #1}\else" + write$ newline$ + " \providecommand{\doi}{doi: \begingroup \urlstyle{rm}\Url}\fi" + write$ newline$ +} + +EXECUTE {begin.bib} + +EXECUTE {init.state.consts} + +ITERATE {call.type$} + +FUNCTION {end.bib} +{ newline$ + "\end{thebibliography}" write$ newline$ +} + +EXECUTE {end.bib} diff --git a/icml2026/icml2026.sty b/icml2026/icml2026.sty new file mode 100644 index 0000000000000000000000000000000000000000..47f1fae8400f54698eb5b3b245da8ea458dac536 --- /dev/null +++ b/icml2026/icml2026.sty @@ -0,0 +1,767 @@ +% File: icml2026.sty (LaTeX style file for ICML-2026, version of 2025-10-29) + +% This file contains the LaTeX formatting parameters for a two-column +% conference proceedings that is 8.5 inches wide by 11 inches high. +% +% Modified by Hanze Dong, Alberto Bietti, and Felix Berkenkamp, 2025 +% - Revert to times for better compatibility +% - Updated years, volume, location +% - Added preprint version +% - Based on the suggestion from Johan Larsson: +% 1. Added an end-of-document safety check to ensure the affiliations or notice footnote is printed: +% (1) Introduces a flag \newif\ificml@noticeprinted and sets it false by default. +% (2) At end of document, emits a package warning if \printAffiliationsAndNotice{...} was never called. +% 2. \printAffiliationsAndNotice now sets the flag when called: Begins with \global\icml@noticeprintedtrue. +% - Migrated to more recent version of fancyhdr for running title in header +% +% Modified by Johan Larsson, 2025 +% - Use newtx instead of times, aligning serif, sans-serif, typerwriter, +% and math fonts. +% - Use caption package to setup captions instead of manually defining themanually defining them. +% - Formatted icml2026.sty and example_paper.tex +% - Use title case for section title to 2.9 +% - Replace subfigure package with subcaption in example, since it is +% designed to work together with the caption package (which is now required). +% - Remove unused label in example +% +% Modified by Tegan Maharaj and Felix Berkenkamp 2025: changed years, volume, location +% +% Modified by Jonathan Scarlett 2024: changed years, volume, location +% +% Modified by Sivan Sabato 2023: changed years and volume number. +% Modified by Jonathan Scarlett 2023: added page numbers to every page +% +% Modified by Csaba Szepesvari 2022: changed years, PMLR ref. Turned off checking marginparwidth +% as marginparwidth only controls the space available for margin notes and margin notes +% will NEVER be used anyways in submitted versions, so there is no reason one should +% check whether marginparwidth has been tampered with. +% Also removed pdfview=FitH from hypersetup as it did not do its job; the default choice is a bit better +% but of course the double-column format is not supported by this hyperlink preview functionality +% in a completely satisfactory fashion. +% Modified by Gang Niu 2022: Changed color to xcolor +% +% Modified by Iain Murray 2018: changed years, location. Remove affiliation notes when anonymous. +% Move times dependency from .tex to .sty so fewer people delete it. +% +% Modified by Daniel Roy 2017: changed byline to use footnotes for affiliations, and removed emails +% +% Modified by Percy Liang 12/2/2013: changed the year, location from the previous template for ICML 2014 + +% Modified by Fei Sha 9/2/2013: changed the year, location form the previous template for ICML 2013 +% +% Modified by Fei Sha 4/24/2013: (1) remove the extra whitespace after the +% first author's email address (in %the camera-ready version) (2) change the +% Proceeding ... of ICML 2010 to 2014 so PDF's metadata will show up % +% correctly +% +% Modified by Sanjoy Dasgupta, 2013: changed years, location +% +% Modified by Francesco Figari, 2012: changed years, location +% +% Modified by Christoph Sawade and Tobias Scheffer, 2011: added line +% numbers, changed years +% +% Modified by Hal Daume III, 2010: changed years, added hyperlinks +% +% Modified by Kiri Wagstaff, 2009: changed years +% +% Modified by Sam Roweis, 2008: changed years +% +% Modified by Ricardo Silva, 2007: update of the ifpdf verification +% +% Modified by Prasad Tadepalli and Andrew Moore, merely changing years. +% +% Modified by Kristian Kersting, 2005, based on Jennifer Dy's 2004 version +% - running title. If the original title is to long or is breaking a line, +% use \icmltitlerunning{...} in the preamble to supply a shorter form. +% Added fancyhdr package to get a running head. +% - Updated to store the page size because pdflatex does compile the +% page size into the pdf. +% +% Hacked by Terran Lane, 2003: +% - Updated to use LaTeX2e style file conventions (ProvidesPackage, +% etc.) +% - Added an ``appearing in'' block at the base of the first column +% (thus keeping the ``appearing in'' note out of the bottom margin +% where the printer should strip in the page numbers). +% - Added a package option [accepted] that selects between the ``Under +% review'' notice (default, when no option is specified) and the +% ``Appearing in'' notice (for use when the paper has been accepted +% and will appear). +% +% Originally created as: ml2k.sty (LaTeX style file for ICML-2000) +% by P. Langley (12/23/99) + +%%%%%%%%%%%%%%%%%%%% +%% This version of the style file supports both a ``review'' version +%% and a ``final/accepted'' version. The difference is only in the +%% text that appears in the note at the bottom of the first column of +%% the first page. The default behavior is to print a note to the +%% effect that the paper is under review and don't distribute it. The +%% final/accepted version prints an ``Appearing in'' note. To get the +%% latter behavior, in the calling file change the ``usepackage'' line +%% from: +%% \usepackage{icml2025} +%% to +%% \usepackage[accepted]{icml2025} +%%%%%%%%%%%%%%%%%%%% + +\NeedsTeXFormat{LaTeX2e} +\ProvidesPackage{icml2026}[2025/10/29 v2.0 ICML Conference Style File] + +% Before 2018, \usepackage{times} was in the example TeX, but inevitably +% not everybody did it. +% \RequirePackage[amsthm]{newtx} +% 2025.11.6 revert to times for better compatibility +\RequirePackage{times} + +% Use fancyhdr package +\RequirePackage{fancyhdr} +\RequirePackage{xcolor} % changed from color to xcolor (2021/11/24) +\RequirePackage{algorithm} +\RequirePackage{algorithmic} +\RequirePackage{natbib} +\RequirePackage{eso-pic} % used by \AddToShipoutPicture +\RequirePackage{forloop} +\RequirePackage{url} +\RequirePackage{caption} + +%%%%%%%% Options +\DeclareOption{accepted}{% + \renewcommand{\Notice@String}{\ICML@appearing} + \gdef\isaccepted{1} +} + +% === Preprint option === +\DeclareOption{preprint}{%% + \renewcommand{\Notice@String}{\ICML@preprint}%% + \gdef\ispreprint{1}%% +} + +% Distinct preprint footer text +\newcommand{\ICML@preprint}{% + \textit{Preprint. \today.}% +} + +\DeclareOption{nohyperref}{% + \gdef\nohyperref{1} +} + +% Helper flag: show real authors for accepted or preprint +\newif\ificmlshowauthors +\icmlshowauthorsfalse + +%%%%%%%%%%%%%%%%%%%% +% This string is printed at the bottom of the page for the +% final/accepted version of the ``appearing in'' note. Modify it to +% change that text. +%%%%%%%%%%%%%%%%%%%% +\newcommand{\ICML@appearing}{\textit{Proceedings of the +$\mathit{43}^{rd}$ International Conference on Machine Learning}, +Seoul, South Korea. PMLR 306, 2026. +Copyright 2026 by the author(s).} + +%%%%%%%%%%%%%%%%%%%% +% This string is printed at the bottom of the page for the draft/under +% review version of the ``appearing in'' note. Modify it to change +% that text. +%%%%%%%%%%%%%%%%%%%% +\newcommand{\Notice@String}{Preliminary work. Under review by the +International Conference on Machine Learning (ICML)\@. Do not distribute.} + +% Cause the declared options to actually be parsed and activated +\ProcessOptions\relax + +% After options are processed, decide if authors should be visible +\ifdefined\isaccepted \icmlshowauthorstrue \fi +\ifdefined\ispreprint \icmlshowauthorstrue \fi + +\ifdefined\isaccepted\else\ifdefined\ispreprint\else\ifdefined\hypersetup + \hypersetup{pdfauthor={Anonymous Authors}} +\fi\fi\fi + +\ifdefined\nohyperref\else\ifdefined\hypersetup + \definecolor{mydarkblue}{rgb}{0,0.08,0.45} + \hypersetup{ % + pdftitle={}, + pdfsubject={Proceedings of the International Conference on Machine Learning 2026}, + pdfkeywords={}, + pdfborder=0 0 0, + pdfpagemode=UseNone, + colorlinks=true, + linkcolor=mydarkblue, + citecolor=mydarkblue, + filecolor=mydarkblue, + urlcolor=mydarkblue, + } + \fi +\fi + + + +% Uncomment the following for debugging. It will cause LaTeX to dump +% the version of the ``appearing in'' string that will actually appear +% in the document. +%\typeout{>> Notice string='\Notice@String'} + +% Change citation commands to be more like old ICML styles +\newcommand{\yrcite}[1]{\citeyearpar{#1}} +\renewcommand{\cite}[1]{\citep{#1}} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% to ensure the letter format is used. pdflatex does compile the +% page size into the pdf. This is done using \pdfpagewidth and +% \pdfpageheight. As Latex does not know this directives, we first +% check whether pdflatex or latex is used. +% +% Kristian Kersting 2005 +% +% in order to account for the more recent use of pdfetex as the default +% compiler, I have changed the pdf verification. +% +% Ricardo Silva 2007 +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\paperwidth=8.5in +\paperheight=11in + +% old PDFLaTex verification, circa 2005 +% +%\newif\ifpdf\ifx\pdfoutput\undefined +% \pdffalse % we are not running PDFLaTeX +%\else +% \pdfoutput=1 % we are running PDFLaTeX +% \pdftrue +%\fi + +\newif\ifpdf %adapted from ifpdf.sty +\ifx\pdfoutput\undefined +\else + \ifx\pdfoutput\relax + \else + \ifcase\pdfoutput + \else + \pdftrue + \fi + \fi +\fi + +\ifpdf +% \pdfpagewidth=\paperwidth +% \pdfpageheight=\paperheight + \setlength{\pdfpagewidth}{8.5in} + \setlength{\pdfpageheight}{11in} +\fi + +% Physical page layout + +\evensidemargin -0.23in +\oddsidemargin -0.23in +\setlength\textheight{9.0in} +\setlength\textwidth{6.75in} +\setlength\columnsep{0.25in} +\setlength\headheight{10pt} +\setlength\headsep{10pt} +\addtolength{\topmargin}{-20pt} +\addtolength{\topmargin}{-0.29in} + +% Historically many authors tried to include packages like geometry or fullpage, +% which change the page layout. It either makes the proceedings inconsistent, or +% wastes organizers' time chasing authors. So let's nip these problems in the +% bud here. -- Iain Murray 2018. +%\RequirePackage{printlen} +\AtBeginDocument{% +\newif\ifmarginsmessedwith +\marginsmessedwithfalse +\ifdim\oddsidemargin=-16.62178pt \else oddsidemargin has been altered.\\ \marginsmessedwithtrue\fi +\ifdim\headheight=10.0pt \else headheight has been altered.\\ \marginsmessedwithtrue\fi +\ifdim\textheight=650.43pt \else textheight has been altered.\\ \marginsmessedwithtrue\fi +\ifdim\marginparsep=11.0pt \else marginparsep has been altered.\\ \marginsmessedwithtrue\fi +\ifdim\footskip=25.0pt \else footskip has been altered.\\ \marginsmessedwithtrue\fi +\ifdim\hoffset=0.0pt \else hoffset has been altered.\\ \marginsmessedwithtrue\fi +\ifdim\paperwidth=614.295pt \else paperwidth has been altered.\\ \marginsmessedwithtrue\fi +\ifdim\topmargin=-24.95781pt \else topmargin has been altered.\\ \marginsmessedwithtrue\fi +\ifdim\headsep=10.0pt \else headsep has been altered.\\ \marginsmessedwithtrue\fi +\ifdim\textwidth=487.8225pt \else textwidth has been altered.\\ \marginsmessedwithtrue\fi +\ifdim\marginparpush=5.0pt \else marginparpush has been altered.\\ \marginsmessedwithtrue\fi +\ifdim\voffset=0.0pt \else voffset has been altered.\\ \marginsmessedwithtrue\fi +\ifdim\paperheight=794.96999pt \else paperheight has been altered.\\ \marginsmessedwithtrue\fi +\ifmarginsmessedwith + +\textbf{\large \em The page layout violates the ICML style.} + +Please do not change the page layout, or include packages like geometry, +savetrees, or fullpage, which change it for you. + +We're not able to reliably undo arbitrary changes to the style. Please remove +the offending package(s), or layout-changing commands and try again. + +\fi} + + +%% The following is adapted from code in the acmconf.sty conference +%% style file. The constants in it are somewhat magical, and appear +%% to work well with the two-column format on US letter paper that +%% ICML uses, but will break if you change that layout, or if you use +%% a longer block of text for the copyright notice string. Fiddle with +%% them if necessary to get the block to fit/look right. +%% +%% -- Terran Lane, 2003 +%% +%% The following comments are included verbatim from acmconf.sty: +%% +%%% This section (written by KBT) handles the 1" box in the lower left +%%% corner of the left column of the first page by creating a picture, +%%% and inserting the predefined string at the bottom (with a negative +%%% displacement to offset the space allocated for a non-existent +%%% caption). +%%% +\def\ftype@copyrightbox{8} +\def\@copyrightspace{ +\@float{copyrightbox}[b] +\begin{center} +\setlength{\unitlength}{1pc} +\begin{picture}(20,1.5) +\put(0,2.5){\line(1,0){4.818}} +\put(0,0){\parbox[b]{19.75pc}{\small \Notice@String}} +\end{picture} +\end{center} +\end@float} + +\setlength\footskip{25.0pt} +\flushbottom \twocolumn +\sloppy + +% Clear out the addcontentsline command +\def\addcontentsline#1#2#3{} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% commands for formatting paper title, author names, and addresses. + +% box to check the size of the running head +\newbox\titrun + +% general page style +\pagestyle{fancy} +\fancyhf{} +\fancyfoot[C]{\thepage} +% set the width of the head rule to 1 point +\renewcommand{\headrulewidth}{1pt} + +% definition to set the head as running head in the preamble +\def\icmltitlerunning#1{\gdef\@icmltitlerunning{#1}} + +% main definition adapting \icmltitle from 2004 +\long\def\icmltitle#1{% + + %check whether @icmltitlerunning exists + % if not \icmltitle is used as running head + \ifx\undefined\@icmltitlerunning% + \gdef\@icmltitlerunning{#1} + \fi + + %add it to pdf information + \ifdefined\nohyperref\else\ifdefined\hypersetup + \hypersetup{pdftitle={#1}} + \fi\fi + + %get the dimension of the running title + \global\setbox\titrun=\vbox{\small\bf\@icmltitlerunning} + + % error flag + \gdef\@runningtitleerror{0} + + % running title too long + \ifdim\wd\titrun>\textwidth% + \gdef\@runningtitleerror{1}% + % running title breaks a line + \else \ifdim\ht\titrun>6.25pt + \gdef\@runningtitleerror{2}% + \fi + \fi + + % if there is somthing wrong with the running title + \ifnum\@runningtitleerror>0 + \typeout{}% + \typeout{}% + \typeout{*******************************************************}% + \typeout{Title exceeds size limitations for running head.}% + \typeout{Please supply a shorter form for the running head} + \typeout{with \string\icmltitlerunning{...}\space prior to \string\begin{document}}% + \typeout{*******************************************************}% + \typeout{}% + \typeout{}% + % set default running title + \gdef\@icmltitlerunning{Title Suppressed Due to Excessive Size} + \fi + + % no running title on the first page of the paper + \thispagestyle{plain} + + {\center\baselineskip 18pt + \toptitlebar{\Large\bf #1}\bottomtitlebar} +} + +% set running title header +\fancyhead[C]{\small\bf\@icmltitlerunning} + +\gdef\icmlfullauthorlist{} +\newcommand\addstringtofullauthorlist{\g@addto@macro\icmlfullauthorlist} +\newcommand\addtofullauthorlist[1]{% + \ifdefined\icmlanyauthors% + \addstringtofullauthorlist{, #1}% + \else% + \addstringtofullauthorlist{#1}% + \gdef\icmlanyauthors{1}% + \fi% + \ifdefined\hypersetup% + \hypersetup{pdfauthor=\icmlfullauthorlist}% + \fi +} + +\def\toptitlebar{\hrule height1pt \vskip .25in} +\def\bottomtitlebar{\vskip .22in \hrule height1pt \vskip .3in} + +\newenvironment{icmlauthorlist}{% + \setlength\topsep{0pt} + \setlength\parskip{0pt} + \begin{center} + }{% + \end{center} +} + +\newcounter{@affiliationcounter} +\newcommand{\@pa}[1]{% + \ifcsname the@affil#1\endcsname + % do nothing + \else + \ifcsname @icmlsymbol#1\endcsname + % nothing + \else + \stepcounter{@affiliationcounter}% + \newcounter{@affil#1}% + \setcounter{@affil#1}{\value{@affiliationcounter}}% + \fi + \fi% + \ifcsname @icmlsymbol#1\endcsname + \textsuperscript{\csname @icmlsymbol#1\endcsname\,}% + \else + \textsuperscript{\arabic{@affil#1}\,}% + \fi +} + +\newcommand{\icmlauthor}[2]{% + \ificmlshowauthors + \mbox{\bf #1}\,\@for\theaffil:=#2\do{\@pa{\theaffil}} \addtofullauthorlist{#1}% + \else + \ifdefined\@icmlfirsttime\else + \gdef\@icmlfirsttime{1} + \mbox{\bf Anonymous Authors}\@pa{@anon} \addtofullauthorlist{Anonymous Authors} + \fi + \fi +} + +\newcommand{\icmlsetsymbol}[2]{% + \expandafter\gdef\csname @icmlsymbol#1\endcsname{#2} +} + +\newcommand{\icmlaffiliation}[2]{% + \ificmlshowauthors + \ifcsname the@affil#1\endcsname + \expandafter\gdef\csname @affilname\csname the@affil#1\endcsname\endcsname{#2}% + \else + {\bf AUTHORERR: Error in use of \textbackslash{}icmlaffiliation command. Label ``#1'' not mentioned in some \textbackslash{}icmlauthor\{author name\}\{labels here\} command beforehand. } + \typeout{}% + \typeout{}% + \typeout{*******************************************************}% + \typeout{Affiliation label undefined. }% + \typeout{Make sure \string\icmlaffiliation\space follows }% + \typeout{all of \string\icmlauthor\space commands}% + \typeout{*******************************************************}% + \typeout{}% + \typeout{}% + \fi + \else + \expandafter\gdef\csname @affilname1\endcsname{Anonymous Institution, Anonymous City, Anonymous Region, Anonymous Country} + \fi +} + +\newcommand{\icmlcorrespondingauthor}[2]{% + \ificmlshowauthors + \ifdefined\icmlcorrespondingauthor@text + \g@addto@macro\icmlcorrespondingauthor@text{, #1 \textless{}#2\textgreater{}} + \else + \gdef\icmlcorrespondingauthor@text{#1 \textless{}#2\textgreater{}} + \fi + \else + \gdef\icmlcorrespondingauthor@text{Anonymous Author \textless{}anon.email@domain.com\textgreater{}} + \fi +} + +\newcommand{\icmlEqualContribution}{\textsuperscript{*}Equal contribution } + + +% --- ICML 2026: ensure authors do not omit the affiliations/notice footnote --- +\newif\ificml@noticeprinted +\icml@noticeprintedfalse +\AtEndDocument{% + \ificml@noticeprinted\relax\else + \PackageWarningNoLine{icml2026}{% + You did not call \string\printAffiliationsAndNotice{}. If you have no notice,% + call \string\printAffiliationsAndNotice\string{} (empty braces).% + }% + \fi +} + + +\newcounter{@affilnum} +\newcommand{\printAffiliationsAndNotice}[1]{\global\icml@noticeprintedtrue% + \stepcounter{@affiliationcounter}% + {\let\thefootnote\relax\footnotetext{\hspace*{-\footnotesep}\ificmlshowauthors #1\fi% + \forloop{@affilnum}{1}{\value{@affilnum} < \value{@affiliationcounter}}{ + \textsuperscript{\arabic{@affilnum}}\ifcsname @affilname\the@affilnum\endcsname% + \csname @affilname\the@affilnum\endcsname% + \else + {\bf AUTHORERR: Missing \textbackslash{}icmlaffiliation.} + \fi + }.% + \ifdefined\icmlcorrespondingauthor@text + { }Correspondence to: \icmlcorrespondingauthor@text. + \else + {\bf AUTHORERR: Missing \textbackslash{}icmlcorrespondingauthor.} + \fi + + \ \\ + \Notice@String + } + } +} + +\long\def\icmladdress#1{% + {\bf The \textbackslash{}icmladdress command is no longer used. See the example\_paper PDF .tex for usage of \textbackslash{}icmlauther and \textbackslash{}icmlaffiliation.} +} + +%% keywords as first class citizens +\def\icmlkeywords#1{% + \ifdefined\nohyperref\else\ifdefined\hypersetup + \hypersetup{pdfkeywords={#1}} + \fi\fi +} + +% modification to natbib citations +\setcitestyle{authoryear,round,citesep={;},aysep={,},yysep={;}} + +% Redefinition of the abstract environment. +\renewenvironment{abstract} +{% + \centerline{\large\bf Abstract} + \vspace{-0.12in}\begin{quote}} + {\par\end{quote}\vskip 0.12in} + +% numbered section headings with different treatment of numbers + +\def\@startsection#1#2#3#4#5#6{\if@noskipsec \leavevmode \fi + \par \@tempskipa #4\relax + \@afterindenttrue + \ifdim \@tempskipa <\z@ \@tempskipa -\@tempskipa \fi + \if@nobreak \everypar{}\else + \addpenalty{\@secpenalty}\addvspace{\@tempskipa}\fi \@ifstar + {\@ssect{#3}{#4}{#5}{#6}}{\@dblarg{\@sict{#1}{#2}{#3}{#4}{#5}{#6}}}} + +\def\@sict#1#2#3#4#5#6[#7]#8{\ifnum #2>\c@secnumdepth + \def\@svsec{}\else + \refstepcounter{#1}\edef\@svsec{\csname the#1\endcsname}\fi + \@tempskipa #5\relax + \ifdim \@tempskipa>\z@ + \begingroup #6\relax + \@hangfrom{\hskip #3\relax\@svsec.~}{\interlinepenalty \@M #8\par} + \endgroup + \csname #1mark\endcsname{#7}\addcontentsline + {toc}{#1}{\ifnum #2>\c@secnumdepth \else + \protect\numberline{\csname the#1\endcsname}\fi + #7}\else + \def\@svsechd{#6\hskip #3\@svsec #8\csname #1mark\endcsname + {#7}\addcontentsline + {toc}{#1}{\ifnum #2>\c@secnumdepth \else + \protect\numberline{\csname the#1\endcsname}\fi + #7}}\fi + \@xsect{#5}} + +\def\@sect#1#2#3#4#5#6[#7]#8{\ifnum #2>\c@secnumdepth + \def\@svsec{}\else + \refstepcounter{#1}\edef\@svsec{\csname the#1\endcsname\hskip 0.4em }\fi + \@tempskipa #5\relax + \ifdim \@tempskipa>\z@ + \begingroup #6\relax + \@hangfrom{\hskip #3\relax\@svsec}{\interlinepenalty \@M #8\par} + \endgroup + \csname #1mark\endcsname{#7}\addcontentsline + {toc}{#1}{\ifnum #2>\c@secnumdepth \else + \protect\numberline{\csname the#1\endcsname}\fi + #7}\else + \def\@svsechd{#6\hskip #3\@svsec #8\csname #1mark\endcsname + {#7}\addcontentsline + {toc}{#1}{\ifnum #2>\c@secnumdepth \else + \protect\numberline{\csname the#1\endcsname}\fi + #7}}\fi + \@xsect{#5}} + +% section headings with less space above and below them +\def\thesection {\arabic{section}} +\def\thesubsection {\thesection.\arabic{subsection}} +\def\section{\@startsection{section}{1}{\z@}{-0.12in}{0.02in} + {\large\bf\raggedright}} +\def\subsection{\@startsection{subsection}{2}{\z@}{-0.10in}{0.01in} + {\normalsize\bf\raggedright}} +\def\subsubsection{\@startsection{subsubsection}{3}{\z@}{-0.08in}{0.01in} + {\normalsize\sc\raggedright}} +\def\paragraph{\@startsection{paragraph}{4}{\z@}{1.5ex plus + 0.5ex minus .2ex}{-1em}{\normalsize\bf}} +\def\subparagraph{\@startsection{subparagraph}{5}{\z@}{1.5ex plus + 0.5ex minus .2ex}{-1em}{\normalsize\bf}} + +% Footnotes +\footnotesep 6.65pt % +\skip\footins 9pt +\def\footnoterule{\kern-3pt \hrule width 0.8in \kern 2.6pt } +\setcounter{footnote}{0} + +% Lists and paragraphs +\parindent 0pt +\topsep 4pt plus 1pt minus 2pt +\partopsep 1pt plus 0.5pt minus 0.5pt +\itemsep 2pt plus 1pt minus 0.5pt +\parsep 2pt plus 1pt minus 0.5pt +\parskip 6pt + +\leftmargin 2em \leftmargini\leftmargin \leftmarginii 2em +\leftmarginiii 1.5em \leftmarginiv 1.0em \leftmarginv .5em +\leftmarginvi .5em +\labelwidth\leftmargini\advance\labelwidth-\labelsep \labelsep 5pt + +\def\@listi{\leftmargin\leftmargini} +\def\@listii{\leftmargin\leftmarginii + \labelwidth\leftmarginii\advance\labelwidth-\labelsep + \topsep 2pt plus 1pt minus 0.5pt + \parsep 1pt plus 0.5pt minus 0.5pt + \itemsep \parsep} +\def\@listiii{\leftmargin\leftmarginiii + \labelwidth\leftmarginiii\advance\labelwidth-\labelsep + \topsep 1pt plus 0.5pt minus 0.5pt + \parsep \z@ \partopsep 0.5pt plus 0pt minus 0.5pt + \itemsep \topsep} +\def\@listiv{\leftmargin\leftmarginiv + \labelwidth\leftmarginiv\advance\labelwidth-\labelsep} +\def\@listv{\leftmargin\leftmarginv + \labelwidth\leftmarginv\advance\labelwidth-\labelsep} +\def\@listvi{\leftmargin\leftmarginvi + \labelwidth\leftmarginvi\advance\labelwidth-\labelsep} + +\abovedisplayskip 7pt plus2pt minus5pt% +\belowdisplayskip \abovedisplayskip +\abovedisplayshortskip 0pt plus3pt% +\belowdisplayshortskip 4pt plus3pt minus3pt% + +% Less leading in most fonts (due to the narrow columns) +% The choices were between 1-pt and 1.5-pt leading +\def\@normalsize{\@setsize\normalsize{11pt}\xpt\@xpt} +\def\small{\@setsize\small{10pt}\ixpt\@ixpt} +\def\footnotesize{\@setsize\footnotesize{10pt}\ixpt\@ixpt} +\def\scriptsize{\@setsize\scriptsize{8pt}\viipt\@viipt} +\def\tiny{\@setsize\tiny{7pt}\vipt\@vipt} +\def\large{\@setsize\large{14pt}\xiipt\@xiipt} +\def\Large{\@setsize\Large{16pt}\xivpt\@xivpt} +\def\LARGE{\@setsize\LARGE{20pt}\xviipt\@xviipt} +\def\huge{\@setsize\huge{23pt}\xxpt\@xxpt} +\def\Huge{\@setsize\Huge{28pt}\xxvpt\@xxvpt} + +% Revised formatting for figure captions and table titles. +\captionsetup{ + skip=0.1in, + font=small, + labelfont={it,small}, + labelsep=period +} +\captionsetup[table]{position=above} +\captionsetup[figure]{position=below} + +\def\fnum@figure{Figure \thefigure} +\def\fnum@table{Table \thetable} + +% Strut macros for skipping spaces above and below text in tables. +\def\abovestrut#1{\rule[0in]{0in}{#1}\ignorespaces} +\def\belowstrut#1{\rule[-#1]{0in}{#1}\ignorespaces} + +\def\abovespace{\abovestrut{0.20in}} +\def\aroundspace{\abovestrut{0.20in}\belowstrut{0.10in}} +\def\belowspace{\belowstrut{0.10in}} + +% Various personal itemization commands. +\def\texitem#1{\par\noindent\hangindent 12pt + \hbox to 12pt {\hss #1 ~}\ignorespaces} +\def\icmlitem{\texitem{$\bullet$}} + +% To comment out multiple lines of text. +\long\def\comment#1{} + +%% Line counter (not in final version). Adapted from NIPS style file by Christoph Sawade + +% Vertical Ruler +% This code is, largely, from the CVPR 2010 conference style file +% ----- define vruler +\makeatletter +\newbox\icmlrulerbox +\newcount\icmlrulercount +\newdimen\icmlruleroffset +\newdimen\cv@lineheight +\newdimen\cv@boxheight +\newbox\cv@tmpbox +\newcount\cv@refno +\newcount\cv@tot +% NUMBER with left flushed zeros \fillzeros[] +\newcount\cv@tmpc@ \newcount\cv@tmpc +\def\fillzeros[#1]#2{\cv@tmpc@=#2\relax\ifnum\cv@tmpc@<0\cv@tmpc@=-\cv@tmpc@\fi + \cv@tmpc=1 % + \loop\ifnum\cv@tmpc@<10 \else \divide\cv@tmpc@ by 10 \advance\cv@tmpc by 1 \fi + \ifnum\cv@tmpc@=10\relax\cv@tmpc@=11\relax\fi \ifnum\cv@tmpc@>10 \repeat + \ifnum#2<0\advance\cv@tmpc1\relax-\fi + \loop\ifnum\cv@tmpc<#1\relax0\advance\cv@tmpc1\relax\fi \ifnum\cv@tmpc<#1 \repeat + \cv@tmpc@=#2\relax\ifnum\cv@tmpc@<0\cv@tmpc@=-\cv@tmpc@\fi \relax\the\cv@tmpc@}% +% \makevruler[][][][][] +\def\makevruler[#1][#2][#3][#4][#5]{ + \begingroup\offinterlineskip + \textheight=#5\vbadness=10000\vfuzz=120ex\overfullrule=0pt% + \global\setbox\icmlrulerbox=\vbox to \textheight{% + { + \parskip=0pt\hfuzz=150em\cv@boxheight=\textheight + \cv@lineheight=#1\global\icmlrulercount=#2% + \cv@tot\cv@boxheight\divide\cv@tot\cv@lineheight\advance\cv@tot2% + \cv@refno1\vskip-\cv@lineheight\vskip1ex% + \loop\setbox\cv@tmpbox=\hbox to0cm{\hfil {\hfil\fillzeros[#4]\icmlrulercount}}% + \ht\cv@tmpbox\cv@lineheight\dp\cv@tmpbox0pt\box\cv@tmpbox\break + \advance\cv@refno1\global\advance\icmlrulercount#3\relax + \ifnum\cv@refno<\cv@tot\repeat + } + } + \endgroup +}% +\makeatother +% ----- end of vruler + +% \makevruler[][][][][] +\def\icmlruler#1{\makevruler[12pt][#1][1][3][\textheight]\usebox{\icmlrulerbox}} +\AddToShipoutPicture{% + \icmlruleroffset=\textheight + \advance\icmlruleroffset by 5.2pt % top margin + \color[rgb]{.7,.7,.7} + \ificmlshowauthors\else + \AtTextUpperLeft{% + \put(\LenToUnit{-35pt},\LenToUnit{-\icmlruleroffset}){%left ruler + \icmlruler{\icmlrulercount}} + %\put(\LenToUnit{1.04\textwidth},\LenToUnit{-\icmlruleroffset}){%right ruler + % \icmlruler{\icmlrulercount}} + } + \fi +} +\endinput diff --git a/icml2026/icml2026.tex b/icml2026/icml2026.tex new file mode 100644 index 0000000000000000000000000000000000000000..d4ffbad2ec121ac7aa098259fde5e00f532c830d --- /dev/null +++ b/icml2026/icml2026.tex @@ -0,0 +1,758 @@ +%%%%%%%% ICML 2026 EXAMPLE LATEX SUBMISSION FILE %%%%%%%%%%%%%%%%% + +\documentclass{article} + +% Recommended, but optional, packages for figures and better typesetting: +\usepackage{microtype} +\usepackage{graphicx} +\usepackage{subcaption} +\usepackage{booktabs} % for professional tables + +% hyperref makes hyperlinks in the resulting PDF. +% If your build breaks (sometimes temporarily if a hyperlink spans a page) +% please comment out the following usepackage line and replace +% \usepackage{icml2026} with \usepackage[nohyperref]{icml2026} above. +\usepackage{hyperref} +\usepackage{url} + +% Attempt to make hyperref and algorithmic work together better: +\newcommand{\theHalgorithm}{\arabic{algorithm}} + +% Use the following line for the initial blind version submitted for review: +\usepackage{icml2026} + +% For preprint, use +% \usepackage[preprint]{icml2026} + +% If accepted, instead use the following line for the camera-ready submission: +% \usepackage[accepted]{icml2026} + +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{mathtools} +\usepackage{amsthm} + +\usepackage{natbib} +\usepackage{subcaption} +\usepackage{graphicx} +% algorithm and algorithmic already loaded by icml2026.sty +\usepackage[utf8]{inputenc} % allow utf-8 input +\usepackage[T1]{fontenc} % use 8-bit T1 fonts +\usepackage{hyperref} % hyperlinks +\usepackage{url} % simple URL typesetting +\usepackage{booktabs} % professional-quality tables +\usepackage{amsfonts} % blackboard math symbols +\usepackage{nicefrac} % compact symbols for 1/2, etc. +\usepackage{microtype} % microtypography +\usepackage{lipsum} +\usepackage{booktabs} +\usepackage{multirow} +\usepackage{graphicx} +\usepackage{wrapfig} +\usepackage[table]{xcolor} +\usepackage{colortbl} +\definecolor{lightgray}{gray}{0.9} +\newcommand{\cellgray}{\cellcolor{lightgray}} +\usepackage{verbatim} +% \usepackage{amsthm} +% \newtheorem{theorem}{Theorem}[section] +% \newtheorem{proposition}[theorem]{Proposition} +% \newtheorem*{remark}{Remark} +\usepackage{enumitem} +\usepackage{makecell} +% \newtheorem{lemma}{Lemma} % Defines the lemma environment +% \setlength{\parskip}{0pt} +% \setlength{\parindent}{0pt} + +\newcommand{\etal}{\textit{et al}.} +\newcommand{\ie}{\textit{i}.\textit{e}.} +\newcommand{\Eg}{\textit{e}.\textit{g}.} +\newcommand{\etc}{\textit{etc}.} +% --- Math packages (likely already present) +% \usepackage{amsmath,amssymb,amsfonts,mathtools} +\DeclareMathOperator{\Lip}{Lip} +\newcommand{\ind}{\mathbf{1}} +\newcommand{\E}{\mathbb{E}} +\newcommand{\R}{\mathbb{R}} +\DeclareMathOperator{\softmax}{softmax} +\newcommand{\avgq}[1]{\langle #1\rangle_q} % e.g., \avgq{t} = \sum_k q_k t_k + + +% if you use cleveref.. +\usepackage[capitalize,noabbrev]{cleveref} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% THEOREMS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\theoremstyle{plain} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{assumption}[theorem]{Assumption} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} + +% Todonotes is useful during development; simply uncomment the next line +% and comment out the line below the next line to turn off comments +%\usepackage[disable,textsize=tiny]{todonotes} +\usepackage[textsize=tiny]{todonotes} + +% The \icmltitle you define below is probably too long as a header. +% Therefore, a short form for the running title is supplied here: +\icmltitlerunning{Submission and Formatting Instructions for ICML 2026} + +\begin{document} + +\twocolumn[ + \icmltitle{Sample Margin-Aware Recalibration of Temperature} + + % It is OKAY to include author information, even for blind submissions: the + % style file will automatically remove it for you unless you've provided + % the [accepted] option to the icml2026 package. + + % List of affiliations: The first argument should be a (short) identifier you + % will use later to specify author affiliations Academic affiliations + % should list Department, University, City, Region, Country Industry + % affiliations should list Company, City, Region, Country + + % You can specify symbols, otherwise they are numbered in order. Ideally, you + % should not use this facility. Affiliations will be numbered in order of + % appearance and this is the preferred way. + \icmlsetsymbol{equal}{*} + + \begin{icmlauthorlist} + \icmlauthor{Firstname1 Lastname1}{equal,yyy} + \icmlauthor{Firstname2 Lastname2}{equal,yyy,comp} + \icmlauthor{Firstname3 Lastname3}{comp} + \icmlauthor{Firstname4 Lastname4}{sch} + \icmlauthor{Firstname5 Lastname5}{yyy} + \icmlauthor{Firstname6 Lastname6}{sch,yyy,comp} + \icmlauthor{Firstname7 Lastname7}{comp} + %\icmlauthor{}{sch} + \icmlauthor{Firstname8 Lastname8}{sch} + \icmlauthor{Firstname8 Lastname8}{yyy,comp} + %\icmlauthor{}{sch} + %\icmlauthor{}{sch} + \end{icmlauthorlist} + + \icmlaffiliation{yyy}{Department of XXX, University of YYY, Location, Country} + \icmlaffiliation{comp}{Company Name, Location, Country} + \icmlaffiliation{sch}{School of ZZZ, Institute of WWW, Location, Country} + + \icmlcorrespondingauthor{Firstname1 Lastname1}{first1.last1@xxx.edu} + \icmlcorrespondingauthor{Firstname2 Lastname2}{first2.last2@www.uk} + + % You may provide any keywords that you find helpful for describing your + % paper; these are used to populate the "keywords" metadata in the PDF but + % will not be shown in the document + \icmlkeywords{Machine Learning, ICML} + + \vskip 0.3in +] + +% this must go after the closing bracket ] following \twocolumn[ ... + +% This command actually creates the footnote in the first column listing the +% affiliations and the copyright notice. The command takes one argument, which +% is text to display at the start of the footnote. The \icmlEqualContribution +% command is standard text for equal contribution. Remove it (just {}) if you +% do not need this facility. + +% Use ONE of the following lines. DO NOT remove the command. +% If you have no special notice, KEEP empty braces: +\printAffiliationsAndNotice{} % no special notice (required even if empty) +% Or, if applicable, use the standard equal contribution text: +% \printAffiliationsAndNotice{\icmlEqualContribution} + +\begin{abstract} +% Deep neural networks often exhibit overconfidence despite their high accuracy. Such miscalibration limits reliability in safety-critical domains where trustworthiness are crucial. Post-hoc calibration methods offer a practical solution where popular approaches like Temperature Scaling (TS) apply a single corrective parameter to all samples, failing to address the sample-dependent nature of miscalibration. While more advanced methods attempt to adapt to sample difficulty, they often rely on complex and indirectly learned indicators. +% In this work, we identify the \emph{logit margin} as a direct, simple, and principled indicator of sample hardness. We provide empirical and theoretical evidence that it serves as a more effective indicator of sample hardness than existing proxies. Meanwhile, we identify a fundamental flaw in current methods that optimizing Negative Log-Likelihood (NLL) can paradoxically degrade calibration. To resolve this, we introduce Charbonnier–SoftECE, a novel and theoretically guaranteed objective that directly minimizes calibration error. + + +Deep neural networks frequently exhibit overconfidence, undermining reliability in safety-critical applications. Existing adaptive methods rely on indirectly learned proxies of sample difficulty. +We establish the \emph{logit margin} as a direct and principled hardness indicator. We prove that margin tightly bounds the feasible temperature range for any target confidence. Empirically, margin strongly correlates with decision boundary proximity and reveals systematic calibration patterns across difficulty levels. We further identify a fundamental flaw in NLL-based optimization: minimizing NLL can paradoxically increase calibration error. To address this, we introduce Charbonnier–SoftECE, a smooth objective that provably upper-bounds the smooth calibration error (smCE). +Building on these insights, we propose SMART (Sample Margin-Aware Recalibration of Temperature), a lightweight method that learns a sample-wise margin-to-temperature mapping guided by our calibration-centric objective. Experiments demonstrate state-of-the-art calibration across CNNs and ViTs on standard, long-tailed, and distribution-shifted benchmarks, with a minimal inference-time data consumption. +% Code: \url{https://anonymous.4open.science/r/SMART-8B11}. + +\end{abstract} + + + + + + +\section{Introduction} +\label{sec:intro} + + +Deep neural networks have achieved remarkable success across diverse domains, yet their deployment in safety-critical applications such as autonomous driving~\cite{feng2019trust} and medical diagnosis~\cite{zwaan2019bridging} demands more than just high predictive accuracy. These high-stakes scenarios require models to provide reliable uncertainty estimates that accurately reflect the true likelihood of prediction correctness, i.e., \emph{calibration}~\cite{guo2017calibration}. A well-calibrated model ensures informed decision-making and appropriate deferral to human experts when uncertainty is high. However, current models commonly suffer from severe miscalibration~\cite{guo2017calibration}, primarily overconfidence~\cite{guo2017calibration,wei2022mitigating,luo2025beyond}, where models assign high confidence scores to predictions that are frequently incorrect. The real-world consequences of such overconfident behavior can be catastrophic, such as wrong diagnostic decisions with high confidence. + +To address miscalibration, the research community has developed two primary streams of solutions. \emph{Train-time calibration} methods integrate calibration directly into the learning process via specialized data~\cite{hendrycksaugmix}, regularizations~\cite{muller2019does}, and designed loss objectives~\cite{mukhoti2020calibrating, tao2023dual}. However, these methods hardly apply to trained models. +In contrast, \emph{post-hoc calibration} methods~\cite{zadrozny2002transforming, zadrozny2001obtaining} operate easily on large pretrained models. Due to its simplicity and effectiveness, Temperature Scaling (TS)~\cite{guo2017calibration} has become the most widespread post-hoc method that learns a single scaling value on the validation set.However, this one-size-fits-all approach is inherently problematic, as miscalibration is not uniform across samples. +To address this, several methods have been proposed to learn separate temperatures per class~\cite{frenkel2021network} or semantic-aware groupings through clustering~\cite{yang2023beyond}. To facilitate more fine-grained temperature scaling, sample-adaptive methods propose to operate on distinctive sample-wise information~\cite{ding2021local, tomani2022parameterized}. + + +\begin{figure*}[t] + \centering + + \begin{subfigure}[b]{0.24\textwidth} + \centering + \includegraphics[width=\textwidth]{figs/hardness_vs_margin_cifar10_airplane_vs_automobile_filtered.pdf} + \caption{Attack vs Margin} + \label{fig:hardness_margin} + \end{subfigure} + \hfill + \begin{subfigure}[b]{0.24\textwidth} + \centering + \includegraphics[width=\textwidth]{figs/group_3_reliability.pdf} + \caption{Margin range 1.78-2.90} + \label{fig:group3_reliability} + \end{subfigure} + \hfill + \begin{subfigure}[b]{0.24\textwidth} + \centering + \includegraphics[width=\textwidth]{figs/group_5_reliability.pdf} + \caption{Margin range 4.00-4.99} + \label{fig:group5_reliability} + \end{subfigure} + \hfill + \begin{subfigure}[b]{0.24\textwidth} + \centering + \includegraphics[width=\textwidth]{figs/group_8_reliability.pdf} + \caption{Margin range 6.76-8.68} + \label{fig:group7_reliability} + \end{subfigure} + % \vspace{-10pt} + \caption{Relationship between min perturbation of FGSM~\cite{goodfellow2014explaining} and logit margin on CIFAR-10, with reliability diagrams for various margin groups from left to right.} + \vspace{-0.2in} + \label{fig:margin_reliability} +\end{figure*} + + + +~\citet{xiong2024proximity} calibrate predictions based on sample proximity, assigning larger temperatures to less proximate samples; ~\citet{yang2023beyond} apply larger temperatures to groups that are harder to distinguish (e.g., birds and airplanes sharing the same background in CIFAR-10); and ~\citet{ding2021local} exploit feature-space sparsity to adaptively guide temperature. Despite their methodological differences, these approaches share the same underlying motivation: sample hardness drives calibration. However, they rely on \emph{indirectly learned} proxies of difficulty. In contrast, we propose a direct and simple measure—the \emph{logit margin}, defined as the gap between the largest and second-largest logits. Empirical results (Figure~\ref{fig:margin_reliability}b–d) show that larger-margin samples are systematically easier and more under-confident, even when their confidence levels are identical. Moreover, the strong correlation between the margin and the minimum perturbation required to reach the decision boundary under attack (Figure~\ref{fig:margin_reliability}a) highlights the margin’s reliability as a hardness indicator. Finally, our theoretical analysis in Appendix~\ref{subsec:confidence-relation} demonstrates that the optimal temperature for any target confidence is tightly bounded by the margin, underscoring its effectiveness as a principled signal of sample difficulty for post-hoc calibration. + + +Another limitation inherent in current scaling-based methods is that they focus on optimizing the NLL loss, which theoretically does not guarantee a reduction in the calibration errors. In fact, as we prove in Appendix~\ref{subsec:Charbonnier-vs-nll}, certain scenarios can lead to a paradoxical outcome where NLL decreases while ECE simultaneously increases, thereby defeating the primary goal of calibration. To address this fundamental misalignment, we adopt a novel scaling objective function, Charbonnier–SoftECE. This new objective directly targets the calibration error. As also established by our theoretical analysis in Appendix~\ref{subsec:Charbonnier-soft-controls-smce}, optimizing with Charbonnier–SoftECE provably resolves the issue inherent in the NLL loss, ensuring that the optimization process aligns directly with the goal of improving calibration. + + + + +Building on these validated insights, we introduce \textbf{S}ample \textbf{M}argin-\textbf{A}ware \textbf{R}ecalibration of \textbf{T}emperature (SMART), a lightweight post-hoc calibration method that aims to learn a direct and minimalistic mapping from logit margins to temperatures: $T(\cdot): \mathbb{R}^+ \to \mathbb{R}^+$. Using Charbonnier–SoftECE as its learning objective, SMART is theoretically guaranteed to yield superior calibration. Experiments on various benchmarks and architectures validate the state-of-the-art effectiveness and efficiency of SMART, even with a minimal validation set. + + +% \vspace{-0.3in} + +% \paragraph{Contributions} +Our work is theory-driven and makes three key contributions: we first provide formal and empirical analysis showing that logit margin is a principled hardness indicator that tightly bounds the feasible temperature range, outperforming existing proxies with minimal computation; second, we prove a fundamental mismatch between NLL optimization and calibration quality, and resolve it through a novel Charbonnier–SoftECE objective that provably upper-bounds smooth calibration error; finally, building on these theoretical insights, we develop SMART, a lightweight margin-aware temperature mapping that achieves state-of-the-art calibration on CNNs and ViTs across long-tail and out-of-distribution datasets, remaining effective with as few as 50 validation samples. + + +\section{Related Work} + +Many techniques have been proposed in recent years to address the network miscalibration problem. These methods can be divided into two categories. The first category is post-hoc calibration techniques that adjust model predictions after training by optimizing additional parameters on a held-out validation set. Non-parametric methods include Histogram Binning~\citep{zadrozny2001obtaining}, Bayesian Binning into Quantiles~\citep{naeini2015obtaining}, Isotonic Regression~\citep{zadrozny2002transforming}, and Spline calibration~\citep{gupta2021calibration}. Parametric methods include Platt Scaling~\citep{platt1999probabilistic}, Beta calibration~\citep{kull2017beta} and its multi-class extension with Dirichlet distributions~\citep{kull2019beyond}, and Temperature Scaling~\citep{guo2017calibration} with enhanced variants~\citep{tomani2022parameterized, frenkel2021network, yang2023beyond, xiong2024proximity, tao2025feature, zhang2020mix}. + +The second category comprises training-based calibration methods that modify the learning process. Regularization approaches include data augmentation methods~\citep{thulasidasan2019mixup, hendrycksaugmix}, model ensembles~\citep{lakshminarayanan2017simple, zhang2020mix, rahaman2021uncertainty}, efficient Batch Ensemble~\citep{wen2020batchensemble}, Label Smoothing~\citep{muller2019does, szegedy2016rethinking}, and weight decay~\citep{guo2017calibration}. Loss modification methods include Brier Loss~\citep{brier1950verification}, differentiable auxiliary losses for calibration error~\citep{karandikar2021soft, kumar2018trainable, krishnan2020improving, bohdal2021meta}, and focal loss variants~\citep{gupta2020calibration, mukhoti2020calibrating, tao2023dual, hui2020evaluation, wang2021rethinking}, which relate to margin-based losses~\citep{cortes1995support, hadsell2006dimensionality, schroff2015facenet, weston2011wsabie}. + +Expected Calibration Error (ECE)~\citep{guo2017calibration} is a widely accepted metric in the literature. However, recent works~\citep{nixon2019measuring, kumar2019verified, roelofs2022mitigating, gupta2020calibration} point out the limitations of ECE, such as its sensitivity to bin size. For a fair comparison, we report the results in terms of four calibration metrics, i.e., ECE, Adaptive-ECE~\citep{ding2020revisiting} and classwise-ECE~\citep{kull2019beyond}. + +\section{Methodology} +\label{sec:methodology} + +We first present preliminaries in Section~\ref{subsec:preliminaries}, then establish margin as a principled hardness indicator in Section~\ref{subsec:margin_motivation}. We identify fundamental limitations of NLL-based calibration objectives in Section~\ref{subsec:nll_mismatch}, introduce our Charbonnier-SmoothSoftECE objective in Section~\ref{subsec:Charbonnier_softece}, and present the SMART framework in Section~\ref{subsec:smart_framework}. + + +\subsection{Preliminaries} +\label{subsec:preliminaries} + +A classification model is \textit{calibrated} if its predictive confidence matches its actual accuracy. For classifier $f_{\theta}$, input $\mathbf{x}$ with true label $y$, and predicted class $\hat{y}$, perfect calibration requires $\mathbb{P}(y=\hat{y} \mid p_{\theta}(\hat{y} \mid \mathbf{x})=p) = p$ for all confidence values $p \in [0,1]$. + +\textbf{Expected Calibration Error (ECE).} For classification model $f_{\theta}$ producing logits $\mathbf{z}_i \in \mathbb{R}^K$, the predictive probability for class $k$ is $p_{\theta}(y_i = k \mid \mathbf{x}_i)= \frac{\exp(z_{i,k})}{\sum_{j=1}^K \exp(z_{i,j})}$. To quantify calibration error, we partition samples into $B$ bins based on predicted confidence, compute average accuracy $\hat{a}_b$ and confidence $\hat{p}_b$ within each bin $b$, and measure their difference: +\begin{equation} +\mathrm{ECE} = \sum_{b=1}^{B} \frac{|I_b|}{N} \bigl| \hat{p}_b - \hat{a}_b \bigr|, +\end{equation} +where $I_b$ is the set of indices in bin $b$ and $N$ is the total number of samples. + +\textbf{Smooth Calibration Error (smCE).} Beyond binned ECE which suffers from discretization artifacts, we also consider the smooth calibration error~\citep{blasiok2023does}, defined as the worst-case correlation between the calibration residual and 1-Lipschitz probes of predicted confidence: +\begin{equation} +\mathrm{smCE}(f) := \sup_{\varphi \in \mathcal{H}} \left| \mathbb{E}\big[(a(X) - p(X))\varphi(p(X))\big] \right|, +\label{eq:smce_def} +\end{equation} +where $\mathcal{H} = \{\varphi : [0,1] \to [-1,1] \mid \mathrm{Lip}(\varphi) \leq 1\}$ is the class of 1-Lipschitz continuous functions, $p(X)$ denotes the predicted confidence (maximum softmax probability), and $a(X) = \mathbb{I}\{\hat{y}(X) = y\}$ is the correctness indicator. This continuous metric avoids binning artifacts and provides theoretical foundation for our objective design in Section~\ref{subsec:Charbonnier_softece}. + +\textbf{Temperature Scaling.} Temperature scaling (TS)~\citep{guo2017calibration} introduces positive scalar $T$ to adjust logit distribution before softmax: $p_{\theta,T}(y_i = k \mid \mathbf{x}_i) = \frac{\exp\bigl(z_{i,k} / T\bigr)}{\sum_{j=1}^K \exp\bigl(z_{i,j} / T\bigr)}$. Smaller temperature $T < 1$ sharpens the distribution, while larger $T > 1$ flattens it. Vanilla TS finds global $\hat{T} = \arg\min_{T>0} \mathcal{L}_{\text{NLL}}(\mathcal{D}_{val}, f_{\theta}, T)$ by minimizing NLL on a validation set. + + + +\subsection{Margin as a Principled Hardness Indicator} +\label{subsec:margin_motivation} + +% \begin{figure*}[t] +% \centering +% \begin{subfigure}[b]{0.25\textwidth} +% \includegraphics[height=3cm]{figs/numerical_margin.pdf} +% \end{subfigure}% +% \begin{subfigure}[b]{0.25\textwidth} +% \includegraphics[height=3cm]{figs/numerical_max_logit.pdf} +% \end{subfigure}% +% \begin{subfigure}[b]{0.25\textwidth} +% \includegraphics[height=3cm]{figs/numerical_normed_max_logit.pdf} +% \end{subfigure}% +% \begin{subfigure}[b]{0.25\textwidth} +% \includegraphics[height=3cm]{figs/sgd_loss_comparison_imagenet_vit_b_32.pdf} +% \end{subfigure}% +% \caption{\textbf{Numerical study of temperature adjustment indicators.} The \textbf{left} three panels show joint distributions of solved temperature $T$ versus candidate indicators across 1,000 sampled logit vectors. \textbf{Right:} Test ECE (dashed) and NLL (solid) during SMART training on ImageNet ViT-B/32.} +% \vspace{-0.2in} +% \label{fig:numerical_study} +% \end{figure*} + + +\begin{figure*}[t] + \centering + \begin{subfigure}[b]{0.25\textwidth} + \includegraphics[height=3.5cm]{figs/numerical_margin.pdf} + \end{subfigure}% + \begin{subfigure}[b]{0.25\textwidth} + \includegraphics[height=3.5cm]{figs/numerical_max_logit.pdf} + \end{subfigure}% + \begin{subfigure}[b]{0.25\textwidth} + \includegraphics[height=3.5cm]{figs/numerical_normed_max_logit.pdf} + \end{subfigure}% + \begin{subfigure}[b]{0.25\textwidth} + \includegraphics[height=3.5cm]{figs/sgd_loss_comparison_imagenet_vit_b_32.pdf} + \end{subfigure}% + \caption{\textbf{Numerical study of temperature adjustment indicators.} The \textbf{left} three panels show joint distributions of solved temperature $T$ versus candidate indicators across 1,000 sampled logit vectors. \textbf{Right:} Test ECE and NLL during training on ImgNet ViT-B/32.} + \vspace{-0.2in} + \label{fig:numerical_study} +\end{figure*} + + +Effective post-hoc calibration requires distinguishing between easy and hard samples to apply appropriate confidence adjustments. While existing methods~\citep{xiong2024proximity, yang2023beyond} recognize this need, they rely on indirectly learned proxies such as feature-space proximity or semantic clustering. We propose using the logit margin $m = z_{\max} - z_{2\text{nd}}$ as a direct hardness indicator, where $z_{\max}$ and $z_{2\text{nd}}$ are the largest and second-largest logits. + +As demonstrated in Figure~\ref{fig:margin_reliability}, samples with different margins exhibit systematically different calibration patterns even when sharing identical predicted confidence levels. Small-margin samples tend toward overconfidence while large-margin samples become underconfident, and margin correlates strongly with adversarial robustness ($r = 0.756$), confirming it captures proximity to decision boundaries. We now establish theoretically why margin provides superior temperature control compared to alternative indicators. + +For a given logit vector $\mathbf{z} \in \mathbb{R}^K$ and target confidence $\hat{p} \in (0,1)$, the temperature-confidence relationship $\frac{e^{z_{\max}/T}}{\sum_{k=1}^K e^{z_k/T}} = \hat{p}$ can be rearranged as $\sum_{k \neq M} e^{(z_k - z_{\max})/T} = S$ where $S := \frac{1}{\hat{p}} - 1$ and $M = \arg\max_k z_k$. Given only $z_{\max}$ and target $\hat{p} > 1/K$, we can construct configurations where all non-maximum logits equal $z_{\max} - \delta$ for varying $\delta > 0$, yielding $T = -\delta/\log(S/(K-1))$ which sweeps $(0,\infty)$ as $\delta$ varies. Thus maximum logit alone provides no bound on feasible temperatures. + +In contrast, margin provides tight constraints. For any non-maximum class $k$, we have $z_k \leq z_{2\text{nd}} = z_{\max} - m$, leading to bounds $e^{-m/T} \leq S \leq (K-1)e^{-m/T}$. Solving for $T$ yields: when $\hat{p} > 1/2$, $T \in [\frac{m}{-\log(S/(K-1))}, \frac{m}{-\log S}]$ (finite interval); when $1/K < \hat{p} \leq 1/2$, $T \in [\frac{m}{-\log(S/(K-1))}, +\infty)$ (finite lower bound). The interval width decreases as $m$ grows, and for binary classification the bounds coincide to uniquely determine $T$. Complete derivations appear in Appendix~\ref{subsec:confidence-relation}. + +Figure~\ref{fig:numerical_study} (left three panels) validates these results empirically. We sample 1,000 random logit vectors and numerically solve for temperatures achieving $\hat{p} = 0.8$. Margin exhibits clear functional structure with $T$ tightly constrained, while maximum logit and normalized maximum logit display scattered patterns spanning orders of magnitude. This establishes margin as the optimal scalar indicator for temperature-based calibration. + + +\subsection{The NLL-Calibration Mismatch} +\label{subsec:nll_mismatch} + +Current post-hoc calibration methods optimize negative log-likelihood (NLL) under the assumption that minimizing NLL improves calibration. We demonstrate this assumption can fail. Figure~\ref{fig:numerical_study} (rightmost panel) illustrates the phenomenon through a controlled experiment where we train SMART's margin-based temperature network on ImageNet ViT-B/32 using NLL as the training objective. While NLL decreases monotonically throughout 80 epochs, ECE begins increasing after epoch 30, creating clear divergence between objectives. By epoch 80, NLL has decreased by 15\% while ECE has increased by 8\% relative to epoch 30. This shows that following NLL gradients can actively worsen calibration despite improving likelihood. + +We formalize conditions under which NLL and calibration objectives have opposing gradients. Consider a margin slice $G \subset [m_{\min}, m_{\max}]$ defining sample region $A := \{x : m(x) \in G\}$ where $m(x) = z_{(1)}(x) - z_{(2)}(x)$ is the margin between top two logits. We study local temperature scaling by factor $s$ applied only to samples in $A$: $T_s(x) = T(x)/s$ if $x \in A$ and $T_s(x) = T(x)$ otherwise. At baseline $s=1$, analyzing how objectives change as $s$ varies reveals their directional preferences. + +Define $t_k := z_k / T$ (scaled logit), $q_k := \frac{e^{t_k}}{\sum_j e^{t_j}}$ (predicted probability), $\langle t \rangle_q := \sum_k q_k t_k$ (expected scaled logit), and $r_X(X) := \mathbb{P}(Y = M(X) \mid X)$ (pointwise top-class probability). For NLL $L_{\mathrm{nll}}(h_s)$ and calibration functional $\mathcal{C}[\psi] = \mathbb{E}[(a(X) - p_s)\psi(p_s)]$ with smooth probe $\psi$, the directional derivatives at $s=1$ are: +\begin{align} +\left.\frac{d}{ds}L_{\mathrm{nll}}(h_s)\right|_{s=1} &= \mathbb{E}\big[\mathbb{I}_A (t_Y - \langle t \rangle_q)\big], \label{eq:nll_deriv}\\ +\left.\frac{d}{ds}\mathcal{C}[\psi]\right|_{s=1} &= \mathbb{E}\Big[\mathbb{I}_A \, p(t_M - \langle t \rangle_q)\big(\psi'(p)(r_X(X) - p) - \psi(p)\big)\Big], \label{eq:calib_deriv} +\end{align} +where $\mathbb{I}_A$ indicates the margin slice, $t_M$ is the scaled logit of the top class, and $t_Y$ is that of the true class. The NLL gradient depends only on whether $t_Y$ exceeds $\langle t\rangle_q$, while the calibration gradient depends on the calibration gap $r_X(X) - p$ weighted by probe sensitivity. These different sensitivities create potential for directional conflict. + +Consider a margin slice $A$ with underconfident region $J_U$ having average calibration gap $\rho_U := \mathbb{E}[r_X(X) - p(X) \mid X \in J_U] > 0$ and overconfident region $J_O$ with gap $\rho_O := \mathbb{E}[p(X) - r_X(X) \mid X \in J_O] > 0$. Let $\mu_U, \mu_O$ denote relative proportions, $\gamma_{\min}, \gamma_{\max}$ be bounds on margin-to-temperature ratios in $A$, and $\Delta_G$ control logit spread. When underconfidence dominates NLL sensitivity such that $\rho_U \gamma_{\min} \mu_U > \gamma_{\max}\rho_O \mu_O + \Delta_G \mu_A$, there exists a sharpening direction where $\frac{d}{ds}L_{\mathrm{nll}}|_{s=1} < 0$ (NLL decreases) yet $\frac{d}{ds}\mathrm{smCE}|_{s=1} > 0$ (calibration worsens). The condition ensures that while sharpening helps underconfident samples in $J_U$, it harms overconfident samples in $J_O$ more severely, causing net calibration degradation despite NLL improvement. Detailed analysis appears in Appendix~\ref{subsec:Charbonnier-vs-nll}. + +This fundamental misalignment explains why NLL-based methods can achieve good likelihood while maintaining poor calibration. The mismatch occurs when calibration benefits from sharpening underconfident predictions are outweighed by costs from further sharpening overconfident predictions, yet NLL gradients favor overall sharpening due to different sensitivity to margin patterns. This motivates developing objectives that directly target calibration error rather than likelihood. + +\subsection{Charbonnier-Smoothed SoftECE Objective} +\label{subsec:Charbonnier_softece} + +\begin{figure}[t] + \centering + \begin{minipage}{0.48\linewidth} + \centering + \includegraphics[width=\linewidth]{figs/train_loss_convergence_vit_b_16.pdf} + \end{minipage}\hfill + \begin{minipage}{0.48\linewidth} + \centering + \includegraphics[width=\linewidth]{figs/gradient_norm_evolution_vit_b_16.pdf} + \end{minipage} + \caption{Post-hoc calibration on ImageNet ViT-B/16: training loss (left) and gradient norm (right) over training epochs SoftECE, and Charbonnier-SoftECE (ours).} + \vspace{-0.2in} + \label{fig:vitb16-hubersoeftece} +\end{figure} + +Section~\ref{subsec:nll_mismatch} demonstrated that NLL optimization can conflict with calibration goals. We require a differentiable objective that directly targets calibration error while remaining statistically efficient with limited validation data. Current approaches face a bias-variance tradeoff: binned ECE has low variance but high bias from fixed binning, while point-wise losses have low bias but high variance from binary correctness indicators. + +Following \citet{karandikar2021soft}, we adopt soft-binned ECE which balances this tradeoff through kernel smoothing. For sample $i$ with confidence $\hat{p}_i$ and bin centers $\{c_b\}_{b=1}^B$, soft membership weights $w_{i,b} = \frac{\exp(-\alpha(\hat{p}_i - c_b)^2)}{\sum_{b'} \exp(-\alpha(\hat{p}_i - c_{b'})^2)}$ distribute each sample's contribution across neighboring bins, creating smooth gradients. In continuous formulation with Gaussian kernel $k_\lambda(t) = e^{-\lambda t^2}$ and reference density $\rho(u)$ on $[0,1]$, this becomes: +\begin{equation} +\text{SoftECE}(f) := \mathbb{E}_X\!\left[\int_0^1 K_\lambda(p(X),u)|a(X)-u|\rho(u)du\right], +\label{eq:softece} +\end{equation} +where $K_\lambda(p,u) = \frac{k_\lambda(p-u)}{\int_0^1 k_\lambda(p-v)\rho(v)dv}$ is the normalized kernel, $p(X)$ is predicted confidence, and $a(X) = \mathbb{I}\{\hat{y}(X)=y\}$ is the correctness indicator. + +We enhance SoftECE with Charbonnier smoothing to achieve theoretical control over calibration quality. Replacing the absolute value with Charbonnier function $\phi_\delta(r) = \sqrt{r^2 + \delta^2}$ yields: +\begin{equation} +\mathcal{H}_{\lambda,\delta}(f) := \mathbb{E}_X\!\left[\int_0^1 K_\lambda(p(X),u)\phi_\delta(a(X)-u)\rho(u)du\right]. +\label{eq:charbonnier_softece} +\end{equation} +The Charbonnier function provides $C^\infty$ smoothness while satisfying $\phi_\delta(r) \geq |r|$, ensuring that minimizing $\mathcal{H}_{\lambda,\delta}$ never weakens calibration control compared to the absolute value formulation. Our key theoretical contribution establishes that this objective provides an upper bound on smooth calibration error. + +\begin{theorem}[Charbonnier-SoftECE Upper Bounds smCE] +\label{thm:Charbonnier_controls_smce} +Assume reference density $\rho$ satisfies $0 < \rho_{\min} \leq \rho(u) \leq \rho_{\max} < \infty$ for all $u \in [0,1]$ with condition number $\kappa := \rho_{\max}/\rho_{\min}$. Then for all classifiers $f$ and smoothing parameters $\delta \geq 0$: +\begin{equation} +\mathrm{smCE}(f) \leq \mathcal{H}_{\lambda,\delta}(f) + 2B_\lambda, +\label{eq:main_bound} +\end{equation} +where $B_\lambda := \sup_{p\in[0,1]}\int_0^1 |p-u|K_\lambda(p,u)\rho(u)du$ represents kernel approximation error. For Gaussian kernels with $\lambda \geq 1$, $B_\lambda \leq \frac{C_\kappa}{\sqrt{\lambda}}$ where $C_\kappa := \frac{2\kappa}{\sqrt{\pi}\,\mathrm{erf}(1)} \approx 1.339\kappa$. +\end{theorem} + +The proof (Appendix~\ref{subsec:Charbonnier-soft-controls-smce}) decomposes smCE using mollification: for any 1-Lipschitz probe $\varphi$, we write $\mathbb{E}[(a-p)\varphi(p)]$ as a smooth term controlled by $\mathcal{H}_{\lambda,\delta}$ plus approximation error bounded by $B_\lambda$. The bound splits into a model-dependent term $\mathcal{H}_{\lambda,\delta}(f)$ that can be optimized and a design-only term $2B_\lambda$ that tightens as $O(1/\sqrt{\lambda})$. Thus minimizing $\mathcal{H}_{\lambda,\delta}$ directly minimizes an upper bound on calibration error, resolving the NLL mismatch from Section~\ref{subsec:nll_mismatch}. + + +Figure~\ref{fig:vitb16-hubersoeftece} demonstrates the practical benefits of Charbonnier smoothing. On ImageNet ViT-B/16, Charbonnier-SoftECE achieves faster training convergence (left panel) while maintaining stable gradient norms throughout optimization (right panel). Standard SoftECE exhibits oscillations in later training epochs. The Charbonnier enhancement thus provides both theoretical guarantees and improved optimization stability. + + +In practice, we discretize Equation~\eqref{eq:charbonnier_softece} using $B=15$ soft bins with Gaussian kernel bandwidth $\sigma=0.05$ (corresponding to $\lambda=200$) and Charbonnier parameter $\delta=10^{-3}$. The bandwidth controls bias-variance tradeoff, the choice of hyperparameters $\lambda$ and $\delta$ exhibits stability across a reasonable range, as detailed in Appendix~\ref{subsec:hyperparameter_sensitivity}. + + + +\subsection{The SMART Framework} +\label{subsec:smart_framework} + +Building on the theoretical foundations established in Sections~\ref{subsec:margin_motivation}--\ref{subsec:Charbonnier_softece}, we introduce SMART (Sample Margin-Aware Recalibration of Temperature), which learns a direct mapping from margin to temperature. The framework combines margin as the input indicator (Section~\ref{subsec:margin_motivation}) with Charbonnier-SoftECE as the training objective (Section~\ref{subsec:Charbonnier_softece}). + +SMART implements a lightweight two-layer MLP that maps logit margin $m = z_{\max} - z_{2\text{nd}}$ to sample-specific temperature $T(m)$: $h = \text{ReLU}(W_1 m + b_1)$ and $T(m) = \text{softplus}(W_2 h + b_2) + \epsilon$, where the hidden dimension is 16 and $\epsilon = 10^{-1}$ ensures numerical stability. The softplus activation guarantees positive temperatures. This architecture requires only 49 trainable parameters regardless of the number of classes $K$, substantially fewer than existing parametric approaches: vector scaling requires $2K$ parameters, matrix scaling $K^2 + K$, class-dependent temperature scaling (CTS)~\citep{frenkel2021network} requires $K$, and spline calibration~\citep{gupta2021calibration} requires $13K$. For ImageNet with $K=1000$, these methods require thousands of parameters while SMART maintains minimal constant size. + +Training minimizes the Charbonnier-SoftECE objective $\mathcal{H}_{\lambda,\delta}$ from Equation~\eqref{eq:charbonnier_softece} on a validation set using Adam optimizer with initial learning rate $5 \times 10^{-3}$. For each sample, we compute its margin, predict temperature via the network, apply temperature scaling to logits, and compute the soft-binned calibration loss with Charbonnier smoothing. At inference, SMART computes the margin for each test sample, predicts its temperature through the trained network, and applies temperature scaling to obtain calibrated predictions. Complete training and inference procedures are detailed in Algorithm~\ref{alg:smart} (Appendix~\ref{Appendix:SMART_framework}). + + + +\begin{table*}[t] + +\centering +\scriptsize +\caption{\textbf{Comparison of Post-Hoc Calibration Methods in ECE (\%, $\downarrow$, 15 bins) Across Various Datasets and Models} (mean across 5 runs). The best-performing method for each dataset-model combination is in bold, and our method is highlighted. Full results in App. \ref{Appendix:full_calibration_performance}.} +\setlength{\tabcolsep}{2pt} +\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}cccccccccccc} +\toprule +\textbf{Dataset} & \textbf{Model} & \textbf{Vanilla} & \textbf{TS} & \textbf{PTS} & \textbf{CTS} & \textbf{Spline} & \textbf{GC} & \textbf{ProCal} & \textbf{FC} & \textbf{SMART} \\ + & & & \citeyear{guo2017calibration} & \citeyear{tomani2022parameterized} & \citeyear{frenkel2021network} & \citeyear{gupta2021calibration} & \citeyear{yang2023beyond} & \citeyear{xiong2024proximity} & \citeyear{tao2025feature} & \textbf{ours}\\ +\midrule +\multirow{2}{*}{CIFAR-10} + & ResNet-50 & $4.34$ & $1.38$ & $1.10$ & $0.83$ & $1.52$ & $1.37$ & $4.17$ & $1.66$ & $\cellgray\mathbf{0.76}$ \\ + & Wide-ResNet & $3.24$ & $0.93$ & $0.90$ & $0.81$ & $1.74$ & $0.89$ & $2.81$ & $1.12$ & $\cellgray\mathbf{0.43}$ \\ +\midrule +\multirow{2}{*}{CIFAR-100} + & ResNet-50 & $17.53$ & $5.61$ & $1.96$ & $3.67$ & $3.48$ & $5.70$ & $9.71$ & $2.91$ & $\cellgray\mathbf{1.37}$ \\ + & Wide-ResNet & $15.34$ & $4.50$ & $1.96$ & $3.01$ & $3.76$ & $4.55$ & $9.44$ & $4.49$ & $\cellgray\mathbf{1.80}$ \\ +\midrule +\multirow{6}{*}{ImageNet-1K} + & ResNet-50 & $3.65$ & $2.17$ & $0.95$ & $2.17$ & $0.62$ & $2.44$ & $1.08$ & $1.71$ & $\cellgray\mathbf{0.52}$ \\ + & DenseNet-121 & $2.53$ & $1.85$ & $1.02$ & $1.86$ & $0.81$ & $2.20$ & $1.52$ & $1.35$ & $\cellgray\mathbf{0.57}$ \\ + & Wide-ResNet & $5.43$ & $2.89$ & $1.14$ & $3.27$ & $0.66$ & $3.66$ & $1.57$ & $1.62$ & $\cellgray\mathbf{0.52}$ \\ + & Swin-B & $5.05$ & $3.91$ & $1.05$ & $1.53$ & $0.88$ & $4.95$ & $1.00$ & $5.05$ & $\cellgray\mathbf{0.46}$ \\ + & ViT-B-16 & $5.62$ & $3.60$ & $1.23$ & $4.65$ & $0.91$ & $4.39$ & $0.97$ & $5.65$ & $\cellgray\mathbf{0.48}$ \\ + & ViT-B-32 & $6.39$ & $3.93$ & $1.27$ & $2.12$ & $0.81$ & $4.67$ & $0.88$ & $6.39$ & $\mathbf{\cellgray{0.71}}$ \\ +\midrule +\multirow{5}{*}{ImageNet-C} + & ResNet-50 & $13.82$ & $1.97$ & $1.12$ & $1.69$ & $5.61$ & $2.69$ & $5.79$ & $2.51$ & $\cellgray\mathbf{0.62}$ \\ + & DenseNet-121 & $12.57$ & $1.58$ & $1.19$ & $1.44$ & $5.18$ & $2.01$ & $9.88$ & $9.44$ & $\cellgray\mathbf{0.63}$ \\ + & Swin-B & $12.03$ & $5.82$ & $1.53$ & $3.05$ & $2.58$ & $6.92$ & $2.53$ & $5.18$ & $\cellgray\mathbf{1.23}$ \\ + & ViT-B-16 & $8.28$ & $5.24$ & $1.27$ & $2.76$ & $1.71$ & $5.95$ & $1.96$ & $5.37$ & $\cellgray\mathbf{1.06}$ \\ + & ViT-B-32 & $7.69$ & $5.10$ & $1.07$ & $2.97$ & $1.43$ & $6.40$ & $1.55$ & $5.50$ & $\cellgray\mathbf{0.96}$ \\ +\midrule +\multirow{5}{*}{ImageNet-LT} + & ResNet-50 & $3.63$ & $2.01$ & $0.99$ & $2.17$ & $0.56$ & $2.20$ & $1.12$ & $1.80$ & $\cellgray\mathbf{0.56}$ \\ + & DenseNet-121 & $2.50$ & $1.80$ & $1.20$ & $1.88$ & $\mathbf{0.79}$ & $2.05$ & $1.79$ & $1.76$ & $\cellgray{0.81}$ \\ + & Wide-ResNet & $5.40$ & $2.99$ & $1.21$ & $2.87$ & $0.81$ & $3.59$ & $1.28$ & $1.68$ & $\cellgray\mathbf{0.53}$ \\ + & Swin-B & $4.69$ & $3.98$ & $1.21$ & $1.50$ & $0.79$ & $4.79$ & $0.95$ & $4.82$ & $\cellgray\mathbf{0.58}$ \\ + & ViT-B-16 & $5.58$ & $3.73$ & $1.14$ & $1.43$ & $0.66$ & $4.34$ & $0.77$ & $5.72$ & $\cellgray\mathbf{0.56}$ \\ + & ViT-B-32 & $6.28$ & $3.98$ & $1.35$ & $2.12$ & $0.72$ & $4.76$ & $0.83$ & $6.26$ & $\mathbf{\cellgray{0.60}}$ \\ +\midrule +\multirow{5}{*}{ImageNet-S} + & ResNet-50 & $22.32$ & $2.06$ & $1.69$ & $1.48$ & $9.76$ & $1.99$ & $9.52$ & $12.58$ & $\cellgray\mathbf{0.92}$ \\ + & DenseNet-121 & $20.13$ & $1.67$ & $1.93$ & $1.16$ & $9.20$ & $1.77$ & $12.93$ & $22.67$ & $\cellgray\mathbf{0.59}$ \\ + & Swin-B & $24.61$ & $6.50$ & $1.53$ & $3.62$ & $8.66$ & $6.92$ & $8.05$ & $1.70$ & $\cellgray\mathbf{1.26}$ \\ + & ViT-B-16 & $16.57$ & $5.75$ & $1.33$ & $2.84$ & $5.70$ & $6.36$ & $5.67$ & $1.93$ & $\cellgray\mathbf{0.98}$ \\ + & ViT-B-32 & $14.22$ & $4.99$ & $1.67$ & $3.25$ & $4.07$ & $6.23$ & $4.44$ & $1.56$ & $\cellgray\mathbf{0.87}$ \\ +\bottomrule +\end{tabular*} + +\label{table:comparison_ECE} +\vspace{-0.1in} +\end{table*} + + +\section{Experiments} +\label{section:Experiments} + + + +\subsection{Experimental Setup} + +\paragraph{Datasets} +We conduct experiments on several benchmark datasets, including CIFAR-10, CIFAR-100~\citep{krizhevsky2009learning}, and ImageNet~\citep{deng2009imagenet}. To probe robustness under common corruptions and distribution shifts, we include \textit{ImageNet-C} (All corruption type averaged, severity 5)~\citep{hendrycks2019benchmarking}, \textit{ImageNet-LT} (a long‐tailed variant with power‐law class imbalance)~\citep{liu2019large}, and \textit{ImageNet-Sketch} (sketch‐based OOD variant)~\citep{wang2019learning}. All experiments employ a training‐time batch size of 1024. CIFAR-10 and CIFAR-100 contain 60,000 images of size $32 \times 32$ pixels, with 10 and 100 classes, respectively, split into 45,000 training, 5,000 for validation and 10,000 test images. For ImageNet related dataset, we use 20\% of the official validation set, as the new validation set, with the remainder used as the test set. The testing batch size for all datasets is set to 128. + +\paragraph{Model Architectures.} +To demonstrate the generality of our calibration methods, we evaluate across a diverse collection of convolutional and transformer-based networks. For CIFAR-10 and CIFAR-100, we employ ResNet-50 and ResNet-110~\citep{he2016deep}, Wide-ResNet~\citep{zagoruyko2016wide}, and DenseNet-121~\citep{huang2017densely}, initialized with pretrained weights~\citep{mukhoti2020calibrating}. Each model is trained for 350 epochs using stochastic gradient descent with momentum 0.9, weight decay $5\times10^{-4}$, and a piecewise‐constant learning‐rate schedule $(0.1/0.01/0.001\ \text{ over } 150/100/100\ \text{ epochs})$. ImageNet and its variants are evaluated on PyTorch’s pretrained ResNet‐50 and DenseNet‐121 ~\citep{paszke2019pytorch}, the transformer designs Swin‐B~\citep{liu2021swin}, ViT‐B/16 and ViT‐B/32~\citep{dosovitskiy2021an}, and Wide‐ResNet‐50. This suite spans from compact CNNs to large‐capacity transformers, allowing us to assess calibration robustness under varying architectural inductive biases and model complexities. Calibration performance is primarily evaluated using ECE, with additional metrics including AdaECE and top-1 accuracy. All experiments are conducted on a NVIDIA 3090 GPU, with results averaged over 5 seeds. + + + + + +\subsection{Calibration Performance} +\label{Sec:Calibration_performance} + + + + +\begin{figure*}[th] +\centering +\includegraphics[width=\linewidth]{figs/adaece_main_figure.pdf} +% \caption{\textbf{Comparison of calibration methods using AdaECE\(\downarrow\) across various datasets and models.} From left to right: CIFAR-10 (ResNet-50), CIFAR-100 (ResNet-50), ImageNet (ResNet-50), ImageNet (ViT-B/16), ImageNet-C (ResNet-50), and ImageNet-C (ViT-B/16). Results are averaged.} +\caption{\textbf{Comparison of calibration methods using AdaECE(\(\downarrow\) across various datasets and models.} From left to right: CIFAR-10/100 (ResNet-50), ImageNet (ResNet-50, ViT-B/16), and ImageNet-C (ResNet-50, ViT-B/16), Averaged over 5 runs.} +\label{fig:adaece_comparison} +\vspace{-0.1in} +\end{figure*} + +\begin{table*}[th] +\centering +\scriptsize +\caption{\textbf{Comparison of Train-time Calibration Methods Using ECE(\(\downarrow\), \%, 15 bins) Across Various Datasets and Models.} The best-performing method for each dataset-model combination is in bold, and our method is highlighted, Averaged over 5 runs.} +\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}cccccccccccccc} +\toprule +\textbf{Dataset} & \textbf{Model} &\multicolumn{2}{c}{\textbf{NLL}} &\multicolumn{2}{c}{\textbf{Brier Loss}} &\multicolumn{2}{c}{\textbf{MMCE}} &\multicolumn{2}{c}{\textbf{LS-0.05}} &\multicolumn{2}{c}{\textbf{FLSD-53}} &\multicolumn{2}{c}{\textbf{FL-3}} \\ + & & base & ours & base & ours & base & ours & base & ours & base & ours & base & ours \\ +\midrule +\multirow{4}{*}{CIFAR10} + & ResNet-50 & 4.34 & \cellgray{\textbf{0.75}} & 1.81 & \cellgray{\textbf{0.96}} & 4.57 & \cellgray{\textbf{0.53}} & 2.97 & \cellgray{\textbf{0.51}} & 1.56 & \cellgray{\textbf{0.42}} & 1.47 & \cellgray{\textbf{0.43}} \\ + & ResNet-110 & 4.41 & \cellgray{\textbf{0.44}} & 2.56 & \cellgray{\textbf{0.60}} & 5.07 & \cellgray{\textbf{0.38}} & 2.09 & \cellgray{\textbf{0.28}} & 1.87 & \cellgray{\textbf{0.45}} & 1.54 & \cellgray{\textbf{0.54}} \\ + & DenseNet-121 & 4.51 & \cellgray{\textbf{0.53}} & 1.52 & \cellgray{\textbf{0.31}} & 5.10 & \cellgray{\textbf{0.66}} & 1.89 & \cellgray{\textbf{0.51}} & 1.23 & \cellgray{\textbf{0.62}} & 1.31 & \cellgray{\textbf{1.02}} \\ + & Wide-ResNet & 3.24 & \cellgray{\textbf{0.30}} & 1.25 & \cellgray{\textbf{0.38}} & 3.30 & \cellgray{\textbf{0.34}} & 4.25 & \cellgray{\textbf{0.36}} & 1.58 & \cellgray{\textbf{0.39}} & 1.68 & \cellgray{\textbf{0.54}} \\ +\midrule +\multirow{4}{*}{CIFAR100} + & ResNet-50 & 17.53 & \cellgray{\textbf{0.99}} & 6.54 & \cellgray{\textbf{1.01}} & 15.31 & \cellgray{\textbf{0.86}} & 7.81 & \cellgray{\textbf{1.50}} & 4.49 & \cellgray{\textbf{1.26}} & 5.16 & \cellgray{\textbf{0.56}} \\ + & ResNet-110 & 19.06 & \cellgray{\textbf{0.98}} & 7.87 & \cellgray{\textbf{0.87}} & 19.13 & \cellgray{\textbf{1.42}} & 11.03 & \cellgray{\textbf{1.01}} & 8.54 & \cellgray{\textbf{0.85}} & 8.65 & \cellgray{\textbf{0.73}} \\ + & DenseNet-121 & 20.99 & \cellgray{\textbf{1.86}} & 5.22 & \cellgray{\textbf{0.59}} & 19.10 & \cellgray{\textbf{1.34}} & 12.87 & \cellgray{\textbf{1.02}} & 3.70 & \cellgray{\textbf{0.91}} & 4.14 & \cellgray{\textbf{0.98}} \\ + & Wide-ResNet & 15.34 & \cellgray{\textbf{1.38}} & 4.35 & \cellgray{\textbf{1.00}} & 13.17 & \cellgray{\textbf{0.98}} & 4.88 & \cellgray{\textbf{1.24}} & 3.02 & \cellgray{\textbf{0.79}} & 2.14 & \cellgray{\textbf{1.12}} \\ +\bottomrule +\end{tabular*} + +\label{table:SMART_ece_compare_with_training_time_methods} +\vspace{-0.1in} +\end{table*} + + + + +We evaluate SMART against leading post-hoc calibration approaches including TS \citep{guo2017calibration}, PTS \citep{tomani2022parameterized}, CTS ~\citep{frenkel2021network}, and spline‐based calibration \citep{gupta2021calibration}, Group Calibration \citep{yang2023beyond}, ProCal \citep{xiong2024proximity} , Feature Clipping (FC) \citep{tao2025feature}, as well as uncalibrated (Vanilla) models across both standard settings and distribution shift scenarios. + +% \paragraph{Calibration on Standard Datasets} +SMART consistently outperforms these methods across CIFAR10, CIFAR-100, and ImageNet-1K (Table~\ref{table:comparison_ECE}), significantly reducing calibration error. The most notable improvement is seen in CIFAR-100, where SMART excels while Spline, despite its strong performance on other datasets, struggles. This highlights SMART's robustness across datasets with varying complexities. CNNs, which often suffer from overconfidence, are generally well-calibrated with TS-based methods. However, transformers see limited calibration improvements from TS-based methods, with SMART outperforming them by a large margin. On larger datasets like ImageNet-1K, SMART maintains its advantage with consistently lower ECE values. SMART works well on both CNN and ViTs where GC and FC failed. + +\paragraph{Robustness under Class Imbalance and Distribution Shift} +Across long‐tailed (ImageNet-LT) and corrupted scenarios (ImageNet-Sketch, ImageNet-C), SMART's sample‐wise temperature adaptation consistently outperforms global and class‐wise scalers. Uniform approaches such as TS struggle to accommodate underrepresented classes or severe input degradations, leading to pronounced calibration drift. Spline, FC and ProCal failed on Imagenet-S with CNNs where SMART still performs robustly. + + + + + +\paragraph{Calibration Performance on AdaECE} +We also evaluate SMART using Adaptive Expected Calibration Error (AdaECE) to provide a comprehensive view of its performance, shown in Figure~\ref{fig:adaece_comparison}, with additional results available in Appendix~\ref{Appendix:Calibration_Performance_on_Other_Metrics}. SMART demonstrates superior performance on AdaECE compared to traditional calibration methods across diverse settings. AdaECE addresses limitations of standard ECE by accounting for uneven confidence distributions, providing a more reliable measure of calibration quality. SMART consistently achieves the lowest AdaECE values and variance across CNN and ViT architectures and datasets (CIFAR and ImageNet variants), demonstrating its robustness to dataset shifts and model architectures. Notably, SMART outperforms more complex methods like Spline calibration and CTS in calibration error and variance while requiring fewer parameters. + +By leveraging instance‐level temperature through logit margins, SMART yields stable calibration gains across diverse distribution shifts. Its lightweight per‐sample inference preserves efficiency while delivering robustness that neither fixed nor ensemble temperature schemes can match. In contrast, Spline collapses on particularly challenging shifts such as ImageNet-S and ImageNet-C — whereas our method consistently sustains the lowest and most stable calibration error even under these adverse conditions. + + +\subsection{Comparison With Training-Time Calibration Methods} +\label{sec:training_time_calibration} + + +We evaluate SMART alongside training-time calibration techniques in Table \ref{table:SMART_ece_compare_with_training_time_methods}, including Brier Loss~\citep{brier1950verification}, Maximum Mean Calibration Error (MMCE) \citep{kumar2018trainable}, Label Smoothing (LS-0.05)~\citep{szegedy2016rethinking}, and Focal Loss variants (FLSD-53 and FL-3)~\citep{mukhoti2020calibrating}. This shows that combining SMART with these methods consistently enhances calibration performance across various models and datasets, further validating SMART’s effectiveness alongside training-time approaches. Moreover, as seen in Table \ref{table:comparison_ECE}, SMART alone, as a post-hoc calibration method, already outperforms these train-time techniques with minimal computational overhead, while train-time methods require significantly more resources. + + + + +\subsection{Scalability with Validation Data} +\label{SMART_is_Data_Efficient} + + +\paragraph{Scalability with Validation Data} + + +SMART demonstrates superior ability to leverage increasing validation sample sizes compared to competing calibration methods, shown in Figure \ref{fig:ece_vs_valsize}. While all approaches struggle with minimal validation data, SMART exhibits continuous performance improvement throughout the entire range of sample sizes tested, ultimately achieving the lowest calibration error. In contrast, the alternative methods display more limited utilization of additional validation samples. TS reaches a performance plateau at moderate sample sizes and fails to improve further, while PTS exhibits concerning instability in the mid-range sample sizes, implicitly reflecting the NLL mismatch. GC demonstrates the most problematic behavior, with significant performance spikes that indicate poor robustness to varying validation set sizes. The consistent improvement trajectory of SMART manifests the margin provides a robust signal that enables more effective temperature estimation as additional validation samples become available. This superior sample utilization capability makes SMART particularly valuable in practical applications where validation data availability may vary. + +\begin{figure}[ht] + % \vspace{-0.15in} + \centering + \includegraphics[width=0.9\linewidth]{figs/calibration_methods_comparison_imagenet_vit_b_32.pdf} + \caption{\textbf{ECE(\(\downarrow\), \%, 15 bins) versus validation sample size.} +Comparison of calibration methods on ImageNet (ViT-B/32), averaged over five runs.} + % \vspace{-0.2in} + \label{fig:ece_vs_valsize} +\end{figure} + +% \begin{wrapfigure}{r}{0.6\linewidth} +% \centering +% % \vspace{-2in} +% \includegraphics[width=\linewidth]{figs/calibration_methods_comparison_imagenet_vit_b_32.pdf} +% \caption{\textbf{ECE(\(\downarrow\), \%, 15 bins) versus validation sample size.} +% Comparison of calibration methods on ImageNet (ViT-B/32), averaged over five runs.} +% \label{fig:ece_vs_valsize} +% % \vspace{-0.2in} +% \end{wrapfigure} + + + +% \paragraph{Bin‐Size Sensitivity} +% Figure \ref{fig:ece_vs_valsize} (right) illustrates the mean ECE across the 6 architectures trained on ImageNet and calibrated using TS, PTS, Spline and SMART. While small bin sizes result in a systematically smaller ECE - a known bias \cite{Kumar2019} - SMART outperforms the other TS-based methods and Spline for all bin sizes with rankings being unchanged. + + + + +\subsection{Ablation Studies} +\label{ablation_Studies} + +\paragraph{Choice of Calibration Objective} +We compare how various calibration objective influence SMART's calibration performance (Table~\ref{tab:imagenet_ece_losses}). While all tested objectives enable significant improvements over vanilla, they exhibit distinct behavior patterns across architectures. NLL and label smoothing losses, despite their prevalence in classification tasks, demonstrate suboptimal calibration performance due to their indirect relationship with confidence estimation objectives. MSE and Brier score offer more reliable improvements by directly penalizing squared confidence errors, yet their effectiveness fluctuates between CNN and transformer architectures. Charbonnier-SoftECE emerges as the superior choice by directly optimizing the calibration metric itself, achieving both the lowest average error and the smallest variance across diverse model architectures, making it the most stable choice for SMART’s temperature mapping. + + +\begin{table}[ht] +\centering +\scriptsize +\caption{\textbf{Different Calibration Objective.} ECE (\%, $\downarrow$, 15 bins) on ImageNet averaged over 5 runs.} +\resizebox{\linewidth}{!}{% +\begin{tabular}{cccccccc} +\toprule +\textbf{Architecture} & \textbf{Method} & \textbf{NLL} & \textbf{LS} & \textbf{MSE} & \textbf{Brier} & \textbf{SoftECE} & \textbf{Charbonnier-SoftECE} \\ +\midrule +\multirow{3}{*}{\begin{tabular}[c]{@{}l@{}}ResNet-50\\(Top-1\,=\,0.761)\end{tabular}} + & TS & $2.04$ & $14.33$ & $3.69$ & $2.31$ & $3.16$ & $2.12$ \\ + & PTS & $1.04$ & $1.87$ & $1.89$ & $1.88$ & $1.88$ & $0.94$ \\ + & \cellgray\textbf{SMART} + & \cellgray\textbf{0.93} + & \cellgray\textbf{1.09} + & \cellgray\textbf{1.39} + & \cellgray\textbf{1.38} + & \cellgray\textbf{0.65} + & \cellgray\textbf{0.52} \\ +\midrule +\multirow{3}{*}{\begin{tabular}[c]{@{}l@{}}ViT-B/16\\(Top-1\,=\,0.810)\end{tabular}} + & TS & $3.73$ & $6.05$ & $5.58$ & $3.11$ & $3.10$ & $3.08$ \\ + & PTS & $5.69$ & $3.22$ & $2.40$ & $2.57$ & $1.15$ & $0.77$ \\ + & \cellgray\textbf{SMART} + & \cellgray\textbf{3.62} + & \cellgray\textbf{3.11} + & \cellgray\textbf{0.84} + & \cellgray\textbf{0.80} + & \cellgray\textbf{0.89} + & \cellgray\textbf{0.48} \\ +\bottomrule +\end{tabular}} + +\vspace{-0.2in} +\label{tab:imagenet_ece_losses} +\end{table} + + + +\paragraph{Choice of Indicators} +We evaluated six candidate uncertainty signals as inputs to our temperature network on ImageNet-1K (Table~\ref{table:Ablation_study_for_alternative_input}): predictive entropy, predicted confidence, full logit vectors, maximum logit, mean-normalized logit deviation, and our proposed margin. The margin consistently achieves the lowest calibration error across all tested architectures, outperforming alternative indicators by substantial margins. While full logit vectors contain rich information, they introduce excessive noise that degrades performance in limited-data scenarios. Simpler scalar measures like maximum logit or predicted confidence fail to adequately capture the competitive dynamics between top classes that drive miscalibration. The margin's superior performance stems from its ability to distill prediction uncertainty into a minimal yet complete representation that directly reflects decision boundary proximity, enabling robust calibration across diverse model architectures. + + +\begin{table}[ht] + \centering + \scriptsize + \caption{\textbf{Comparison on alternative on-the-shelf indicator}.} + % \vspace{-0.1in} + \resizebox{\linewidth}{!}{% + \begin{tabular}{ccccccc} + \toprule + \textbf{Model} & \textbf{Entropy} & \textbf{Conf.} & \textbf{All Logits} & \textbf{Logit$_{\text{max}}$} & \textbf{Logit$_{\text{max}}$ - $\overline{\text{Logits}}$} & \cellgray\textbf{Margin (ours)} \\ + \midrule + ResNet-50 & 0.87 & 0.97 & 0.87 & 0.91 & 0.85 & \cellgray\textbf{0.52} \\ + DenseNet-121 & 0.62 & 0.89 & 0.79 & 0.80 & 0.84 & \cellgray\textbf{0.57} \\ + Wide-ResNet & 1.00 & 1.22 & 0.92 & 0.57 & 0.63 & \cellgray\textbf{0.52} \\ + Swin-B & 0.62 & 0.81 & 0.89 & 0.78 & 0.87 & \cellgray\textbf{0.46} \\ + ViT-B/16 & 0.90 & 0.75 & 0.97 & 0.91 & 1.20 & \cellgray\textbf{0.48} \\ + \bottomrule + \end{tabular}} + + \vspace{-0.2in} + \label{table:Ablation_study_for_alternative_input} +\end{table} + + +\section{Conclusion and Limitation} +\label{sec:conclusion} + +We introduced SMART, a lightweight recalibration method leveraging the logit margin as a principled calibration indicator for precise temp adjustment. By capturing sample hardness through this indicative signal, SMART achieves SOTA calibration performance with minimal parameters compared to existing methods. Our Charbonnier-Smoothed SoftECE objective enables stable optimization as validation data scales. Extensive experiments confirm SMART's robustness across diverse architectures, datasets, and challenging distribution shifts, consistently outperforming current post-hoc and even training-based methods. Future work could explore integrating SMART with other uncertainty quantification methods or investigate other hardness indicator to further improve calibration and robustness in safety-critical applications. +\paragraph{Limitation} While SMART demonstrates excellent performance across tested scenarios, its effectiveness may vary slightly for extremely specialized domains with highly skewed class distributions. Additionally, though our method requires minimal validation data, performance could degrade in zero-shot scenarios where no domain-specific calibration samples are available. + + + +% Acknowledgements should only appear in the accepted version. +% \section*{Acknowledgements} + +% \textbf{Do not} include acknowledgements in the initial version of the paper +% submitted for blind review. + +% If a paper is accepted, the final camera-ready version can (and usually should) +% include acknowledgements. Such acknowledgements should be placed at the end of +% the section, in an unnumbered section that does not count towards the paper +% page limit. Typically, this will include thanks to reviewers who gave useful +% comments, to colleagues who contributed to the ideas, and to funding agencies +% and corporate sponsors that provided financial support. + +\section*{Impact Statement} + +% Authors are \textbf{required} to include a statement of the potential broader +% impact of their work, including its ethical aspects and future societal +% consequences. This statement should be in an unnumbered section at the end of +% the paper (co-located with Acknowledgements -- the two may appear in either +% order, but both must be before References), and does not count toward the paper +% page limit. In many cases, where the ethical impacts and expected societal +% implications are those that are well established when advancing the field of +% Machine Learning, substantial discussion is not required, and a simple +% statement such as the following will suffice: + +This paper presents work whose goal is to advance the field of Machine +Learning. There are many potential societal consequences of our work, none +which we feel must be specifically highlighted here. + +% The above statement can be used verbatim in such cases, but we encourage +% authors to think about whether there is content which does warrant further +% discussion, as this statement will be apparent if the paper is later flagged +% for ethics review. + +% In the unusual situation where you want a paper to appear in the +% references without citing it in the main text, use \nocite +% \nocite{langley00} + +\bibliography{icml2026} +\bibliographystyle{icml2026} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% APPENDIX +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\newpage +\appendix +\onecolumn +\input{appendix} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\end{document} + +% This document was modified from the file originally made available by +% Pat Langley and Andrea Danyluk for ICML-2K. This version was created +% by Iain Murray in 2018, and modified by Alexandre Bouchard in +% 2019 and 2021 and by Csaba Szepesvari, Gang Niu and Sivan Sabato in 2022. +% Modified again in 2023 and 2024 by Sivan Sabato and Jonathan Scarlett. +% Previous contributors include Dan Roy, Lise Getoor and Tobias +% Scheffer, which was slightly modified from the 2010 version by +% Thorsten Joachims & Johannes Fuernkranz, slightly modified from the +% 2009 version by Kiri Wagstaff and Sam Roweis's 2008 version, which is +% slightly modified from Prasad Tadepalli's 2007 version which is a +% lightly changed version of the previous year's version by Andrew +% Moore, which was in turn edited from those of Kristian Kersting and +% Codrina Lauth. 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