Transactions on Machine Learning Research (TMLR)

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• Starter files
• main.tex5.0 KB

  %% Transactions on Machine Learning Research (TMLR).
  %%   \usepackage{tmlr}            anonymous submission (default)
  %%   \usepackage[accepted]{tmlr}  camera-ready with authors + OpenReview link
  %%   \usepackage[preprint]{tmlr}  de-anonymized, no TMLR branding (arXiv)
  \documentclass[10pt]{article}
  \usepackage{tmlr}
  
  \usepackage{amsmath,amssymb,amsthm}
  \usepackage{booktabs}
  \usepackage{graphicx}
  \usepackage{tikz}
  \usepackage{algorithm}
  \usepackage{algpseudocode}
  \usepackage{hyperref}
  \usepackage{url}
  
  \newtheorem{theorem}{Theorem}
  \newtheorem{assumption}{Assumption}
  
  \title{Implicit Regularization of Gradient Descent in Wide Networks}
  
  % Authors are hidden while the package is used without [accepted] or [preprint].
  \author{\name First Author \email first@university.edu \\
        \addr Department of Computer Science\\
        University Name
        \AND
        \name Second Author \email second@lab.org \\
        \addr Research Lab}
  
  \def\month{MM}  % camera-ready: month of publication
  \def\year{YYYY} % camera-ready: year of publication
  \def\openreview{\url{https://openreview.net/forum?id=XXXX}} % camera-ready: OpenReview link
  
  \begin{document}
  
  \maketitle
  
  \begin{abstract}
  We study the implicit bias of gradient descent on two-layer networks in the
  infinite-width limit. We show that, under a separability assumption, the
  learned predictor converges in direction to the minimum-norm interpolant of
  the associated kernel, and we validate the prediction on synthetic data.
  \end{abstract}
  
  \section{Introduction}
  
  Overparameterized networks generalize despite interpolating the training
  data; a leading explanation is the implicit regularization of the optimizer
  \citep{neyshabur2015search}. In the infinite-width regime the dynamics
  linearize around initialization \citep{jacot2018ntk}, which makes the
  question tractable. Our contributions:
  \begin{itemize}
    \item a characterization of the limit predictor (Theorem~\ref{thm:main});
    \item a finite-width approximation bound;
    \item experiments on synthetic and image data (Section~\ref{sec:exp}).
  \end{itemize}
  
  \section{Setting}
  
  Let $f_\theta(x) = \frac{1}{\sqrt{m}} \sum_{j=1}^m a_j\, \sigma(w_j^\top x)$
  with $\theta = (a, W)$ and data $\{(x_i, y_i)\}_{i=1}^n$, $\|x_i\| = 1$.
  
  \begin{assumption}\label{ass:sep}
  The kernel matrix $K_{ij} = \mathbb{E}_w[\sigma(w^\top x_i)\sigma(w^\top x_j)]$
  is positive definite.
  \end{assumption}
  
  \begin{algorithm}[t]
  \caption{Gradient descent on the squared loss}\label{alg:gd}
  \begin{algorithmic}[1]
  \State initialize $\theta_0$ with i.i.d.\ $\mathcal{N}(0,1)$ entries
  \For{$t = 0, \dots, T-1$}
    \State $\theta_{t+1} \gets \theta_t - \eta \nabla_\theta \frac{1}{2n}\sum_i (f_{\theta_t}(x_i) - y_i)^2$
  \EndFor
  \end{algorithmic}
  \end{algorithm}
  
  \section{Main result}
  
  \begin{theorem}\label{thm:main}
  Under Assumption~\ref{ass:sep}, with step size $\eta < 1/\lambda_{\max}(K)$,
  Algorithm~\ref{alg:gd} satisfies
  $\lim_{m \to \infty} \lim_{t \to \infty} f_{\theta_t}(x) = k(x)^\top K^{-1} y$
  for every $x$, where $k(x)_i = \mathbb{E}_w[\sigma(w^\top x)\sigma(w^\top x_i)]$.
  \end{theorem}
  
  \begin{proof}[Proof sketch]
  In the limit the network is linear in $\theta$ around $\theta_0$, so gradient
  descent on the squared loss converges to the minimum-norm interpolant in the
  tangent feature space, whose kernel is $K$. Full details are in
  Appendix~\ref{app:proof}.
  \end{proof}
  
  \section{Experiments}\label{sec:exp}
  
  Figure~\ref{fig:width} shows the gap between the finite-width predictor and
  the kernel interpolant shrinking as width grows; Table~\ref{tab:results}
  reports test accuracy.
  
  \begin{figure}[t]
  \centering
  \begin{tikzpicture}[x=1.1cm,y=1.1cm]
    \draw[->] (0,0) -- (5,0) node[right] {\small $\log_2 m$};
    \draw[->] (0,0) -- (0,3) node[above] {\small gap};
    \draw[thick] plot coordinates {(0.5,2.6) (1.5,1.9) (2.5,1.3) (3.5,0.8) (4.5,0.45)};
    \foreach \x/\y in {0.5/2.6,1.5/1.9,2.5/1.3,3.5/0.8,4.5/0.45} \fill (\x,\y) circle (1.4pt);
  \end{tikzpicture}
  \caption{Distance between the trained network and the kernel interpolant as a function of width.}
  \label{fig:width}
  \end{figure}
  
  \begin{table}[t]
  \caption{Test accuracy (\%) over 5 seeds.}
  \label{tab:results}
  \centering
  \begin{tabular}{lccc}
  \toprule
  Method & $m = 2^8$ & $m = 2^{12}$ & kernel \\
  \midrule
  GD, $\eta = 0.1$ & $91.2 \pm 0.4$ & $93.0 \pm 0.3$ & $93.1 \pm 0.2$ \\
  GD, $\eta = 0.5$ & $90.8 \pm 0.6$ & $92.9 \pm 0.3$ & $93.1 \pm 0.2$ \\
  \bottomrule
  \end{tabular}
  \end{table}
  
  \section{Related work}
  
  Implicit bias of gradient methods \citep{neyshabur2015search}, the neural
  tangent kernel \citep{jacot2018ntk}, and lazy training.
  
  \section{Conclusion}
  
  We characterized the infinite-width limit of gradient descent and verified
  it empirically. Extending the analysis beyond the lazy regime is open.
  
  \subsubsection*{Broader impact statement}
  This is a theoretical study with no direct societal impact.
  
  \subsubsection*{Acknowledgments}
  Acknowledgments go here for the accepted version.
  
  \bibliography{references}
  \bibliographystyle{tmlr}
  
  \appendix
  \section{Proof of Theorem~\ref{thm:main}}\label{app:proof}
  Full proof.
  
  \end{document}
  

• Libraries and docs
• references.bib980 B

  @inproceedings{jacot2018ntk,
    author    = {Jacot, Arthur and Gabriel, Franck and Hongler, Cl{\'e}ment},
    title     = {Neural Tangent Kernel: Convergence and Generalization in Neural Networks},
    booktitle = {Advances in Neural Information Processing Systems},
    volume    = {31},
    year      = {2018}
  }
  
  @inproceedings{neyshabur2015search,
    author    = {Neyshabur, Behnam and Tomioka, Ryota and Srebro, Nathan},
    title     = {In Search of the Real Inductive Bias: On the Role of Implicit Regularization in Deep Learning},
    booktitle = {International Conference on Learning Representations, Workshop Track},
    year      = {2015}
  }
  
  @article{zhang2021understanding,
    author  = {Zhang, Chiyuan and Bengio, Samy and Hardt, Moritz and Recht, Benjamin and Vinyals, Oriol},
    title   = {Understanding deep learning (still) requires rethinking generalization},
    journal = {Communications of the ACM},
    volume  = {64},
    number  = {3},
    pages   = {107--115},
    year    = {2021}
  }
  

• tmlr.bst26 KBVendored from github.com on 2026-08-19.

  %% File: `tmlr.bst'
  %% A copy of iclm2010.bst, which is 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)
  %%
  %% 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,
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   %
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    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{ }}{  ff{ }}{  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}
  

• tmlr.sty6.4 KBVendored from github.com on 2026-08-19.

  %%%% TMLR Macros (LaTex)
  %%%% Adapted by Hugo Larochelle and Fabian Pedregosa from the 
  %%%% ICLR stylefile Macros and borrowing from the JMLR Macros
  %%%% Style File
  
  %%%% Last edited, January 2021 by Chris J. Maddison
  %%%%      Change font choice
  
  % Change the overall width of the page.  If these parameters are
  %       changed, they will require corresponding changes in the
  %       maketitle section.
  %
  \usepackage{eso-pic} % used by \AddToShipoutPicture
  \RequirePackage{fancyhdr}
  \RequirePackage{natbib}
  
  \usepackage[T1]{fontenc}
  \usepackage{lmodern}
  
  % modification to natbib citations
  \setcitestyle{authoryear,round,citesep={;},aysep={,},yysep={;}}
  
  \renewcommand{\topfraction}{0.95}   % let figure take up nearly whole page
  \renewcommand{\textfraction}{0.05}  % let figure take up nearly whole page
  
  
  %%%%%%%% Options
  \newif\if@accepted\@acceptedfalse
  \DeclareOption{accepted}{%
      \@acceptedtrue
  }
  % declare preprint option, which creates a preprint version ready for
  % upload to, e.g., arXiv
  \newif\if@preprint\@preprintfalse
  \DeclareOption{preprint}{
    \@preprinttrue
    \@acceptedtrue
  }
  
  \DeclareOption*{\PackageWarning{tmlr}{Unknown ‘\CurrentOption’}}
  \ProcessOptions\relax
  
  % Specify the dimensions of each page
  
  \setlength{\paperheight}{11in}
  \setlength{\paperwidth}{8.5in}
  
  
  \oddsidemargin 0in    %   Note \oddsidemargin = \evensidemargin
  \evensidemargin 0in
  \marginparwidth 0.07 true in
  %\marginparwidth 0.75 true in
  %\topmargin 0 true pt           % Nominal distance from top of page to top of
  %\topmargin 0.125in
  \topmargin -0.625in
  \addtolength{\headsep}{0.25in}
  \textheight 9.0 true in       % Height of text (including footnotes & figures)
  \textwidth 6.5 true in        % Width of text line.
  \widowpenalty=10000
  \clubpenalty=10000
  
  
  % \thispagestyle{empty}        \pagestyle{empty}
  \flushbottom \sloppy
  
  % We're never going to need a table of contents, so just flush it to
  % save space --- suggested by drstrip@sandia-2
  \def\addcontentsline#1#2#3{}
  
  % Title stuff, taken from deproc.
  \def\maketitle{\par
  \begingroup
     \def\thefootnote{\fnsymbol{footnote}}
     \def\@makefnmark{\hbox to 0pt{$^{\@thefnmark}$\hss}} % for perfect author
                                                          % name centering
  %   The footnote-mark was overlapping the footnote-text,
  %   added the following to fix this problem               (MK)
     \long\def\@makefntext##1{\parindent 1em\noindent
                              \hbox to1.8em{\hss $\m@th ^{\@thefnmark}$}##1}
     \@maketitle \@thanks
  \endgroup
  \setcounter{footnote}{0}
  \let\maketitle\relax \let\@maketitle\relax
  \gdef\@thanks{}\gdef\@author{}\gdef\@title{}\let\thanks\relax}
  
  \newlength\aftertitskip     \newlength\beforetitskip
  \newlength\interauthorskip  \newlength\aftermaketitskip
  
  %% Changeable parameters.
  \setlength\aftertitskip{0.3in plus 0.2in minus 0.2in}
  \setlength\beforetitskip{0.05in plus 0.08in minus 0.08in}
  \setlength\interauthorskip{0.08in plus 0.1in minus 0.1in}
  \setlength\aftermaketitskip{0.3in plus 0.1in minus 0.1in}
  
  \def\@startauthor{\noindent \normalsize\bf}
  \def\@endauthor{}
  
  \def\addr{\small\it}%
  \def\email{\hfill\small\it}%
  \def\name{\normalsize\bf}%
  \def\name{\normalsize\bf}%
  \def\AND{\@endauthor\rm\hss \vskip \interauthorskip \@startauthor}
  
  % The toptitlebar has been raised to top-justify the first page
  
  \usepackage{fancyhdr}
  \pagestyle{fancy}
  \fancyhead{}
  
  \if@accepted
      \if@preprint
          \lhead{}
      \else
          \lhead{Published in Transactions on Machine Learning Research (\month/\year)}
      \fi
  \else
         \lhead{Under review as submission to TMLR}
  \fi
  
  % Title (includes both anonymized and non-anonymized versions)
  \def\@maketitle{\vbox{\hsize\textwidth
  %\linewidth\hsize \vskip 0.1in \toptitlebar \centering
  {\LARGE\bf\sffamily \@title\par}\vskip \aftertitskip
  %\bottomtitlebar % \vskip 0.1in %  minus
  \if@accepted
      \if@preprint
          \@startauthor \@author \@endauthor
      \else
          \@startauthor \@author \\ \\ {\bf Reviewed on OpenReview:} \openreview \@endauthor
      \fi
  \else
      \@startauthor Anonymous authors\\Paper under double-blind review \@endauthor
  \fi
  \vskip 0.3in minus 0.1in}}
  
  \renewenvironment{abstract}{\vskip.075in\centerline{\large\bf\sffamily
  Abstract}\vspace{0.5ex}\begin{quote}}{\par\end{quote}\vskip 1ex}
  
  % sections with less space
  \def\section{\@startsection {section}{1}{\z@}{-2.0ex plus
      -0.5ex minus -.2ex}{1.5ex plus 0.3ex
  minus0.2ex}{\large\bf\raggedright\sffamily}}
  
  \def\subsection{\@startsection{subsection}{2}{\z@}{-1.8ex plus
  -0.5ex minus -.2ex}{0.8ex plus .2ex}{\normalsize\bf\raggedright\sffamily}}
  \def\subsubsection{\@startsection{subsubsection}{3}{\z@}{-1.5ex
  plus      -0.5ex minus -.2ex}{0.5ex plus
  .2ex}{\normalsize\bf\raggedright\sffamily}}
  \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}}
  \def\subsubsubsection{\vskip
  5pt{\noindent\normalsize\rm\raggedright\sffamily}}
  
  
  % Footnotes
  \footnotesep 6.65pt %
  \skip\footins 9pt plus 4pt minus 2pt
  \def\footnoterule{\kern-3pt \hrule width 12pc \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 .5pc
  
  
  %\leftmargin2em
  \leftmargin3pc
  \leftmargini\leftmargin \leftmarginii 2em
  \leftmarginiii 1.5em \leftmarginiv 1.0em \leftmarginv .5em
  
  %\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%
  
  
  \def\toptitlebar{\hrule height4pt\vskip .25in\vskip-\parskip}
  
  \def\bottomtitlebar{\vskip .29in\vskip-\parskip\hrule height1pt\vskip
  .09in} %
  

• LICENSE11 KBVendored from github.com on 2026-08-19.

                                   Apache License
                             Version 2.0, January 2004
                          http://www.apache.org/licenses/
  
     TERMS AND CONDITIONS FOR USE, REPRODUCTION, AND DISTRIBUTION
  
     1. Definitions.
  
        "License" shall mean the terms and conditions for use, reproduction,
        and distribution as defined by Sections 1 through 9 of this document.
  
        "Licensor" shall mean the copyright owner or entity authorized by
        the copyright owner that is granting the License.
  
        "Legal Entity" shall mean the union of the acting entity and all
        other entities that control, are controlled by, or are under common
        control with that entity. For the purposes of this definition,
        "control" means (i) the power, direct or indirect, to cause the
        direction or management of such entity, whether by contract or
        otherwise, or (ii) ownership of fifty percent (50%) or more of the
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