%\documentclass{uai2024} % for initial submission
\documentclass[accepted]{uai2024} % after acceptance, for a revised version; 
% also before submission to see how the non-anonymous paper would look like 
                        
%% There is a class option to choose the math font
% \documentclass[mathfont=ptmx]{uai2024} % ptmx math instead of Computer
                                         % Modern (has noticeable issues)
% \documentclass[mathfont=newtx]{uai2024} % newtx fonts (improves upon
                                          % ptmx; less tested, no support)
% NOTE: Only keep *one* line above as appropriate, as it will be replaced
%       automatically for papers to be published. Do not make any other
%       change above this note for an accepted version.

%% Choose your variant of English; be consistent
\usepackage[american]{babel}
% \usepackage[british]{babel}

%% Some suggested packages, as needed:
\usepackage{natbib} % has a nice set of citation styles and commands
    \bibliographystyle{plainnat}
    \renewcommand{\bibsection}{\subsubsection*{References}}
\usepackage{booktabs} % commands to create good-looking tables
%% Provided macros
% \smaller: Because the class footnote size is essentially LaTeX's \small,
%           redefining \footnotesize, we provide the original \footnotesize
%           using this macro.
%           (Use only sparingly, e.g., in drawings, as it is quite small.)

\usepackage{amssymb,amsmath,amsthm}
\usepackage{array, bibentry}
\usepackage{hyperref}
\usepackage{mathtools}
\usepackage[parfill]{parskip}
\usepackage{graphicx}
\usepackage{mathrsfs}
\usepackage{caption}
\usepackage{subcaption}
\usepackage{bm}
\usepackage{amsfonts, amscd}
\usepackage{courier}
\usepackage{multicol}
\usepackage{tcolorbox}
\usepackage{float}
\usepackage{color}
\usepackage[nameinlink,capitalize]{cleveref}
\usepackage{svg}
\usepackage[ruled, noend, noline]{algorithm2e}
\usepackage{multirow}
\usepackage{bbding}

%% Self-defined macros
\newcommand{\T}{\mathsf{T}}

\newcommand{\brho}{\boldsymbol{\rho}}
\newcommand{\blambda}{\boldsymbol{\lambda}} 
\newcommand{\balpha}{\boldsymbol{\alpha}}
\newcommand{\btheta}{\boldsymbol{\theta}}
\newcommand{\bmu}{\boldsymbol{\mu}}
\newcommand{\bc}{\boldsymbol{c}}
\newcommand{\bd}{{d}}
\newcommand{\bs}{\boldsymbol{s}}
\newcommand{\bg}{{g}}
\newcommand{\bp}{\boldsymbol{p}}
\newcommand{\bv}{{v}}
\newcommand{\bF}{\boldsymbol{F}}
\newcommand{\bG}{\boldsymbol{G}}
\newcommand{\bH}{{H}}
\newcommand{\bI}{{I}}
\newcommand{\bV}{\boldsymbol{V}}
\newcommand{\bQ}{\boldsymbol{Q}}
\newcommand{\boldm}{\boldsymbol{m}}
\newcommand{\EE}{\mathbb{E}}
\newcommand{\Ts}{\mathsf{T}}
\newcommand{\KL}{\text{K\L{}}}
\newcommand{\bx}{{x}}
\newcommand{\by}{{y}}
\newcommand{\bbR}{\mathbb{R}}
\newcommand{\mycite}[1]{\citeauthor{#1} (\citeyear{#1})}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{definition}{Definition}[section]
\newtheorem{assumption}{Assumption}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{remark}{Remark}[section]
\newtheorem{example}{Example}[section]
% iteration macros
\newcommand{\xk}{x_k}
\newcommand{\Hk}{H_k}
\newcommand{\Qk}{Q_k}
\newcommand{\gk}{g_k}
\newcommand{\gradk}{\nabla f (\xk)}
\newcommand{\dk}{d_k}
\newcommand{\gkn}{g_{k+1}}
\newcommand{\xkn}{x_{k+1}}
\newcommand{\dkn}{d_{k+1}}
\newcommand{\lk}{\lambda_k}
\newcommand{\ghg}{\gk^TH_kg_k}
\newcommand{\dhd}{\dk^TH_k\dk}
\newcommand{\dhtd}{\dk^T\left(H_k + \lambda_kI\right)\dk}
\newcommand{\hkap}{\tilde{H}_k}
\newcommand{\vk}{V_k}
\newcommand{\spa}{\mathsf{span}}
\newcommand{\spak}{\mathcal L_k}
\newcommand{\he}{\theta}
\newcommand{\hv}{\xi}
\newcommand{\her}{\theta^{R}}
\newcommand{\hvr}{\xi^{R}}
\newcommand{\rayxi}{\mathcal R(\Hk, \xi)}
\newcommand{\rayg}{\mathcal R(\Hk, \gk)}
\newcommand{\raygr}{\mathcal R(\Hk^R, \gk)}

\definecolor{salmon}{rgb}{1.0, 0.55, 0.41}
\definecolor{amethyst}{rgb}{0.6, 0.4, 0.8}
\newcommand{\chenyu}[1]{\textcolor{salmon}{[Chenyu: #1]}}
\newcommand{\tank}[1]{\textcolor{red}{[Jiyuan: #1]}}
\newcommand{\cz}[1]{\textcolor{amethyst}{[CZ: #1]}}
% \usepackage[inline]{showlabels}
% \definecolor{lavenderpurple}{rgb}{0.59, 0.48, 0.71}
% \renewcommand{\showlabelfont}{\small\color{lavenderpurple}}
% \usepackage[nameinlink,capitalize]{cleveref}
\crefname{assumption}{Assumption}{Assumptions}
\crefname{Assumption}{Assumption}{Assumptions}
\crefname{Lemma}{Lemma}{Lemmata}
\crefname{lemma}{Lemma}{Lemmata}
\crefname{Theorem}{Theorem}{Theorems}
\crefname{theorem}{Theorem}{Theorems}
\crefname{corollary}{Corollary}{Corollaries}
\crefname{proposition}{Proposition}{Propositions}
\crefname{claim}{Claim}{Claims}
\crefname{procedure}{Procedure}{Procedures}
\crefname{algorithm}{Algorithm}{Algorithms}
\crefname{figure}{Figure}{Figures}
\crefname{remark}{Remark}{Remarks}
\crefname{section}{Section}{Sections}
\crefname{procedure}{Procedure}{Procedures}
\crefname{definition}{Definition}{Definitions}
\crefname{example}{Example}{Examples}
\crefname{table}{Table}{Tables}
\crefname{equation}{}{}
\crefname{enumi}{}{}
\crefname{algocf}{Algorithm}{Algorithms}

% \usepackage[inline]{showlabels}
% \definecolor{lavenderpurple}{rgb}{0.59, 0.48, 0.71}
% \renewcommand{\showlabelfont}{\tiny\color{lavenderpurple}}


\title{A Homogenization Approach for Gradient-Dominated Stochastic Optimization}

% The standard author block has changed for UAI 2024 to provide
% more space for long author lists and allow for complex affiliations
%
% All author information is authomatically removed by the class for the
% anonymous submission version of your paper, so you can already add your
% information below.
%
% Add authors
\author[1,*]{Jiyuan Tan}
\author[2,*]{Chenyu Xue}
\author[2]{Chuwen Zhang}
\author[3]{Qi Deng}
\author[3]{Dongdong Ge}
\author[1]{Yinyu Ye}
% Add affiliations after the authors
\affil[*]{%
    Equal Contribution
}
\affil[1]{%
    Department of Management Science and Engineering\\
    Stanford University\\
    Palo Alto, California, USA
}
\affil[2]{%
    Research Institute for Interdisciplinary Sciences, Shanghai University of Finance and Economics\\
    Shanghai, China
}
\affil[3]{%
    Antai College of Economics and Management, Shanghai Jiao Tong University\\
    Shanghai, China
  }

\begin{document}
\maketitle

\input{abstract}
\input{paper}
\clearpage




% References
\bibliography{ref}

\newpage

\onecolumn

\title{A Homogenization Approach for Gradient-Dominated Stochastic Optimization\\(Supplementary Material)}
\maketitle

\appendix
\input{proof}


\end{document}
