\documentclass[accepted]{uai2023} % for initial submission
% \documentclass[accepted]{uai2023} % 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]{uai2023} % ptmx math instead of Computer
% Modern (has noticable issues)
% \documentclass[mathfont=newtx]{uai2023} % 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{mathtools} % amsmath with fixes and additions
% \usepackage{siunitx} % for proper typesetting of numbers and units
\usepackage{booktabs} % commands to create good-looking tables
\usepackage{tikz} % nice language for creating drawings and diagrams

% for cross referencing the main text
% PLEASE ONLY USE xr IN THE SUPPLEMENTARY MATERIAL. 
% In the main paper, hard code any cross-reference to the supplementary material. 
\usepackage{xr} 
\externaldocument{wang_8}

%% 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.)

%% Self-defined macros
\newcommand{\swap}[3][-]{#3#1#2} % just an example


%%%%%%%%%%%%%%%%%% Author's Own Pre


\usepackage{amsfonts,amssymb,amsthm}
\usepackage{textcomp}
\usepackage{xcolor}
\usepackage{dashrule}

\usepackage{xspace}
\usepackage{bm, bbm}
\usepackage{subfig}
\usepackage{multirow}
\usepackage{hyperref}
\hypersetup{
	colorlinks   = true, %Colours links instead of ugly boxes
	urlcolor     = blue, %Colour for external hyperlinks
	linkcolor    = blue, %Colour of internal links
	citecolor   = blue %Colour of citations
}

\usepackage{algorithm}
\usepackage{algorithmic}

% \usepackage{eqparbox}
% \renewcommand{\algorithmiccomment}[1]{\hfill\eqparbox{COMMENT}{\# #1}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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}

\makeatletter
\newcommand\addstarred[1]{%
    \expandafter\let\csname\string#1@nostar\endcsname#1%
    \edef#1{\noexpand\@ifstar\expandafter\noexpand\csname\string#1@star\endcsname\expandafter\noexpand\csname\string#1@nostar\endcsname}%
    \expandafter\newcommand\csname\string#1@star\endcsname%
}
\makeatother

\newcommand{\ceil}[1]{\left\lceil#1\right\rceil}
\newcommand{\floor}[1]{\left\lfloor#1\right\rfloor}
\newcommand{\round}[1]{\lceil#1\rfloor}
% \addstarred\round[1]{\left\lceil#1\right}
\newcommand{\1}[1]{\mathbbm{1}{\{#1\}}}
\addstarred\1[1]{\mathbbm{1}{\left\{#1\right\}}}
\renewcommand{\O}{{O}}
\renewcommand{\ge}{\geqslant}
\renewcommand{\le}{\leqslant}

\DeclarePairedDelimiter\abs{\lvert}{\rvert}%
\DeclarePairedDelimiter\norm{\lVert}{\rVert}%

\newcommand{\N}{\mathbb{N}}
\newcommand{\R}{\mathbb{R}}
\renewcommand{\P}{\mathbb{P}}
\newcommand{\E}{\mathbb{E}}
\newcommand{\brai}{{(i)}}
\newcommand{\braj}{{(j)}}

\DeclareMathOperator*{\argmax}{arg\,max}
\DeclareMathOperator*{\argmin}{arg\,min}
\DeclareMathOperator*{\KL}{{\normalfont KL}}
\DeclareMathOperator*{\kl}{{\normalfont kl}}
\DeclareMathOperator*{\ERT}{\mathbb{E}[\normalfont \text{R}_T (\mathcal{A})]}
% \DeclareMathOperator{\fr}{\texttt{fr}}
% \DeclareMathOperator*{\Kfr}{\mathcal{K}^\fr}
\DeclareMathOperator*{\RT}{\text{\normalfont R}_T}
\DeclareMathOperator*{\RTI}{\text{\normalfont R}_T^{(i)}}

\newcommand{\mo}[1]{{\textcolor{blue}{MH says: #1}}}
\newcommand{\dt}[1]{{\textcolor{brown}{DT: #1}}}

\newcommand{\ACMAB}{\texttt{AC-MA2B}\xspace}
\newcommand{\FreeExp}{\texttt{FreeExp}\xspace}
\newcommand{\NoFreeExp}{\texttt{NoFreeExp}\xspace}
\newcommand{\MATOBHR}{\texttt{MA2B-HR}\xspace}
\newcommand{\MATOB}{\texttt{MA2B}\xspace}
\newcommand{\mK}{\mathcal{K}}
\newcommand{\mM}{\mathcal{M}}

\ifodd 2
\newcommand{\rev}[1]{{\color{blue}#1}}%revise of the text
\newcommand{\com}[1]{\textbf{\color{blue} (COMMENT: #1)}}%comment of the text
\else
\newcommand{\rev}[1]{#1}
\newcommand{\com}[1]{}
\fi

\ifodd 1
\newcommand{\icmlrev}[1]{{\color{blue}#1}}%revise of the text
\newcommand{\icmlcom}[1]{\textbf{\color{blue} (COMMENT: #1)}}%comment of the text
\else
\newcommand{\icmlrev}[1]{#1}
\newcommand{\icmlcom}[1]{}
\fi

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\title{Exploration for Free: How Does Reward Heterogeneity \\ Improve Regret in Cooperative Multi-agent Bandits?\\(Supplementary Material)}

% The standard author block has changed for UAI 2023 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]{Xuchuang Wang}
\author[2]{Lin Yang}
\author[3]{Yu-Zhen Janice Chen}
\author[1]{Xutong Liu}
\author[3]{Mohammad Hajiesmaili}
\author[3]{Don Towsley}
\author[1]{John C.S. Lui}
% Add affiliations after the authors
\affil[1]{%
Department of Computer Science and Engineering\\
The Chinese University of Hong Kong\\
Hong Kong
}
\affil[2]{%
School of Intelligence Science and Technology\\
Nanjing University\\
Jiangsu, China
}
\affil[3]{%
College of Information and Computer Sciences\\
University of Massachusetts Amherst\\
Massachusetts, USA
  }
  
\begin{document}

\onecolumn %% Turn this off if single column is desired for the supplement
\maketitle

% This Supplementary Material should be submitted as a separate file. Please do not append the Supplementary Material to the main paper.

\vspace{5pt}
\appendix
\input{sections/_related_works}
\input{sections/_other_possible_models}
% \input{sections/_application_scenarios.tex}
% \input{sections/_simulation_details}
\input{sections/_proofs.tex}

% \bibliography{freeExp}
% References
\bibliography{bibliography}

\end{document}
