%
% Load some packages
% Cross-cite between different files
% \usepackage {xr-hyper}
% \makeatletter
% \newcommand*{\addFileDependency}[1]{% argument=file name and extension
%   \typeout{(#1)}
%   \@addtofilelist{#1}
%   \IfFileExists{#1}{}{\typeout{No file #1.}}
% }
% \makeatother

% \newcommand*{\myexternaldocument}[1]{%
%     \externaldocument{#1}%
%     \addFileDependency{#1.tex}%
%     \addFileDependency{#1.aux}%
% }

\usepackage[T1]{fontenc}
\usepackage{amsmath, amssymb}
\usepackage{algorithm,algorithmic}
\usepackage{graphicx}
\usepackage{subfigure}
\usepackage{stfloats}
\usepackage{paralist}
\usepackage{placeins}
\usepackage{colortbl}
\definecolor{LightCyan}{rgb}{0.8, 0.9, 1}
\usepackage{multirow}
% \usepackage[colorlinks,
% 	linkcolor=black,
% 	citecolor=black,
% 	urlcolor=magenta,
% 	linktocpage,
% 	plainpages=false,
% 	draft=false]{hyperref}

	
% Some macros for simplicity
\newcommand{\BlackBox}{\rule{1.5ex}{1.5ex}}  % end of proof
\newenvironment{proof}{\par\noindent{\bf Proof\ }}{\hfill\BlackBox\\[2mm]}
\newtheorem{eg}{Example}
\newtheorem{Def}{Definition}
\newtheorem{thm}{Theorem}
\newtheorem{prop}{Proposition}
\newtheorem{lem}{Lemma}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{ass}{Assumption}

% boldsymbol
\def \c {\boldsymbol{c}}
\def \w {\boldsymbol{w}}

% mathbb
\def \Ibb {\mathbb{I}}
\def \N {\mathbb{N}}
\def \R {\mathbb{R}}
\def \E {\mathbb{E}}

% mathbf
\def \a {\mathbf{a}}
\def \b {\mathbf{b}}
\def \d {\mathbf{d}}
\def \e {\mathbf{e}}
\def \f {\mathbf{f}}
\def \g {\mathbf{g}}
\def \h {\mathbf{h}}
\def \m {\mathbf{m}}
\def \p {\mathbf{p}}
\def \q {\mathbf{q}}
\def \s {\mathbf{s}}
\def \t {\mathbf{t}}
\def \u {\mathbf{u}}
\def \v {\mathbf{v}}
\def \x {\mathbf{x}}
\def \y {\mathbf{y}}
\def \z {\mathbf{z}}

% mathcal
\def \A {\mathcal{A}}
\def \B {\mathcal{B}}
\def \C {\mathcal{C}}
\def \D {\mathcal{D}}
\def \Ec {\mathcal{E}}
\def \F {\mathcal{F}}
\def \G {\mathcal{G}}
% \def \H {\mathcal{H}}
\def \I {\mathcal{I}}
\def \K {\mathcal{K}}
\def \L {\mathcal{L}}
\def \M {\mathcal{M}}
\def \Nc {\mathcal{N}}
\def \P {\mathcal{P}}
\def \Q {\mathcal{Q}}
\def \Rc {\mathcal{R}}
\def \S {\mathcal{S}}
\def \T {\mathcal{T}}
\def \V {\mathcal{V}}
\def \W {\mathcal{W}}
\def \X {\mathcal{X}}

% bar
\def \db {\bar{\d}}
% \def \Rb {\bar{\Rt}}
\def \ub {\bar{\u}}
\def \vb {\bar{\v}}
\def \wb {\bar{\w}}
\def \xb {\bar{\x}}

% hat
\def \Sh {\hat{S}}
\def \Uh {\hat{U}}
\def \Vh {\hat{V}}
\def \Wh {\hat{W}}

% widehat
\def \bwh {\widehat{b}}
\def \fwh {\widehat{f}}
\def \Gwh {\widehat{G}}
\def \hwh {\widehat{\h}}
\def \gwh {\widehat{\g}}
\def \pwh {\widehat{p}}
\def \Twh {\widehat{T}}
\def \Uwh {\widehat{U}}
\def \vwh {\widehat{\v}}
\def \wwh {\widehat{\w}}
\def \xwh {\widehat{\x}}

\def \lah {\widehat{\lambda}}
\def \alh {\widehat{\alpha}}
\def \muh {\widehat{\mu}}
\def \nah {\widehat{\nabla}}
\def \rhh {\widehat{\rho}}

% tilde
\def \Otc {\tilde{\mathcal{O}}}

% widetilde
\def \bt {\widetilde{\b}}
\def \bts {\widetilde{b}}
\def \ft {\widetilde{\f}}
\def \fts {\widetilde{f}}
\def \Ot {\widetilde{O}}
\def \Ut {\widetilde{U}}
\def \wt {\widetilde{\w}}
\def \xt {\widetilde{\x}}
\def \zt {\widetilde{\z}}
\def \zts {\widetilde{z}}

% others
\def \Pb {\overline{P}}
\def \bmeta {\bm \eta}
\def \d {\mathrm{d}}

% \DeclareMathOperator*{\Reg}{Regret}
% \DeclareMathOperator*{\WAReg}{A-Regret}
% \DeclareMathOperator*{\SAReg}{SA-Regret}
% \DeclareMathOperator*{\DReg}{D-Regret}
% \DeclareMathOperator*{\poly}{poly}
\DeclareMathOperator*{\argmin}{argmin}
\DeclareMathOperator*{\argmax}{argmax}

% % Build 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}}

% \def\abovespaceLarge{\abovestrut{0.30in}}
% \def\belowspaceLarge{\belowstrut{0.15in}}
