
\newcommand{\todo}[1]{{\color{blue} {\sc} [TODO: #1]}}
\newcommand{\note}[1]{{\color{red} {\sc} [Comment: #1]}}

\newcommand{\fw}[1]{{\color{purple}\bf [FW: #1]}}
\newcommand{\cg}[1]{{\color{brown}\bf [CG: #1]}}
\newcommand{\rk}[1]{{\color{green}\bf [RK: #1]}}

\newcommand{\methodoneshort}{DGM-OLS}
\newcommand{\methodoneupper}{\sf dgm}
\newcommand{\methodonelong}{De-biased Gaussian Mechanism for Ordinary Least Squares }
\newcommand{\methodonebrev}{DGM}
\newcommand{\methodtwoshort}{RMGM-OLS}
\newcommand{\methodtwoupper}{\sf rmgm}
\newcommand{\methodtwolong}{Random Mixing prior to Gaussian Mechanism for Ordinary Least Squares }
\newcommand{\methodtwobrev}{RMGM}
\newcommand{\methodthreeshort}{BGM-OLS}
\newcommand{\methodthreeupper}{\sf bgm}
\newcommand{\methodthreelong}{Biased Gaussian mechanism }
\newcommand{\methodthreebrev}{BGM}

\newcommand{\tabincell}[2]{\begin{tabular}{@{}#1@{}}#2\end{tabular}}

\newtheorem{assumption}{Assumption}
\newtheorem{lemma}{Lemma}
\newtheorem{theorem}{Theorem}
\newtheorem{corollary}{Corollary}
\newtheorem{definition}{Definition}
\newcommand{\assumptionautorefname}{Assumption}
\newcommand{\corollaryautorefname}{Corollary}
\newcommand{\lemmaautorefname}{Lemma}
\newcommand{\algorithmautorefname}{Algorithm}

\newcommand{\bu}{\mathbf{u}}
\newcommand{\bbE}{\mathbb{E}}
\newcommand{\bbR}{\mathbb{R}}
\newcommand{\bbP}{\mathbb{P}}
\newcommand{\bv}{\mathbf{v}}
\newcommand{\bw}{\mathbf{w}}
\newcommand{\bx}{\mathbf{x}}
\newcommand{\by}{\mathbf{y}}
\newcommand{\bz}{\mathbf{z}}
\newcommand{\bg}{\mathbf{g}}
\newcommand{\ba}{\mathbf{a}}
\newcommand{\bb}{\mathbf{b}}
\newcommand{\bc}{\mathbf{c}}
\newcommand{\calR}{\mathcal{R}}
\newcommand{\calO}{\mathcal{O}}
\newcommand{\calA}{\mathcal{A}}
\newcommand{\calB}{\mathcal{B}}
\newcommand{\calD}{\mathcal{D}}
\newcommand{\calM}{\mathcal{M}}
\newcommand{\calP}{\mathcal{P}}
\newcommand{\calY}{\mathcal{Y}}
\newcommand{\calX}{\mathcal{X}}
\newcommand{\calN}{\mathcal{N}}
\newcommand{\calU}{\mathcal{U}}
\newcommand{\calI}{\mathcal{I}}
\newcommand{\calS}{\mathcal{S}}
\newcommand*\diff{\mathop{}\!\mathrm{d}}

\DeclareMathOperator*{\argmin}{arg\,min}
\DeclareMathOperator*{\plim}{plim}
\newtheorem{claim}{Claim}


\newcommand{\hfl}{hFL}
\newcommand{\vfl}{vFL}

\newcommand{\xin}[1]{\textcolor{red}{[Xin: #1]}}
\newcommand{\ruihan}[1]{\textcolor{blue}{[Ruihan: #1]}}
\newcommand{\kevin}[1]{\textcolor{cyan}{Kevin: #1}}
\newcommand{\jiankai}[1]{\textcolor{red}{Jiankai: #1}}
\newcommand{\tianyi}[1]{\textcolor{red}{Tianyi: #1}}
% \newcommand{\kevin}[1]{{\color{cyan} #1\xspace}}
