\newcommand{\PRcomment}[1]{\textcolor{teal}{[PRD: #1]}}
\newcommand{\TODO}[1]{\textcolor{red}{TODO: #1}}
\newcommand{\fix}[1]{\textcolor{orange}{ #1}}

\newcommand{\w}{w}
\newcommand{\cost}{c}
\newcommand{\A}{A}
\newcommand{\bvec}{b}

\newcommand{\R}{\mathbb{R}}
\newcommand{\E}{\mathbb{E}}
\newcommand{\Et}{\mathbb{E}_{(x,y) \sim \Pd_T} \ }
\newcommand{\Etf}{\mathbb{E}_{(x,y,z) \sim \Pd_{T}^{f}} \ }
\newcommand{\Etstar}{\mathbb{E}_{(x,y,z) \sim \Pd_{S}^{*}} \ }
\newcommand{\Es}{\mathbb{E}_{(x,y) \sim \Pd_S} \ }

\newcommand{\D}{\mathcal{D}}
\newcommand{\X}{\mathcal{X}}
\newcommand{\Y}{\mathcal{Y}}
\newcommand{\Z}{\mathcal{W}}
\newcommand{\C}{\mathcal{Y}}
\newcommand{\Pd}{\mathcal{P}}
\newcommand{\St}{\mathcal{S}}
\newcommand{\Ac}{\mathcal{A}}
\newcommand{\Hyp}{\mathcal{H}}
\newcommand{\loss}{\mathcal{L}}
\newcommand{\OT}{\text{OT}}
\newcommand*\dt{\mathop{}\!\mathrm{d}}
\newcommand{\dtproof}{\dt\Pi^*((x_s,y_s,z_s),(x_t,y_t,z_{t}^{f}))}
\newcommand{\dtproofshort}{\dt\Pi^*(\vb{w}_{s},\vb{w}_{t}^{f})}

\newcommand{\dq}{q}
\newcommand{\dqreg}{\dq_{\text{reg}}}
\newcommand{\ldq}{l_{g}}
\newcommand{\lspo}{l_{SPO}}
\newcommand{\ldqsym}{\Tilde{l}_{g}}



\newcommand{\loadeq}[1]{%
    \ExecuteMetaData[equations.tex]{eq#1}%
}


\newcommand{\sth}{\text{s.t.}}
\newcommand{\argmax}{\text{\normalfont argmax}}
\newcommand{\argmin}{\text{\normalfont argmin}}