\usepackage{amsthm}

\newtheorem{remark}{Remark}
\newtheorem{cor}{Corollary}
\newtheorem{condition}{Condition}
\newtheorem{lemma}{Lemma}
\newtheorem{theorem}{Theorem}
\newcommand{\tmix}{t_{\text{mix}}}
\newcommand{\ar}[1]{\textcolor{blue}{AR:{#1}}}

\newcommand{\lambdamax}{\lambda_{\text{max}}}
\newcommand{\onehot}{\text{one-hot}}
\newcommand{\lambdamin}{\lambda_{\text{min}}}
\newcommand{\fisherinfj}{I_{j,\thetastar}}
\newcommand{\fisherinfk}{I_{k,\thetastar}}
\newcommand{\conditionnum}{\gamma_{\thetastar}}
\newcommand{\fn}{f_n}
\newcommand{\rnthetastar}{r_n(\thetastar)}
\newcommand{\rntheta}{r_n(\theta)}
\newcommand{\dsname}{GINC\xspace}
\newcommand{\Topic}{Concept\xspace}
\newcommand{\Topics}{Concepts\xspace}
\newcommand{\topic}{concept\xspace}
\newcommand{\topics}{concepts\xspace}
\newcommand{\hiddenseg}{H}
\newcommand{\hiddensegstart}{h^{\text{start}}}
\newcommand{\hiddensegstarttest}{h^{\text{start}}_{\text{test}}}
\newcommand{\hiddensegstarttestprime}{{{h^{\text{start}}_{\text{test}}}'}}
\newcommand{\hiddensegend}{H^{\text{seg}}}
\newcommand{\hiddensegendtest}{H^{\text{seg}}_{\text{test}}}
\newcommand{\promptseq}{S_n}
\newcommand{\obsex}{O^{\text{ex}}}
\newcommand{\Lzeroone}{L_{\text{0-1}}}
\newcommand{\LCE}{L_{\text{CE}}}
\newcommand{\indicator}{\mathbf{1}}
\newcommand{\badset}{\sB}
\newcommand{\startconst}{\frac{1}{|\sD|}}
\newcommand{\delimub}{c_2}
\newcommand{\delimlb}{c_1}
\newcommand{\delimstartub}{c_4}
\newcommand{\delimstartlb}{c_3}
\newcommand{\translb}{c_5}
\newcommand{\emitlb}{c_6}
\newcommand{\examplelb}{c_7}
\newcommand{\hiddenstartlb}{c_8}
% \newcommand{\prompthiddenstartlb}{c_8}
% \newcommand{\prompthiddenstartub}{c_8}

\newcommand{\KLk}{KL_k(\thetastar \| \theta)}
\newcommand{\KLj}{KL_j(\thetastar \| \theta)}
\newcommand{\KLjprev}{KL^\theta_{j-1}}
\newcommand{\KLjnext}{KL^\theta_{j+1}}
\newcommand{\ppromptj}{\pprompt^j}
\newcommand{\pjtheta}{p^j_\theta}
\newcommand{\pjnexttheta}{p^{j+1}_\theta}
\newcommand{\pjthetastar}{p^j_\thetastar}
\newcommand{\pjnextthetastar}{p^{j+1}_\thetastar}
\newcommand{\ptwojtheta}{p^{(2,j)}_\theta}
\newcommand{\pijtheta}{p^{(i,j)}_\theta}
\newcommand{\ptwojthetastar}{p^{(2,j)}_\thetastar}
\newcommand{\pijthetastar}{p^{(i,j)}_\thetastar}
\newcommand{\ptwojnexttheta}{p^{(2,j+1)}_\theta}
\newcommand{\pijnexttheta}{p^{(i,j+1)}_\theta}
\newcommand{\ptwojnextthetastar}{p^{(2,j+1)}_\thetastar}
\newcommand{\pijnextthetastar}{p^{(i,j+1)}_\thetastar}

\newcommand{\errstart}{\epsilon^\theta_{\text{start}}}
\newcommand{\errdelim}{\epsilon^\theta_{\text{delim}}}


\newcommand{\h}{h}
\newcommand{\obs}{o}
\newcommand{\obsset}{\sO}
\newcommand{\delim}{h^{\text{delim}}}
\newcommand{\obsseg}{O}
\newcommand{\obsdelim}{o^{\text{delim}}}
\newcommand{\X}{x}
\newcommand{\y}{y}
\newcommand{\Xtest}{x_{\text{test}}}
\newcommand{\ytest}{y_{\text{test}}}
\newcommand{\bigK}{K}

\newcommand{\hatp}{\hat{p}}
\newcommand{\pprompt}{p_{\text{prompt}}}
\newcommand{\ppromptstart}{p_{\text{prompt}}}
\newcommand{\ppromptdelim}{p_{\text{prompt}}^{\text{delim}}}
\newcommand{\unif}{\text{Unif}}
\newcommand{\betax}{\beta(x_{n+1})}



\newcommand{\Tic}{\sT_{\text{in-context}}}

\newcommand{\minv}{{-1}}
\newcommand{\minimize}{\text{minimize}}
\newcommand{\thetastar}{{\theta^*}}
\newcommand{\statdistpiH}{\pi^\thetastar}
\newcommand{\statdistpiHx}{\pi_x^\thetastar}
\newcommand{\statdistpiHtheta}{\pi^\theta}
\newcommand{\statdistpiHthetax}{\pi_x^\theta}
\newcommand{\statdistpiD}{\pi^\thetastar_\sD}
\newcommand{\statdistpiDtheta}{\pi^\theta_\sD}
\newcommand{\conetheta}{C_1^\theta(\epsilon^\theta(t_n))}
\newcommand{\ctwotheta}{C_2^\theta(\epsilon^\theta(t_n))}
\newcommand{\conethetastar}{C_1^\thetastar(\epsilon^\thetastar(t_n))}
\newcommand{\ctwothetastar}{C_2^\thetastar(\epsilon^\thetastar(t_n))}
\newcommand{\conethetat}{C_1^\theta(\epsilon^\theta(t))}
\newcommand{\ctwothetat}{C_2^\theta(\epsilon^\theta(t))}
\newcommand{\conethetastart}{C_1^\thetastar(\epsilon^\thetastar(t))}
\newcommand{\ctwothetastart}{C_2^\thetastar(\epsilon^\thetastar(t))}

\newcommand{\tilf}{\tilde{f}}
\newcommand{\maximize}{\text{maximize}}
\newcommand{\spoc}{\textsc{SPoC}}
\newcommand{\testp}{\textsc{TestP}}
\newcommand{\testw}{\textsc{TestW}}
\newcommand{\Pibeta}{\Pi_{\beta}}
\newcommand{\hatalpha}{\hat{\alpha}}
% \newcommand{\fstd}{f_{\theta_{\text{direct}}}}
% \newcommand{\fstdi}{f_{\theta_{\text{direct},i}}}
\newcommand{\fstd}{f_{\text{direct}}}
\newcommand{\fstdi}{f_{\text{direct},j}}
\newcommand{\fhatpi}{\hat{f}_{\Pi}}
\newcommand{\fpi}{f_{\Pi}}
\newcommand{\hattheta}{\hat{\theta}}
\newcommand{\ftheta}{f_{\theta}}
\newcommand{\fthetai}{f_{\theta_j}}
\newcommand{\fthetamin}{f_{\theta_{\text{min}}}}
\newcommand{\fthetabasemin}{f^*_{\text{base}}}
\newcommand{\fthetabase}{f_{\text{base}}}
\newcommand{\fthetacomp}{f_{\text{comp}}}
\newcommand{\fhattheta}{\hat{f}_{\theta}}
\newcommand{\fhat}{\hat{f}}
\newcommand{\fstar}{f^\star}
\newcommand{\proj}{\text{proj}}
\newcommand{\projC}{\text{proj}_{C}}
\newcommand{\thetareg}{{\theta_{\text{reg}}}}
\newcommand\uy{\tilde y}
\newcommand\noiseuy{\tilde z}

\newcommand{\Tcal}{\mathcal{T}}
\newcommand{\Tcalstar}{\mathcal{T}^\star}
% \newcommand{\tau}{\tau}
\newcommand{\taustar}{\tau^\star}


\newcommand{\yone}{\y_{\text{max}}}


% WIDEBAR COMMAND
\newlength{\widebarargwidth}
\newlength{\widebarargheight}
\newlength{\widebarargdepth}
\DeclareRobustCommand{\widebar}[1]{%
    \settowidth{\widebarargwidth}{\ensuremath{#1}}%
    \settoheight{\widebarargheight}{\ensuremath{#1}}%
    \settodepth{\widebarargdepth}{\ensuremath{#1}}%
    \addtolength{\widebarargwidth}{-0.3\widebarargheight}%
    \addtolength{\widebarargwidth}{-0.3\widebarargdepth}%
    \makebox[0pt][l]{\hspace{0.3\widebarargheight}%
    \hspace{0.3\widebarargdepth}%
    \addtolength{\widebarargheight}{0.3ex}%
    \rule[\widebarargheight]{0.95\widebarargwidth}{0.1ex}}%
{#1}}