\newcommand{\sumlims}[2]{\sum\limits_{#1}\limits^{#2}}
\newcommand{\prodlims}[2]{\prod\limits_{#1}\limits^{#2}}
\newcommand{\bigcuplims}[2]{\bigcup\limits_{#1}\limits^{#2}}
\newcommand{\minlim}[1]{\min\limits_{#1}}
\newcommand{\maxlim}[1]{\max\limits_{#1}}
\DeclareMathOperator*{\argmax}{arg\,max}
\DeclareMathOperator*{\argmin}{arg\,min}
\DeclareMathOperator*{\Proba}{\mathbf{Pr}}
\DeclareMathOperator*{\sg}{sg}
\DeclareMathOperator*{\td}{\rightarrow}
\newcommand{\Probalim}[2]{\Proba\limits_{#1}\left[#2\right]}
\newcommand{\tdv}[2]{\td\limits_{#1 \rightarrow #2}}
\newcommand{\argminlim}[1]{\argmin\limits_{#1}}
\newcommand{\argmaxlim}[1]{\argmax\limits_{#1}}
\newcommand{\seqint}[1]{\llbracket #1 \rrbracket}
\newcommand{\esp}{\mathbf{E}}
\newcommand{\uesp}[1]{\underset{#1}{\esp}}
\newcommand{\prob}{\mathbf{P}}
\newcommand{\uprob}[1]{\underset{#1}{\prob}}
\DeclareMathOperator{\conv}{Conv}
\newcommand{\eq}{\Leftrightarrow}
\newcommand{\imp}{\Rightarrow}
\newcommand{\revimp}{\Leftarrow}
\newcommand{\limlim}[1]{\lim\limits_{#1}}
\newcommand{\bigO}{\mathcal{O}}
\newcommand{\one}[1]{\mathbf{1}\left\{#1\right\}}

\newcommand{\mg}{\mu}

\newcommand{\hx}{x}
\newcommand{\hy}{y}
\newcommand{\ham}{\m_\test}
\newcommand{\m}{m}
\newcommand{\de}{d}
\newcommand{\eu}{e}
\newcommand{\ef}{f}
\newcommand{\Dismat}{\mathbf{A}}
\newcommand{\dis}{\bm{\omega}}
\newcommand{\disti}{\dis_t[i]}
\newcommand{\fun}{\tilde{\dis}}
\newcommand{\dist}{d}
\newcommand{\X}{\mathcal{X}}
\newcommand{\Y}{\mathcal{Y}}
\newcommand{\dataset}{\mathcal{S}}
\newcommand{\train}{\mathcal{L}}
\newcommand{\test}{\mathcal{T}}
\newcommand{\voters}{\mathcal{H}}
\newcommand{\voter}{h}
\newcommand{\dime}{l}
\newcommand{\wei}{\bm{\pi}}
\newcommand{\weiti}{\wei_t(i)}
\newcommand{\weif}{\pi}
\newcommand{\mgs}{\mu^{(\train)}}
\newcommand{\T}{T}
\newcommand{\F}{P}
\newcommand{\eF}{\mathcal{F}}
\newcommand{\hdis}{\hat{\omega}}
\newcommand{\disb}{\overline{\disf}}
\newcommand{\disf}{\omega}
\newcommand{\bH}{\mathbf{h}}
\newcommand{\bA}{A}
\newcommand{\bD}{D}
\newcommand{\supp}{\tilde{\X}}
\newcommand{\edge}{\gamma}
\newcommand{\error}{\epsilon}
\newcommand{\tF}{F}
\newcommand{\Z}{Z}
\newcommand{\Hf}{H}
\newcommand{\tFT}{\tF^{\eF_T}}
\newcommand{\tFt}{\tF^{\eF_t}}
\newcommand{\HfT}{\Hf^{\eF_T}}
\newcommand{\se}{\epsilon}
\newcommand{\sd}{\delta}
\newcommand{\plif}{\hat{\xi}}
\newcommand{\plaf}{\hat{\zeta}}
\newcommand{\np}{n_{+}}
\newcommand{\nm}{n_{-}}
\newcommand{\ratio}{\rho(b)}
\newcommand{\neigh}{\mathcal{N}}
\newcommand{\basedi}{\mathcal{D}}
\newcommand{\basedixy}{\basedi_{\X, \Y}}
\newcommand{\prior}{\mathcal{P}}
\newcommand{\kernel}{K}
\newcommand{\vker}{\bm{k}}
\newcommand{\mgm}{\eta}
\newcommand{\mgn}{\beta}
\newcommand{\empr}{\mathfrak{R}_{\text{emp}}}
\newcommand{\thr}{\mathfrak{R}_{\text{th}}}
\newcommand{\thm}{\mathfrak{M}_{\text{th}}}
\newcommand{\diffv}{\tau}
\newcommand{\normd}{\nu_{\ha,\hb}(\hx)}
\newcommand{\usd}{\frac{1}{2}}
\newcommand{\ha}{a}
\newcommand{\hb}{b}
\newcommand{\predx}{\pred(\hx)}
\newcommand{\predxi}{\F^{\eF_T}_{0,\hb}(x_i)}
\newcommand{\pred}{\F^{\eF_T}_{\ha,\hb}}
\newcommand{\hdist}{\hdis_{t}^{\ha,\hb}}
\newcommand{\fracw}{\frac{\dis_{t,i} \normd}{\ha^\hb + \dist(x_i, \hx)^\hb}}
\newcommand{\sa}{\alpha_{\ha, \hb}}
\newcommand{\sach}[2]{\left.#1\, \right|\, #2}
\newcommand{\neighst}{\neigh_\se^\train}
\newcommand{\ldxm}{\train \sim (\basedixy)^\m}
\newcommand{\ldxi}{x_i \sim (\basedixy)}
\newcommand{\xydx}{(\hx, \hy) \sim \basedixy}
\newcommand{\eldxm}[1]{\uesp{\ldxm}\left[#1\right]}
\newcommand{\eldxi}[1]{\uesp{\ldxi}\left[#1\right]}
\newcommand{\pldxm}[1]{\uprob{\ldxm}\left[#1\right]}
\newcommand{\pldxi}[1]{\uprob{\ldxi}\left[#1\right]}
\newcommand{\exydx}[1]{\uesp{\xydx}\left[#1\right]}
\newcommand{\pxydx}[1]{\uprob{\xydx}\left[#1\right]}
\newcommand{\disip}{\dis_{\min}^{+}}
\newcommand{\disim}{\dis_{\min}^{-}}
\newcommand{\disap}{\dis_{\max}^{+}}
\newcommand{\disam}{\dis_{\max}^{-}}
\newcommand{\usb}{\frac{1}{b}}
\newcommand{\Pp}{P^+_\se}
\newcommand{\Pm}{P^-_\se}
\newcommand{\bays}{\mathcal{B}^*_\basedi}
\newcommand{\disfun}{\phi_\basedi}
\newcommand{\disfund}{\disfun(\dist(x, x'))}
\newcommand{\disfundi}{\disfun(\dist(x, x_i))}
\newcommand{\disfune}{\disfun(\se)}
\newcommand{\disfuned}{\disfun(\se_2)}

\newcommand{\usm}{\frac{1}{\m}}
\newcommand{\sumlimsm}{\sumlims{i=1}{\m}}
\newcommand{\Rplus}{\left[0\right., +\infty\left.\right[}
\newcommand{\Rpluset}{\left]0, +\infty\right[}
\newcommand{\xyi}{\left(x_i, y_i\right)}

\newcommand{\algo}{SamBA\xspace}

% \newcommand{\cecile}[1]{\textcolor{purple}{#1}}

\newtheorem{theorem}{Theorem}
\newtheorem{definition}{Definition}
\newtheorem{hypothesis}{Hypothesis}
\newtheorem{corollary}{Corollary}
\newtheorem{lemma}{Lemma}

