

\usepackage{euscript}	 %  
\usepackage{makecell}

\usepackage{times}
\RequirePackage{fix-cm}
\usepackage{cmap}					%   PDF

\usepackage{comment}
\usepackage{xr}

\usepackage[flushleft]{threeparttable} % http://ctan.org/pkg/threeparttable
\usepackage{booktabs}       % professional-quality tables

%\usepackage[noend]{algpseudocode}

%\usepackage{extsizes} %   14- 
%\usepackage{geometry} %    
\usepackage{url}
\usepackage{csquotes} %   
\usepackage{siunitx}
\usepackage{float}
\restylefloat{table}
\usepackage{array}

%\usepackage{undertilde}

\usepackage{setspace} % 
%\onehalfspacing %  1.5
%\doublespacing %  2
%\singlespacing %  1
\usepackage{mathtools}
\usepackage[export]{adjustbox}
\usepackage{lastpage} % ,     .
\usepackage{soulutf8} %  
\usepackage{icomma} % "" : $0,2$ --- , $0, 2$ --- 

%\newenvironment{solution}
%{\renewcommand\qedsymbol{$\blacksquare$}\begin{proof}[Solution]}{\end{proof}}

\usepackage{etoolbox}
\usepackage{mathtools}
\usepackage{etextools}
\usepackage{ifthen}
\usepackage{caption}
\usepackage{subcaption}
\usepackage{amsmath,amssymb, amsthm}
%\usepackage{charter}
%\usepackage{calc}

\usepackage{calligra}
\usepackage{amsfonts}
\usepackage{layout}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amssymb}
\usepackage{hyperref}%[backref=true]
% \usepackage[colorlinks,linkcolor=magenta,citecolor=blue, pagebackref=true,backref=true]{hyperref}
\usepackage{cleveref}
\usepackage{lipsum}
\usepackage{mathtools}
%\mathtoolsset{showonlyrefs=true}
\usepackage{mathrsfs}
\usepackage{url}
\usepackage{color}
\usepackage{url}
\usepackage{color,xcolor, colortbl}
\usepackage{setspace}

\usepackage{multicol} %  
\usepackage{filecontents}
%nwork with tables
\usepackage{array,tabularx,tabulary,booktabs} %    
%\usepackage{longtable}  %  
\usepackage{multirow} %    
\usepackage{graphicx}
%\usepackage{subfig}
\usepackage{fancyhdr}
\usepackage{hyperref}
\usepackage{theoremref}
\usepackage{bbm}

\usepackage{pifont}%for defining \xmark


%%%%%%%%%%%%%%
%NEW COMMANDS   %%%
%%%%%%%%%%%%%%

\DeclareMathOperator*{\argmax}{arg\,max}
\DeclareMathOperator*{\argmin}{arg\,min}
\DeclareMathOperator*{\sign}{sign}
\DeclarePairedDelimiter\abs{\lvert}{\rvert}
\DeclarePairedDelimiter\floor{\lfloor}{\rfloor}

\newcommand{\z}{\zeta}
\newcommand{\lt}{\left}
\newcommand{\rt}{\right}
\newcommand{\al}{\alpha}
\newcommand{\p}{\partial}
\newcommand{\D}{\Delta}
\newcommand{\nfr}{\nicefrac}
\newcommand{\fr}{\frac}
\newcommand{\dfr}{\dfrac}
\newcommand{\mbf}{\mathbf}
\newcommand{\mds}{\mathds}
\newcommand{\bb}{\mathbb}
\newcommand{\wt}{\widetilde}
\newcommand{\wS}{\wt{S}}
\newcommand{\wA}{\wt{A}}
\newcommand{\wL}{\wt{L}}
\newcommand{\wLi}{\wt{L}_i}
\newcommand{\wwL}{\wt{\wt{L}}}
\newcommand{\wB}{\wt{B}}
\newcommand{\wC}{\wt{C}}
\newcommand{\ws}{\wt{s}}
\newcommand{\wg}{\wt{g}}
\newcommand{\wgix}{\wg_i(x)}
\newcommand{\wgiy}{\wg_i(y)}
\newcommand{\wgik}{\wg_i(x^k)}
\newcommand{\wgikpo}{\wg_i(x^{k+1})}
\newcommand{\wgu}{\wt{g}^u}
\newcommand{\wG}{\wt{G}}
\newcommand{\hG}{\hat{G}}
\newcommand{\hAi}{\hat{A}_i}
\newcommand{\hA}{\hat{A}}
\newcommand{\pA}{A^{\prime}}
\newcommand{\pB}{B^{\prime}}
\newcommand{\pC}{C^{\prime}}
\newcommand{\wpA}{\wA^{\prime}}
\newcommand{\wpB}{\wB^{\prime}}
\newcommand{\wpC}{\wC^{\prime}}
\newcommand{\ppA}{A^{\prime \prime}}
\newcommand{\ppB}{B^{\prime \prime}}
\newcommand{\ppC}{C^{\prime \prime}}
\newcommand{\one}{{\mathbbm{1}}}
\newcommand{\prob}{{\mathbb{P}}}


\newcommand{\hg}{\hat{g}}
\newcommand{\hgi}{\hat{g}_i}
\newcommand{\hgix}{\hat{g}_i(x)}
\newcommand{\hgiy}{\hat{g}_i(y)}
\newcommand{\hgik}{\hat{g}_i(x^k)}
\newcommand{\hgikpo}{\hat{g}_i(x^{k+1})}
\newcommand{\hgu}{\hat{g}^u}
\newcommand{\hs}{\hat{s}}


\newcommand{\eik}{e_i^t}
\newcommand{\gik}{g_i^t}
\newcommand{\gikpo}{g_i^{t+1}}
\newcommand{\eikpo}{e_i^{t+1}}
\newcommand{\vik}{v_i^t}
\newcommand{\vikpo}{v_i^{t+1}}

\newcommand{\eit}{e_i^t}
\newcommand{\eitpo}{e_i^{t+1}}
%\newcommand{\vit}{v_i^t}
%\newcommand{\vitpo}{v_i^{t+1}}

\newcommand{\wcC}{\wt{\cC}}
\newcommand{\La}{\Lambda}
\newcommand{\lam}{\lambda}
\newcommand{\lami}{\lam_i}
\newcommand{\opn}{\operatorname}
\newcommand{\vp}{\varphi}
\newcommand{\bM}{\mbf M}
\newcommand{\bG}{\mbf G}
\newcommand{\bI}{\mbf I}
\newcommand{\bD}{\mbf D}
\newcommand{\bA}{\mbf A}
\newcommand{\bC}{\mbf C}
\newcommand{\bP}{\mbf P}
\newcommand{\bB}{\bb B}
\newcommand{\bU}{\bb U}
\newcommand{\bLa}{\mbf {\La}}
\newcommand{\R}{\bb R}
\newcommand{\bE}{\mbf E}
\newcommand{\hS}{\hat{S}}
\newcommand{\eps}{{\varepsilon}}
\newcommand{\bL}{\mathbf{L}}
\newcommand{\stepsize}{\eta}
\newcommand{\cE}{{\cal E}}
\newcommand{\norm}[1]{\left\| #1 \right\|}
\newcommand{\lin}[1]{\left\langle #1\right\rangle} % product
\newcommand{\cD}{\mathcal{D}}
\newcommand{\cC}{\mathcal{C}}
\newcommand{\cQ}{\mathcal{Q}}
\newcommand{\cCw}{\cC_{\omega}}
\newcommand{\cCd}{\cC_{\delta}}
\newcommand{\Ctk}[1]{\cC_{Tk}\rb{#1}}
\newcommand{\gixk}{g_{i}(x_k)}
\newcommand{\cS}{\mathcal{S}}
\newcommand{\mM}{\mathbf{M}}
\newcommand{\mN}{\mathbf{N}}
\newcommand{\bN}{\mathbf{N}}
\newcommand{\mI}{\mathbf{I}}
\newcommand{\mL}{\mathbf{L}}
\newcommand{\mD}{\mathbf{D}}
\newcommand{\bx}{\bar{x}}
\newcommand{\by}{\bar{y}}
\newcommand{\barB}{\overline{B}}
\newcommand{\barL}{\overline{L}}
\newcommand{\barC}{\overline{C}}
\newcommand{\eqtext}[1]{\ensuremath{\stackrel{\text{#1}}{=}}}
\newcommand{\letext}[1]{\ensuremath{\stackrel{\text{#1}}{\le}}}
\newcommand{\getext}[1]{\ensuremath{\stackrel{\text{#1}}{\ge}}}
\newcommand{\succtext}[1]{\ensuremath{\stackrel{\text{#1}}{\succ}}}
\newcommand{\preceqtext}[1]{\ensuremath{\stackrel{\text{#1}}{\preceq}}}
\newcommand{\onenorm}[1]{\left\| #1 \right\|_1}      % norm 
\newcommand{\twonorm}[1]{\left\| #1 \right\|_2}      % norm 
\newcommand{\threenorm}[1]{\left\| #1 \right\|_3}      % norm 
\newcommand{\gnorm}[1]{\left\| #1 \right\|^{1 + \gamma }}      % norm 
\newcommand{\sqnorm}[1]{\left\| #1 \right\|^{2}}
\newcommand{\anorm}[1]{\left\| #1 \right\|^{\al}}      % norm 
\newcommand{\bnorm}[1]{\left\| #1 \right\|^{\beta}}      % norm 
\newcommand{\ahnorm}[1]{\left\| #1 \right\|^{\al/2}}      % norm 
\newcommand{\bbone}[1]{\mathbb{1}}      % norm 
\newcommand{\g}{\gamma}

\newcommand{\nf}[1]{\nabla f\rb{#1}}      % norm 

\newcommand{\hotidea}{{\color{red}\bf HOT IDEA: }}
\newcommand{\done}{{\color{blue}\bf DONE: }}
\newcommand{\cO}{{\cal O}}

\newcommand{\del}[1]{}

\newcommand{\finf}{f^\mathrm{inf}}
\newcommand{\fiinf}{f_i^\mathrm{inf}}

\newcommand{\EE}[1]{\mathbb{E}\left[ #1 \right]}
\newcommand{\parens}[1]{\left( #1 \right)}
\newcommand{\brac}[1]{\left\{ #1 \right\}}






\let\la=\langle
\let\ra=\rangle

\def\<{\left\langle}
\def\>{\right\rangle}
% \def\[{\left[}
%\def\]{\right]}
\def\({\left(}
\def\){\right)}

%\newcommand{\st}{\;:\;}


\newcommand{\eqdef}{\stackrel{\text{def}}{=}}

\newcommand{\ii}{{}^{(i)}}


\newcommand{\Prob}{\mathbf{Prob}}
\newcommand{\E}{\mathbf{E}}



% sets
\DeclareMathOperator{\card}{card}       % cardinality of a set
\DeclareMathOperator{\diam}{diam}       % diameter of a set
\DeclareMathOperator{\MVEE}{MVEE}       % minim volume enclosing ellipsoid of a set
\DeclareMathOperator{\vol}{vol}         % volume of a set

% statistical
% \DeclareMathOperator{\Exp}{{\rm E}}           % expectation
%\newcommand{\Exp}[1]{{\rm E}\left[#1\right]}
\newcommand{\Exp}[1]{{\bb E}\left[#1\right]}
\newcommand{\ExpSub}[2]{{\bb E}_{#1}\left[#2\right]}
\newcommand{\ExpCond}[2]{{\bb E}\left[\left.#1\right\vert#2\right]}

\newcommand{\samplefunc}{\textit{S}}

\DeclareMathOperator{\Cov}{Cov}         % covariance
\DeclareMathOperator{\Var}{Var}         % variance
\DeclareMathOperator{\Corr}{Corr}       % correlation

% functions and operators
\DeclareMathOperator{\signum}{sign}     % signum/sign of a scalar
\DeclareMathOperator{\dom}{dom}         % domain
\DeclareMathOperator{\epi}{epi}         % epigraph
\DeclareMathOperator{\Ker}{null}        % nullspace/kernel
\DeclareMathOperator{\nullspace}{null}  % nullpsace
\DeclareMathOperator{\range}{range}     % range
\DeclareMathOperator{\Image}{Im}        % image

% topology
\DeclareMathOperator{\interior}{int}    % interior
\DeclareMathOperator{\ri}{rint}         % relative interior
\DeclareMathOperator{\rint}{rint}       % relative interior
\DeclareMathOperator{\bdry}{bdry}       % boundary
\DeclareMathOperator{\cl}{cl}           % closure

% vectors, matrices
\DeclareMathOperator{\linspan}{span}
\DeclareMathOperator{\linspace}{linspace}
\DeclareMathOperator{\cone}{cone}

%\DeclareMathOperator{\tr}{tr}           % trace
\DeclareMathOperator{\rank}{rank}       % rank
\DeclareMathOperator{\conv}{conv}       % convex hull
\DeclareMathOperator{\Diag}{Diag}       % Diag(v) = diagonal matrix with v_i on the diagonal
\DeclareMathOperator{\diag}{diag}       % diag(D) = the diagonal vector of matrix D

\DeclareMathOperator{\Arg}{Arg}         % Argument



\newcommand {\maxin}{\max_{i=1,\dots, n}}
\newcommand {\maxi}{\max_{i}}
\newcommand {\minin}{\min_{i=1,\dots, n}}
\newcommand {\mini}{\min_{i}}
\newcommand {\sumin}{\sum_{i=1}^{n}}
\newcommand {\sumlmi}{\fr{1}{m_i}\sum_{l=1}^{m_i}}
\newcommand {\suminn}{\fr{1}{n}\sum_{i=1}^{n}}
\newcommand {\sumjm}{\sum_{j=1}^{m}}
\newcommand {\sumjmi}{\sum_{j=1}^{m_i}}
\newcommand {\sumjmm}{\fr{1}{m}\sum_{j=1}^{m}}
\newcommand {\sumjnn}{\fr{1}{n}\sum_{j=1}^{n}}
\newcommand {\sumjmmi}{\fr{1}{m_i}\sum_{j=1}^{m_i}}
\newcommand {\sumjn}{\sum_{j=1}^{n}}
\newcommand {\wik}{w_{i}^{k}}
\newcommand {\wjk}{w_{j}^{k}}
\newcommand {\wk}{w^{k}}
\newcommand {\wikpo}{w_{i}^{k+1}}
\newcommand {\wjkpo}{w_{j}^{k+1}}
%\newcommand {\sumin}{\sum_{i=1}^{n}}
\newcommand {\sumid}{\sum_{i=1}^{d}}
\newcommand {\ghx}{\hat{g}(x)}
\newcommand {\gh}{\hat{g}}
\newcommand {\gih}{\hat_i}
\newcommand {\gihx}{\hat{g_i}(x)}
%\newcommand {\sumjn}{\sum_{j=1}^{n}}
\newcommand{\cmark}{\textcolor{green}{\ding{51}}}%
\newcommand{\xmark}{\textcolor{red}{\ding{55}}}%

\DeclareMathOperator{\tr}{Tr}
% \renewcommand{\tr}[1]{\tr\left(#1\right)}

\newtheorem{example}{Example}
\newtheorem{assumption}{Assumption}
\newtheorem{lemma}{Lemma}



%\usepackage[noend]{algpseudocode}
%\usepackage{verbatim}
%\usepackage[ruled,vlined]{algorithm2e}
%\usepackage{algorithm}
%\usepackage{algpseudocode}

\newtheorem{theorem}{Theorem}
\newtheorem{proposition}{Proposition}

\theoremstyle{plain}

\newtheorem{prop}[theorem]{Proposition}
\newtheorem{corollary}{Corollary}
\newtheorem{lem}[theorem]{Lemma}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{remark}{Remark}


\newtheorem{exercise}[theorem]{Exercise}

\newtheorem{rem}[theorem]{Remark}
\newtheorem{que}[theorem]{Question}
\newtheorem{definition}{Definition}


%zerosarah paper symbols

\newcommand*{\qedb}{\hfill\ensuremath{\square}}   % or \blacksquare
\newcommand{\topic}[1]{\vspace{2mm}\noindent{{\bf #1:}}}
\newcommand{\head}[1]{\noindent{{\bf #1:}}}
\newcommand{\red}[1]{\textcolor{red}{#1}}
\newcommand{\blue}[1]{\textcolor{blue}{#1}}


\newcommand\smallO{
  \mathchoice
    {{\scriptstyle\mathcal{O}}}% \displaystyle
    {{\scriptstyle\mathcal{O}}}% \textstyle
    {{\scriptscriptstyle\mathcal{O}}}% \scriptstyle
    {\scalebox{.7}{$\scriptscriptstyle\mathcal{O}$}}%\scriptscriptstyle
  }
  
\usepackage{relsize} %SEB: feel free to revert, I think the font looked a bit too big otherwise
\newcommand{\algname}[1]{{\sf\red\relscale{0.90}#1}\xspace}
\usepackage{float}
\floatstyle{plaintop}
\restylefloat{table}

\usepackage{enumitem}

\definecolor{LightCyan}{rgb}{0.88,1,1}
\usepackage{varwidth}