% \documentclass{uai2023} % for initial submission
\documentclass[accepted]{uai2023} % after acceptance, for a revised
                        % version; also before submission to
                        % see how the non-anonymous paper
                        % would look like

%% There is a class option to choose the math font
% \documentclass[mathfont=ptmx]{uai2023} % ptmx math instead of Computer
% Modern (has noticable issues)
% \documentclass[mathfont=newtx]{uai2023} % newtx fonts (improves upon
% ptmx; less tested, no support)
% NOTE: Only keep *one* line above as appropriate, as it will be replaced
%       automatically for papers to be published. Do not make any other
%       change above this note for an accepted version.

%% Choose your variant of English; be consistent
\usepackage[american]{babel}
% \usepackage[british]{babel}

%% Some suggested packages, as needed:
\usepackage{natbib} % has a nice set of citation styles and commands
\bibliographystyle{plainnat}
\renewcommand{\bibsection}{\subsubsection*{References}}
\usepackage{mathtools} % amsmath with fixes and additions
% \usepackage{siunitx} % for proper typesetting of numbers and units
\usepackage{booktabs} % commands to create good-looking tables
\usepackage{tikz} % nice language for creating drawings and diagrams

% for cross referencing the main text
% PLEASE ONLY USE xr IN THE SUPPLEMENTARY MATERIAL. 
% In the main paper, hard code any cross-reference to the supplementary material. 
\usepackage{xr} 
\externaldocument{uai2023-template}

%% Provided macros
% \smaller: Because the class footnote size is essentially LaTeX's \small,
%           redefining \footnotesize, we provide the original \footnotesize
%           using this macro.
%           (Use only sparingly, e.g., in drawings, as it is quite small.)

%% Self-defined macros
\newcommand{\swap}[3][-]{#3#1#2} % just an example

\title{Nonconvex Stochastic Scaled Gradient Descent \\ and Generalized Eigenvector Problems \\(Supplementary Material)}

% The standard author block has changed for UAI 2023 to provide
% more space for long author lists and allow for complex affiliations
%
% All author information is authomatically removed by the class for the
% anonymous submission version of your paper, so you can already add your
% information below.
%
% Add authors
\author[1]{\href{mailto:<junchili@berkeley.edu>?Subject=Your UAI 2023 paper}{Chris Junchi Li}{}}
\author[1,2]{Michael I.~Jordan}
% Add affiliations after the authors
\affil[1]{%
Department of Electrical Engineering and Computer Sciences%
\\
UC Berkeley%
\\
Berkeley, California, USA
}
\affil[2]{%
Department of Statistics%
\\
UC Berkeley%
\\
Berkeley, California, USA
}

\usepackage{mathrsfs}
%\usepackage{enumerate}
\usepackage{algorithm}
\usepackage{algorithmic}
\usepackage{amsmath}
\usepackage{natbib}
\usepackage{amssymb}
\usepackage{bm}
\def\beq{\begin{equation} }\def\eeq{\end{equation} }\def\ep{\varepsilon}\def\1{\mathbf{1}}\def\cov{{\rm cov}}\def\var{{\rm var}}
\newcommand{\uu}{\bm{u}}
\newcommand{\vv}{{\bm{v}}}
\newcommand{\ww}{\bm{w}}
\newcommand{\cC}{\mathcal{C}}
\def\bzeta{\boldsymbol{\zeta}}
\def\bSigma{\boldsymbol{\mathbf{\Sigma}}}
\newcommand{\Exs}{\mathbb{E}}
\newcommand{\Ab}{\mathbf{A}}
\newcommand{\Bb}{\mathbf{B}}
\newcommand{\real}{\mathbb{R}}
\newcommand{\cS}{{\mathcal{S}}}
\newcommand{\cF}{{\mathcal{F}}}
\newcommand{\cH}{{\mathcal{H}}}
\newcommand{\cM}{\mathcal{M}}
\newcommand{\cT}{{\mathcal{T}}}
\newcommand{\Ib}{\mathbf{I}}
\newcommand{\argmin}{\mathop{\mathrm{argmin}}}
\def\lipr{\delta}
\newcommand{\cV}{\mathcal{V}}
\def\rhoi{\rho}
\def\alphai{\mu}
\def\betai{\beta}
\def\eps{\epsilon}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\bP}{\bm{P}}
\def\diag{\mathop{\mathrm{diag}}}
\def\ivv{\overline{\vv}}
\def\bchi{\boldsymbol{\chi}}
\newcommand{\bS}{\bm{S}}
\def\bxi{\boldsymbol{\xi}}
\newcommand{\bPhi}{\bm{\mathbf{\Phi}}}
\newcommand{\Xb}{\bm{X}}
\newcommand{\Yb}{\bm{Y}}
\def\deltai{\delta}
\newcommand{\cN}{\mathcal{N}}
\newcommand{\bR}{\bm{R}}
\newcommand{\bQ}{\bm{Q}}
\def\tbS{\widetilde{\bS}}
\def\tbP{\widetilde{\bP}}
\def\tDelta{\overline{\bm{\Delta}}}
\def\radius{r}
\newcommand{\ud}{d}
\def\kJ{\mathscr{J}}
\def\gammai{\gamma}
\newcommand{\cA}{\mathcal{A}}
\newcommand{\cI}{\mathcal{I}}
\newcommand{\cL}{\mathcal{L}}
\newcommand{\cO}{\mathcal{O}}
\newcommand{\cQ}{\mathcal{Q}}
\newcommand{\eb}{\mathbf{e}}
\newcommand{\ub}{\mathbf{u}}
\newcommand{\vb}{\mathbf{v}}
\newcommand{\wb}{\mathbf{w}}
\newcommand{\xb}{\mathbf{x}}
\newcommand{\yb}{\mathbf{y}}
\newcommand{\zb}{\mathbf{z}}
\newcommand{\ba}{\bm{a}}
\newcommand{\bb}{\bm{b}}
\newcommand{\bv}{\bm{v}}


\newtheorem{theorem}{Theorem}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{assumption}{Assumption}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}


\usepackage{graphicx,subfigure}

\newcommand{\jmlrBlackBox}{\rule{1.5ex}{1.5ex}}
\providecommand{\BlackBox}{\jmlrBlackBox}
\newcommand{\jmlrQED}{\hfill\jmlrBlackBox\par\bigskip}
\newenvironment{proof}%
{%
\par\noindent{\bfseries\upshape Proof\ }%
}%
{\jmlrQED}


%SPACING ORIGINAL
\setlength{\textfloatsep}{5pt}% Remove \textfloatsep
\setlength\floatsep{3pt}
\setlength\intextsep{1pt}
%\setlength{\bibsep}{0ex}
%\nipsfinalcopy % Uncomment for camera-ready version
\setlength{\abovecaptionskip}{1pt}
\setlength{\belowcaptionskip}{1pt}
\setlength{\parskip}{0.15em}
%\usepackage[compact]{titlesec}
%\titlespacing*{\section}
%{0pt}{1ex}{1ex}
%\titlespacing*{\subsection}
%{0pt}{1ex}{1ex}
%\titlespacing*{\subsubsection}
%{0pt}{1ex}{1ex}


\newtheorem{innercustom}{}
\newenvironment{custom}[1]
{\renewcommand\theinnercustom{#1}\innercustom}
{\endinnercustom}

%\usepackage{enumerate}
%\usepackage[dvipsnames]{xcolor}
%\def\blue#1{}
\def\blue#1{\textcolor{cyan}{#1}}
%\def\green#1{}
\def\green#1{\colorbox{green}{#1}}

%=Alternating
\def\red#1{}\def\pb{}%\usepackage{fullpage}
%\def\red#1{\textcolor{orange}{#1}}\def\pb{\newpage}\addtolength{\oddsidemargin}{-.5in}\addtolength{\evensidemargin}{-.5in}\addtolength{\textwidth}{1in}\addtolength{\topmargin}{-.5in}\addtolength{\textheight}{-4in}   %\usepackage{refcheck}

\newcommand{\cjlcomment}[1]{{\bf{{\color{cyan}{{Junchi {---} #1}}}}}}

\begin{document}

\onecolumn %% Turn this off if single column is desired for the supplement
\maketitle

\appendix\input{real-supp}








































\pb
%\bibliographystyle{plainnat}
%\bibliographystyle{apalike}
\bibliography{SAILreferences}

%\newpage\tableofcontents

\end{document}