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header.tex
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% !TEX root = neurips_2019.tex
% !TeX spellcheck = en-US
\usepackage{times,wrapfig,amsmath,amsfonts,bm,color,enumitem,algorithm,algpseudocode}
\usepackage{times}
\usepackage{amsmath,amssymb,amsthm,bbm,mathtools}
\usepackage{graphicx}
\usepackage{subfigure}
\usepackage{hyperref}
\usepackage{url}
\usepackage{pbox}
\newcommand\tTab{\rule{0pt}{2.5ex}} % Top strut
\newcommand\bTab{\rule[-1.5ex]{0pt}{0pt}} % Bottom strut
\newcommand{\w}{\ensuremath{\mathbf{w}}}
\newcommand{\W}{\ensuremath{\mathbf{W}}}
\renewcommand{\u}{\ensuremath{\mathbf{u}}}
\renewcommand{\v}{\ensuremath{\mathbf{v}}}
\newcommand{\x}{\ensuremath{\mathbf{x}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\argmin}[1]{\underset{#1}{\mathrm{argmin}} \:}
\newcommand{\argmax}[1]{\underset{#1}{\mathrm{argmax}} \:}
\newcommand{\inner}[2]{\left\langle #1, #2 \right\rangle}
\newcommand{\gb}[1]{\boldsymbol{#1}}
\newcommand{\hinge}[1]{\left[#1\right]_+}
\newcommand{\valpha}{\gb{\alpha}}
\newcommand{\vtheta}{\gb{\theta}}
\newcommand{\vmu}{\gb{\mu}}
\newcommand{\vnu}{\gb{\nu}}
\newcommand{\D}{\mathcal{D}}
\newcommand{\proj}{P}
\newcommand{\hbeta}{\hat{\beta}}
\newcommand{\lnd}[1]{\ln\left(\frac{#1}{\delta}\right)}
\newcommand{\slnd}[1]{\ln({#1}/{\delta})}
\newcommand{\one}[1]{\left[\!\left[ #1 \right]\!\right]}
\newcommand{\ones}{\mathbbm{1}}
\newcommand{\E}{\ensuremath{\mathbbf{E}}}
\renewcommand{\P}{\ensuremath{\mathbb{P}}}
\newcommand{\etal}{{\it et al}}
\newcommand{\norm}[1]{\left\lVert{#1}\right\rVert}
\newcommand{\abs}[1]{\left\lvert{#1}\right\rvert}
\newcommand{\vnorm}[1]{\left\lVert{#1}\right\rVert} % vector norm
\newcommand{\mnorm}[1]{\left\lVert{#1}\right\rVert} % matrix norm
\newcommand{\loss}[2]{\ell({#1};{#2})}
\newcommand{\regzr}[1]{R({#1})} % regularizer
\newcommand{\generr}[1]{\ell({#1})} % generalization error of a predictor
\newcommand{\emperr}[1]{{\hat{\ell}}({#1})}
\newcommand{\wml}{\hat{w}} % w minimizing training regularized objective
\newcommand{\waprox}{w^*} % w minimizing regularized gen error
\newcommand{\walg}{\tilde{w}} % output of algorithm
\newcommand{\optacc}{\epsilon_{\text{acc}}}
\newcommand{\apperr}{\epsilon_{\text{aprx}}}
\newcommand{\esterr}{\epsilon_{\text{est}}}
\newcommand{\opterr}{\epsilon_{\text{opt}}}
\newcommand{\rank}{\operatorname{rank}}
\newcommand{\diag}{\operatorname{diag}}
\newcommand{\trace}{\operatorname{trace}}
\newcommand{\sign}{\operatorname{sign}}
\newcommand{\conv}{\operatorname{conv}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\FF}{\mathcal{F}}
\newcommand{\HH}{\mathcal{H}}
\newcommand{\GG}{\mathcal{G}}
\newcommand{\ceil}[1]{\lceil #1\rceil}
\newcommand{\floor}[1]{\lfloor #1\rfloor}
\newcommand{\mip}[2]{{\left\langle{#1},{#2}\right\rangle}}
\newcommand{\defeq}{\triangleq}
\mathchardef\hyphen="2D
\usepackage{times}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{graphicx}
\usepackage{subfigure}
\usepackage{hyperref}
\usepackage{url}
\usepackage{pbox}
\newcommand{\T}{^\top}
\newcommand{\TT}{\top }
\newcommand{\RR}{\mathbb{R}}
\newcommand{\EE}{\mathbb{E}}
\newcommand{\MM}{\mathcal{M}}
% \newtheorem{thm}{Theorem}
% \newtheorem{lem}{Lemma}
\newtheorem{cor}{Corollary}
% \newtheorem{Def}{Definition}
% \newtheorem{note}{Note}
\newcommand{\calA}{\mathcal{A}}
\newcommand{\calB}{\mathcal{B}}
\newcommand{\calC}{\mathcal{C}}
\newcommand{\calD}{\mathcal{D}}
\newcommand{\calE}{\mathcal{E}}
\newcommand{\calF}{\mathcal{F}}
\newcommand{\calG}{\mathcal{G}}
\newcommand{\calH}{\mathcal{H}}
\newcommand{\calI}{\mathcal{I}}
\newcommand{\calJ}{\mathcal{J}}
\newcommand{\calK}{\mathcal{K}}
\newcommand{\calL}{\mathcal{L}}
\newcommand{\calM}{\mathcal{M}}
\newcommand{\calN}{\mathcal{N}}
\newcommand{\calO}{\mathcal{O}}
\newcommand{\calP}{\mathcal{P}}
\newcommand{\calQ}{\mathcal{Q}}
\newcommand{\calR}{\mathcal{R}}
\newcommand{\calS}{\mathcal{S}}
\newcommand{\calT}{\mathcal{T}}
\newcommand{\calU}{\mathcal{U}}
\newcommand{\calV}{\mathcal{V}}
\newcommand{\calW}{\mathcal{W}}
\newcommand{\calX}{\mathcal{X}}
\newcommand{\calY}{\mathcal{Y}}
\newcommand{\calZ}{\mathcal{Z}}
\newcommand{\mathA}{\mathbb{A}}
\newcommand{\mathB}{\mathbb{B}}
\newcommand{\mathC}{\mathbb{C}}
\newcommand{\mathD}{\mathbb{D}}
\newcommand{\mathE}{\mathbb{E}}
\newcommand{\mathF}{\mathbb{F}}
\newcommand{\mathG}{\mathbb{G}}
\newcommand{\mathH}{\mathbb{H}}
\newcommand{\mathI}{\mathbb{I}}
\newcommand{\mathJ}{\mathbb{J}}
\newcommand{\mathK}{\mathbb{K}}
\newcommand{\mathL}{\mathbb{L}}
\newcommand{\mathM}{\mathbb{M}}
\newcommand{\mathN}{\mathbb{N}}
\newcommand{\mathO}{\mathbb{O}}
\newcommand{\mathP}{\mathbb{P}}
\newcommand{\mathQ}{\mathbb{Q}}
\newcommand{\mathR}{\mathbb{R}}
\newcommand{\mathS}{\mathbb{S}}
\newcommand{\mathT}{\mathbb{T}}
\newcommand{\mathU}{\mathbb{U}}
\newcommand{\mathV}{\mathbb{V}}
\newcommand{\mathW}{\mathbb{W}}
\newcommand{\mathX}{\mathbb{X}}
\newcommand{\mathY}{\mathbb{Y}}
\newcommand{\mathZ}{\mathbb{Z}}
\newcommand{\veca}{\mathbf{a}}
\newcommand{\vecd}{\mathbf{d}}
\newcommand{\vece}{\mathbf{e}}
\newcommand{\vecf}{\mathbf{f}}
\newcommand{\vecg}{\mathbf{g}}
\newcommand{\vech}{\mathbf{h}}
\newcommand{\veci}{\mathbf{i}}
\newcommand{\vecj}{\mathbf{j}}
\newcommand{\veck}{\mathbf{k}}
\newcommand{\vecl}{\mathbf{l}}
\newcommand{\vecm}{\mathbf{m}}
\newcommand{\vecn}{\mathbf{n}}
\newcommand{\veco}{\mathbf{o}}
\newcommand{\vecp}{\mathbf{p}}
\newcommand{\vecq}{\mathbf{q}}
\newcommand{\vecr}{\mathbf{r}}
\newcommand{\vecs}{\mathbf{s}}
\newcommand{\vect}{\mathbf{t}}
\newcommand{\vecu}{\mathbf{u}}
\newcommand{\vecv}{\mathbf{v}}
\newcommand{\vecw}{\mathbf{w}}
\newcommand{\vecx}{\mathbf{x}}
\newcommand{\vecy}{\mathbf{y}}
\newcommand{\vecz}{\mathbf{z}}
\newcommand{\numinputs}{d}
\newcommand{\numoutputs}{k}
\newcommand{\numhiddens}{H}
\newcommand{\mU}{\boldsymbol{U}}
% \newcommand{\mV}{\boldsymbol{V}}
\newcommand{\mW}{\boldsymbol{W}}
\newcommand{\rect}{\text{rect}}
\newcommand{\gpath}{\text{path}}
\newcommand{\layer}{\text{layer}}
\newcommand{\DAG}{\text{DAG}}
\newcommand{\dout}{d_\text{out}}
\newcommand{\din}{d_\text{in}}
\newcommand{\vin}{v_{\textrm{in}}}
\newcommand{\vout}{v_{\textrm{out}}}
\newcommand{\relu}{\sigma_{\textsc{relu}}}
\newcommand{\gnorm}[1]{\norm{#1}}
\newcommand{\mugeom}{\mu^{\textit{geo}}}
\newcommand{\pathr}{\phi}
\newcommand{\convexnn}{\nu}
\newcommand{\net}{{\rm net}}
\newcommand{\removed}[1]{}
% \newcommand{\fix}{\marginpar{FIX}}
% \newcommand{\new}{\marginpar{NEW}}
\newcommand{\var}{\mathrm{Var}}
\newcommand{\cov}{\mathrm{Cov}}
\newcommand{\In}{\text{in}}
\newcommand{\Out}{\text{out}}
\newcommand{\vecX}{\mathbf{x}}
\newcommand{\vecH}{\mathbf{h}}
\newcommand{\vecW}{\mathbf{w}}
\newcommand{\vecY}{\mathbf{y}}
\renewcommand{\vec}[1]{\mathrm{vec}(#1)}
\newcommand{\vh}{{\vec h}}
\newcommand{\vw}{{\vec w}}
% \newcommand{\vx}{{\vec x}}
\newcommand{\vz}{{\vec z}}
\newcommand{\vphi}{\boldsymbol{\phi}}
\newcommand{\vpi}{\boldsymbol{\pi}}
\newcommand{\vecGam}{\boldsymbol{\gamma}}
\newcommand{\vecDelta}{\boldsymbol{\Delta}}
\newcommand{\vecb}{\vec b}
\newcommand{\vecc}{\vec c}
\newcommand{\tV}{\widetilde{V}}
% \newcommand{\tW}{\widetilde{\vec W}}
\newcommand{\tH}{\widetilde{\vec h}}
\newcommand{\vecmu}{\boldsymbol{\mu}}
\newcommand{\St}{\ensuremath{\mathrm{St}}}
\newcommand{\Ob}{\ensuremath{\mathrm{Ob}}}
\newcommand{\I}{\ensuremath{\mathbf{I}}}
\newcommand{\Exp}{\ensuremath{\mathrm{Exp}}}
\newcommand{\Rt}{\ensuremath{\mathrm{Rt}}}
\newcommand{\tangent}{\ensuremath{\mathit{T}_{\mathbf{W}}\mathcal{M}}}
\DeclareMathOperator{\Tr}{Tr}
\newcommand{\h}{\mathbf{h}}
\newcommand{\J}{\mathbf{J}}
\newcommand{\A}{\mathbf{A}}
\newcommand{\G}{\mathbf{G}}
% \newcommand{\H}{\mathbf{H}}
% \makeatletter
% \def\vec{\mathop{\operator@font vec}\nolimits}
% \makeatother
\newtheorem*{remark}{Remark}
\newtheorem{proposition}{Proposition}
% \newtheorem{theorem}{Theorem}
\newtheorem{assumption}{Assumption}
% \newtheorem*{definition}{Definition}
% \newtheorem{lemma}{Lemma}
\newtheorem{asu}{Assumption}
\newcounter{subassumption}[asu]
\renewcommand{\thesubassumption}{(\textit{\roman{subassumption}})}
\makeatletter
\renewcommand{\p@subassumption}{\theasu}% Counter prefix.
\makeatother
\newcommand{\subasu}{% Just like \item in a list, but for an asu
\refstepcounter{subassumption}%
\thesubassumption~\ignorespaces}
% Figure reference, lower-case.
\def\figref#1{figure~\ref{#1}}
% Figure reference, capital. For start of sentence
\def\Figref#1{Figure~\ref{#1}}
\def\twofigref#1#2{figures \ref{#1} and \ref{#2}}
\def\quadfigref#1#2#3#4{figures \ref{#1}, \ref{#2}, \ref{#3} and \ref{#4}}
% Section reference, lower-case.
\def\secref#1{section~\ref{#1}}
% Section reference, capital.
\def\Secref#1{Section~\ref{#1}}
% Reference to two sections.
\def\twosecrefs#1#2{sections \ref{#1} and \ref{#2}}
% Reference to three sections.
\def\secrefs#1#2#3{sections \ref{#1}, \ref{#2} and \ref{#3}}
% Reference to an equation, lower-case.
\def\eqref#1{equation~\ref{#1}}
% Reference to an equation, upper case
\def\Eqref#1{Equation~\ref{#1}}
% A raw reference to an equation---avoid using if possible
\def\plaineqref#1{\ref{#1}}
% Reference to a chapter, lower-case.
\def\chapref#1{chapter~\ref{#1}}
% Reference to an equation, upper case.
\def\Chapref#1{Chapter~\ref{#1}}
% Reference to a range of chapters
\def\rangechapref#1#2{chapters\ref{#1}--\ref{#2}}
% Reference to an algorithm, lower-case.
% \def\algref#1{algorithm~\ref{#1}}
% Reference to an algorithm, upper case.
% \def\Algref#1{Algorithm~\ref{#1}}
% \def\twoalgref#1#2{algorithms \ref{#1} and \ref{#2}}
% \def\Twoalgref#1#2{Algorithms \ref{#1} and \ref{#2}}
% Reference to a part, lower case
% \def\partref#1{part~\ref{#1}}
% Reference to a part, upper case
% \def\Partref#1{Part~\ref{#1}}
% \def\twopartref#1#2{parts \ref{#1} and \ref{#2}}