From d8a68c6917f5b6053117e0145f6d4d80a8bec26b Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Tue, 14 Apr 2015 15:20:19 -0400
Subject: Starting paper draft

---
 paper/sections/negative.tex | 113 ++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 113 insertions(+)
 create mode 100644 paper/sections/negative.tex

(limited to 'paper/sections/negative.tex')

diff --git a/paper/sections/negative.tex b/paper/sections/negative.tex
new file mode 100644
index 0000000..f88117a
--- /dev/null
+++ b/paper/sections/negative.tex
@@ -0,0 +1,113 @@
+\subsection{Learnable $\nRightarrow$ Optimizable}
+
+Consider the following function:
+\begin{displaymath}
+    g(S) = \max\left\{\sqrt{n}|X\cap S|, |S|\right\}
+\end{displaymath}
+where $|X|=\sqrt{n}$. Assume that you are given polynomially many samples where
+each element is included with probability $1/2$. Then with high probability all
+the samples will have size roughly $n/2$, so you will observe $g(S)=|S|$ with
+high probability.
+
+\paragraph{Claim 1:} $g$ is PAC-learnable because if you output $|S|$ then you
+will be correct with high probability is $S$ is drawn from the same
+distribution as above.
+
+\paragraph{Claim 2:} $g$ is not optimizable under budget $\sqrt{n}$ because you
+never learn anything about $X$.
+
+\paragraph{Open Question:} Cook a similar example where $g$ is submodular and
+where you are observing sets of the same size as your budget.
+
+\paragraph{Positive results:} Try to obtain guarantees about learning
+parameters of parametric submodular functions and show whether or not these
+guarantees are sufficient for optimization. First look at learning weights in
+a cover function. Maybe facility location? Sums of concave over modular are
+probably too hard because of the connection to neural networks.
+
+\subsection{Optimizable $n\Rightarrow$ Learnable}
+
+Recall the matroid construction from~\cite{balcan2011learning}:
+\begin{theorem}
+  For any $k$ with $k = 2^{o(n^{1/3})}$, there exists a family of sets ${\cal A}
+  \subseteq2^{[n]}$ and a family of matroids $\cal{M} = \{M_{\cal{B}} :
+  \cal{B} \subseteq\cal{A} \}$ such that:
+  \begin{itemize}
+    \item $|{\cal A}| = k$ and $|A| = n^{1/3}$ for every $A \in \cal{A}$
+    \item For every $\cal{B} \subseteq\cal{A}$ and every $A \in \cal{A}$, we
+      have:
+      \begin{align*}
+        \text{rank}_{M_{\cal{B}}}(A) & = 8 \log k \ if A\in{\cal B} \\
+                                     & = |A| \ if A\in {\cal A}\backslash{\cal B}
+      \end{align*}
+  \end{itemize}
+\end{theorem}
+
+Consider the following subset of the above family of matroids: ${\cal
+M}^{\epsilon} = \{M_{\cal B} : {\cal B} \subseteq{\cal A} \wedge |{\cal
+A}\backslash{\cal B}| \geq \epsilon|{\cal A}|\}$ for $k = 2^{n^{1/6}}$.
+Consider an \emph{unknown} function $f$, corresponding to the rank function of
+one of the matroids $M_{\cal B}$ from ${\cal M}^{\epsilon}$.  Note that as long
+as we observe \emph{one} set from ${\cal A} \backslash {\cal B}$, we can
+optimize $f$ exactly! Indeed, the largest possible value for $f$ under
+cardinality constraint $n^{1/3}$ is $\max_{A\in 2^{[n]}} |A| = n^{1/3}$.
+
+One example of a distribution under which this occurs with probability at least
+a constant is $D_u$, the uniform distribution over all sets of ${\cal A}$.  For
+$c>0$, after $n^c$ observations $O \equiv (S_i, f(S_i))_i$ for $S_i \sim D_u$,
+we will observe at least one element from $\cal{A}\backslash\cal{B}$ with
+constant probability:
+
+\begin{equation} \nonumber \mathbb{P}(\bigcup_{S_i} S_i \in {\cal
+A}\backslash{\cal B}) \geq 1 - (1 - \epsilon)^{n^c} \approx \epsilon n^c
+\end{equation}
+
+However, it follows from the analysis of~\cite{balcan2011learning} that we
+cannot learn $f$ under any distribution, even with active value queries!
+Indeed, for any polynomially-sized set of observations, there exists a
+super-polynomially number of functions in ${\cal M^1}$ which coincide on this
+set of observations, but which take very different values outside of this set
+of observations: $8n^{1/6}$ for $A \in {\cal B}$ and $n^{1/3}$ for $A \in {\cal
+A}\backslash {\cal B}$.
+
+{\bf TODO:} A cleaner simpler example would be nice.
+
+\subsection{Which learning guarantees imply optimization?}
+
+Here, we consider the following question: which learning guarantees imply
+optimization? For example, \cite{TODO} provides the following guarantee for
+cover functions:
+
+\begin{theorem}
+There exists an algorithm such that for any $\epsilon>0$ given random and
+uniform examples of a coverage function $c$ outputs a hypothesis, which is also
+a coverage function $h$, such that with probability $2/3$: $\mathbb{E}_{\cal
+U}[|h(x) - c(x)|] \leq \epsilon$. The algorithm runs in time $\tilde {\cal O}(n)
+\cdot \text{poly}(s/\epsilon)$ and uses $\log(n)\cdot \text{poly}(s/epsilon)$
+and examples, where $s = \min\{size(c), (1/\epsilon)^{\log(1/\epsilon)}\}$.
+\end{theorem}
+
+We would like to understand to what extent this $\ell1$-bound allows us to
+optimize the coverage function $c$ under cardinality constraints using the
+hypothesis $h$. Let us consider the simpler case of an additive function, and
+suppose we have a similar guarantee: $\mathbb{E}_{x \sim {\cal U}}[|h(x) -
+c(x)|]$ where $\forall x, c(x) \equiv \sum_i w_i x_i$ and $h(x) \equiv \sum_i
+\hat w_i x_i$. Can we find a bound on $\|w - \hat w\|_{\infty}$?
+
+As it turns out, yes. Let $\forall i, w_i - \hat w_i \equiv \alpha_i$ and let
+$V(x) \equiv |h(x) - c(x)| = |\sum_{i} \alpha_i x_i |$. Consider the collection
+of good sets ${\cal G} \equiv \{S : v(S) < 4\epsilon\}$. We claim that $|{\cal G}|
+\geq \frac{3}{4}\cdot2^n$. Suppose the contrary, there is at least a quarter of
+the sets which have value $v(S) > 4\epsilon$ such that $\mathbb{E}_{x \sim {\cal
+U}}[|v(x)|] \geq \frac{1}{2^n}\sum_{S \in {\cal G}^c} |v(S)| >
+\frac{1}{4}\cdot4\epsilon = \epsilon$ which is a contradiction. Consider element
+$i$ such that $|\alpha_i| \equiv \|\alpha\|_{\infty}$. Consider the collection
+of sets which contain $i$: ${\cal S}_i \equiv \{S : i \in S\}$.  Notice that
+$|{\cal S}_i| = |{\cal S}_i^c| = 2^{n-1}$. Therefore, $|{\cal G} \cap {\cal
+S}_i^c| \geq \frac{1}{4}\cdot 2^n$. For all sets $S$ in ${\cal G} \cap {\cal
+S}_i^c$, $v(S\cup\{j\}) \geq \alpha_i - 4\epsilon$. It follows that
+$\mathbb{E}_{x \sim {\cal U}}[|v(x)|] \geq \frac{1}{4}(\alpha_j - 4\epsilon)$
+and therefore we have $\|w - \hat w\|_{\infty} \leq 8 \epsilon$.
+
+
+
-- 
cgit v1.2.3-70-g09d2


From 90211ba1847251c552361a7d0b2eef1c540ef72a Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Tue, 14 Apr 2015 15:27:24 -0400
Subject: Nips format and minor tweaks. We are already close to the page limit…
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 paper/main.tex              |  10 +-
 paper/nips15submit_e.sty    | 236 ++++++++++++++++++++++++++++++++++++++++++++
 paper/sections/negative.tex |   2 +-
 3 files changed, 244 insertions(+), 4 deletions(-)
 create mode 100644 paper/nips15submit_e.sty

(limited to 'paper/sections/negative.tex')

diff --git a/paper/main.tex b/paper/main.tex
index 58b6b22..0b33bdd 100644
--- a/paper/main.tex
+++ b/paper/main.tex
@@ -1,6 +1,11 @@
 \documentclass[10pt]{article}
-\usepackage{fullpage, amsmath, amssymb, amsthm, bbm}
+\usepackage{nips15submit_e}
+\usepackage{amsmath, amssymb, amsthm, bbm, microtype}
 \usepackage[utf8x]{inputenc}
+\usepackage[capitalize, noabbrev]{cleveref}
+\usepackage[pagebackref=false,breaklinks=true,colorlinks=true]{hyperref}
+\usepackage{natbib}
+\usepackage{url}
 
 \DeclareMathOperator{\E}{\mathbb{E}}
 \let\P\relax
@@ -11,7 +16,6 @@
 \newcommand{\ints}{\mathbb{N}}
 \renewcommand{\O}{\mathcal{O}}
 
-
 \newtheorem{proposition}{Proposition}
 \newtheorem{corollary}{Corollary}
 \newtheorem{problem}{Problem}
@@ -35,7 +39,7 @@
 \section{Positive results}
 \input{sections/positive.tex}
 
-\bibliographystyle{apalike}
+\bibliographystyle{abbrv}
 \bibliography{optimize}
 
 
diff --git a/paper/nips15submit_e.sty b/paper/nips15submit_e.sty
new file mode 100644
index 0000000..75dc8a8
--- /dev/null
+++ b/paper/nips15submit_e.sty
@@ -0,0 +1,236 @@
+%%%% NIPS Macros (LaTex)
+%%%% Style File
+%%%% Dec 12, 1990   Rev Aug 14, 1991; Sept, 1995; April, 1997; April, 1999
+
+% This file can be used with Latex2e whether running in main mode, or
+% 2.09 compatibility mode.
+%
+% If using main mode, you need to include the commands
+%             \documentclass{article}
+%             \usepackage{nips10submit_e,times}
+% as the first lines in your document.  Or, if you do not have Times
+% Roman font available, you can just use
+%             \documentclass{article}
+%             \usepackage{nips10submit_e}
+% instead.
+%
+% If using 2.09 compatibility mode, you need to include the command
+%             \documentstyle[nips10submit_09,times]{article} 
+% as the first line in your document.  Or, if you do not have Times
+% Roman font available, you can include the command
+%             \documentstyle[nips10submit_09]{article}
+% instead.
+
+% Change the overall width of the page.  If these parameters are
+%       changed, they will require corresponding changes in the
+%       maketitle section.
+%
+\usepackage{eso-pic} % used by \AddToShipoutPicture 
+
+\renewcommand{\topfraction}{0.95}   % let figure take up nearly whole page
+\renewcommand{\textfraction}{0.05}  % let figure take up nearly whole page
+
+% Define nipsfinal, set to true if nipsfinalcopy is defined  
+\newif\ifnipsfinal
+\nipsfinalfalse
+\def\nipsfinalcopy{\nipsfinaltrue}
+\font\nipstenhv  = phvb at 8pt % *** IF THIS FAILS, SEE nips10submit_e.sty ***
+
+% Specify the dimensions of each page
+
+\setlength{\paperheight}{11in}
+\setlength{\paperwidth}{8.5in}
+
+\oddsidemargin .5in    %   Note \oddsidemargin = \evensidemargin
+\evensidemargin .5in
+\marginparwidth 0.07 true in
+%\marginparwidth 0.75 true in
+%\topmargin 0 true pt           % Nominal distance from top of page to top of
+%\topmargin 0.125in
+\topmargin -0.625in
+\addtolength{\headsep}{0.25in}
+\textheight 9.0 true in       % Height of text (including footnotes & figures)
+\textwidth 5.5 true in        % Width of text line.
+\widowpenalty=10000
+\clubpenalty=10000
+
+% \thispagestyle{empty}        \pagestyle{empty}
+\flushbottom \sloppy
+
+% We're never going to need a table of contents, so just flush it to 
+% save space --- suggested by drstrip@sandia-2
+\def\addcontentsline#1#2#3{}
+
+% Title stuff, taken from deproc.
+\def\maketitle{\par 
+\begingroup
+   \def\thefootnote{\fnsymbol{footnote}}
+   \def\@makefnmark{\hbox to 0pt{$^{\@thefnmark}$\hss}} % for perfect author
+                                                        % name centering
+%   The footnote-mark was overlapping the footnote-text,
+%   added the following to fix this problem               (MK)
+   \long\def\@makefntext##1{\parindent 1em\noindent
+                            \hbox to1.8em{\hss $\m@th ^{\@thefnmark}$}##1}
+   \@maketitle \@thanks
+\endgroup
+\setcounter{footnote}{0}
+\let\maketitle\relax \let\@maketitle\relax
+\gdef\@thanks{}\gdef\@author{}\gdef\@title{}\let\thanks\relax}
+
+% The toptitlebar has been raised to top-justify the first page
+
+% Title (includes both anonimized and non-anonimized versions)
+\def\@maketitle{\vbox{\hsize\textwidth
+\linewidth\hsize \vskip 0.1in \toptitlebar \centering
+{\LARGE\bf \@title\par}  \bottomtitlebar % \vskip 0.1in %  minus
+\ifnipsfinal
+   \def\And{\end{tabular}\hfil\linebreak[0]\hfil
+            \begin{tabular}[t]{c}\bf\rule{\z@}{24pt}\ignorespaces}% 
+  \def\AND{\end{tabular}\hfil\linebreak[4]\hfil
+            \begin{tabular}[t]{c}\bf\rule{\z@}{24pt}\ignorespaces}% 
+    \begin{tabular}[t]{c}\bf\rule{\z@}{24pt}\@author\end{tabular}% 
+\else 
+     \begin{tabular}[t]{c}\bf\rule{\z@}{24pt}
+Anonymous Author(s) \\
+Affiliation \\
+Address \\
+\texttt{email} \\
+\end{tabular}% 
+\fi
+\vskip 0.3in minus 0.1in}}
+
+\renewenvironment{abstract}{\vskip.075in\centerline{\large\bf
+Abstract}\vspace{0.5ex}\begin{quote}}{\par\end{quote}\vskip 1ex}
+
+% sections with less space
+\def\section{\@startsection {section}{1}{\z@}{-2.0ex plus
+    -0.5ex minus -.2ex}{1.5ex plus 0.3ex
+minus0.2ex}{\large\bf\raggedright}}
+
+\def\subsection{\@startsection{subsection}{2}{\z@}{-1.8ex plus    
+-0.5ex minus -.2ex}{0.8ex plus .2ex}{\normalsize\bf\raggedright}}
+\def\subsubsection{\@startsection{subsubsection}{3}{\z@}{-1.5ex
+plus      -0.5ex minus -.2ex}{0.5ex plus
+.2ex}{\normalsize\bf\raggedright}}
+\def\paragraph{\@startsection{paragraph}{4}{\z@}{1.5ex plus   
+0.5ex minus .2ex}{-1em}{\normalsize\bf}}
+\def\subparagraph{\@startsection{subparagraph}{5}{\z@}{1.5ex plus 
+  0.5ex minus .2ex}{-1em}{\normalsize\bf}}
+\def\subsubsubsection{\vskip
+5pt{\noindent\normalsize\rm\raggedright}}
+
+
+% Footnotes
+\footnotesep 6.65pt %
+\skip\footins 9pt plus 4pt minus 2pt
+\def\footnoterule{\kern-3pt \hrule width 12pc \kern 2.6pt }
+\setcounter{footnote}{0}
+
+% Lists and paragraphs
+\parindent 0pt
+\topsep 4pt plus 1pt minus 2pt
+\partopsep 1pt plus 0.5pt minus 0.5pt
+\itemsep 2pt plus 1pt minus 0.5pt
+\parsep 2pt plus 1pt minus 0.5pt
+\parskip .5pc
+
+
+%\leftmargin2em 
+\leftmargin3pc
+\leftmargini\leftmargin \leftmarginii 2em
+\leftmarginiii 1.5em \leftmarginiv 1.0em \leftmarginv .5em 
+
+%\labelsep \labelsep 5pt
+
+\def\@listi{\leftmargin\leftmargini}
+\def\@listii{\leftmargin\leftmarginii
+   \labelwidth\leftmarginii\advance\labelwidth-\labelsep
+   \topsep 2pt plus 1pt minus 0.5pt
+   \parsep 1pt plus 0.5pt minus 0.5pt
+   \itemsep \parsep}
+\def\@listiii{\leftmargin\leftmarginiii
+    \labelwidth\leftmarginiii\advance\labelwidth-\labelsep
+    \topsep 1pt plus 0.5pt minus 0.5pt 
+    \parsep \z@ \partopsep 0.5pt plus 0pt minus 0.5pt
+    \itemsep \topsep}
+\def\@listiv{\leftmargin\leftmarginiv
+     \labelwidth\leftmarginiv\advance\labelwidth-\labelsep}
+\def\@listv{\leftmargin\leftmarginv
+     \labelwidth\leftmarginv\advance\labelwidth-\labelsep}
+\def\@listvi{\leftmargin\leftmarginvi
+     \labelwidth\leftmarginvi\advance\labelwidth-\labelsep}
+
+\abovedisplayskip 7pt plus2pt minus5pt%
+\belowdisplayskip \abovedisplayskip
+\abovedisplayshortskip  0pt plus3pt%
+\belowdisplayshortskip  4pt plus3pt minus3pt%
+
+% Less leading in most fonts (due to the narrow columns)
+% The choices were between 1-pt and 1.5-pt leading
+%\def\@normalsize{\@setsize\normalsize{11pt}\xpt\@xpt} % got rid of @ (MK)
+\def\normalsize{\@setsize\normalsize{11pt}\xpt\@xpt}
+\def\small{\@setsize\small{10pt}\ixpt\@ixpt}
+\def\footnotesize{\@setsize\footnotesize{10pt}\ixpt\@ixpt}
+\def\scriptsize{\@setsize\scriptsize{8pt}\viipt\@viipt}
+\def\tiny{\@setsize\tiny{7pt}\vipt\@vipt}
+\def\large{\@setsize\large{14pt}\xiipt\@xiipt}
+\def\Large{\@setsize\Large{16pt}\xivpt\@xivpt}
+\def\LARGE{\@setsize\LARGE{20pt}\xviipt\@xviipt}
+\def\huge{\@setsize\huge{23pt}\xxpt\@xxpt}
+\def\Huge{\@setsize\Huge{28pt}\xxvpt\@xxvpt}
+
+\def\toptitlebar{\hrule height4pt\vskip .25in\vskip-\parskip}
+
+\def\bottomtitlebar{\vskip .29in\vskip-\parskip\hrule height1pt\vskip
+.09in} %
+%Reduced second vskip to compensate for adding the strut in \@author
+
+% Vertical Ruler
+% This code is, largely, from the CVPR 2010 conference style file
+% ----- define vruler
+\makeatletter
+\newbox\nipsrulerbox
+\newcount\nipsrulercount
+\newdimen\nipsruleroffset
+\newdimen\cv@lineheight
+\newdimen\cv@boxheight
+\newbox\cv@tmpbox
+\newcount\cv@refno
+\newcount\cv@tot
+% NUMBER with left flushed zeros  \fillzeros[<WIDTH>]<NUMBER>
+\newcount\cv@tmpc@ \newcount\cv@tmpc
+\def\fillzeros[#1]#2{\cv@tmpc@=#2\relax\ifnum\cv@tmpc@<0\cv@tmpc@=-\cv@tmpc@\fi
+\cv@tmpc=1 %
+\loop\ifnum\cv@tmpc@<10 \else \divide\cv@tmpc@ by 10 \advance\cv@tmpc by 1 \fi
+   \ifnum\cv@tmpc@=10\relax\cv@tmpc@=11\relax\fi \ifnum\cv@tmpc@>10 \repeat
+\ifnum#2<0\advance\cv@tmpc1\relax-\fi
+\loop\ifnum\cv@tmpc<#1\relax0\advance\cv@tmpc1\relax\fi \ifnum\cv@tmpc<#1 \repeat
+\cv@tmpc@=#2\relax\ifnum\cv@tmpc@<0\cv@tmpc@=-\cv@tmpc@\fi \relax\the\cv@tmpc@}%
+% \makevruler[<SCALE>][<INITIAL_COUNT>][<STEP>][<DIGITS>][<HEIGHT>]
+\def\makevruler[#1][#2][#3][#4][#5]{\begingroup\offinterlineskip
+\textheight=#5\vbadness=10000\vfuzz=120ex\overfullrule=0pt%
+\global\setbox\nipsrulerbox=\vbox to \textheight{%
+{\parskip=0pt\hfuzz=150em\cv@boxheight=\textheight
+\cv@lineheight=#1\global\nipsrulercount=#2%
+\cv@tot\cv@boxheight\divide\cv@tot\cv@lineheight\advance\cv@tot2%
+\cv@refno1\vskip-\cv@lineheight\vskip1ex%
+\loop\setbox\cv@tmpbox=\hbox to0cm{{\nipstenhv\hfil\fillzeros[#4]\nipsrulercount}}%
+\ht\cv@tmpbox\cv@lineheight\dp\cv@tmpbox0pt\box\cv@tmpbox\break
+\advance\cv@refno1\global\advance\nipsrulercount#3\relax
+\ifnum\cv@refno<\cv@tot\repeat}}\endgroup}%
+\makeatother
+% ----- end of vruler
+
+% \makevruler[<SCALE>][<INITIAL_COUNT>][<STEP>][<DIGITS>][<HEIGHT>]
+\def\nipsruler#1{\makevruler[12pt][#1][1][3][0.993\textheight]\usebox{\nipsrulerbox}}
+\AddToShipoutPicture{%
+\ifnipsfinal\else
+\nipsruleroffset=\textheight
+\advance\nipsruleroffset by -3.7pt
+  \color[rgb]{.7,.7,.7}
+  \AtTextUpperLeft{%
+    \put(\LenToUnit{-35pt},\LenToUnit{-\nipsruleroffset}){%left ruler
+      \nipsruler{\nipsrulercount}}
+  }
+\fi
+}
diff --git a/paper/sections/negative.tex b/paper/sections/negative.tex
index f88117a..2eaeb47 100644
--- a/paper/sections/negative.tex
+++ b/paper/sections/negative.tex
@@ -25,7 +25,7 @@ guarantees are sufficient for optimization. First look at learning weights in
 a cover function. Maybe facility location? Sums of concave over modular are
 probably too hard because of the connection to neural networks.
 
-\subsection{Optimizable $n\Rightarrow$ Learnable}
+\subsection{Optimizable $\nRightarrow$ Learnable}
 
 Recall the matroid construction from~\cite{balcan2011learning}:
 \begin{theorem}
-- 
cgit v1.2.3-70-g09d2


From ead8ef85b50848f078a360b040f59ea962821cb4 Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Tue, 14 Apr 2015 16:46:34 -0400
Subject: Sketch of the introduction

---
 paper/sections/introduction.tex | 81 +++++++++++++++++++++++++++--------------
 paper/sections/negative.tex     | 30 +++++++++++++++
 2 files changed, 84 insertions(+), 27 deletions(-)

(limited to 'paper/sections/negative.tex')

diff --git a/paper/sections/introduction.tex b/paper/sections/introduction.tex
index bcb406f..6eda59d 100644
--- a/paper/sections/introduction.tex
+++ b/paper/sections/introduction.tex
@@ -1,27 +1,54 @@
-Let $\Omega$ be the universe of elements and $f$ a function defined on subsets
-of $\Omega$: $f : S \in 2^{[\Omega]} \mapsto f(S) \in \mathbb{R}$. Let $K$ be a
-collection of sets of $2^{[\Omega]}$, which we call \emph{constraints}. Let
-$S^*_K$ be any solution to $\max_{S \in K} f(S)$, which we will also denote by
-$S^*$ when there is no ambiguity. Let $L$ be the problem size, which is often
-(but not always) equal to $|\Omega|$.
-
-In general, we say we can efficiently optimize a function $f$ under constraint
-$K$ when we have a polynomial-time algorithm making adaptive value queries to
-$f$,which returns a set $S$ such that $S \in K$ and $f(S) \geq \alpha f(S^*)$
-with high probability and $\alpha$ an absolute constant.
-
-Here, we consider the scenario where we cannot make adaptive value queries, and
-in fact, where we cannot make queries at all! Instead, we suppose that we
-observe a polynomial number of set-value pairs $(S, f(S))$ where $S$ is taken
-from a known distribution $D$. We say we can efficiently \emph{passively
-optimize} $f$ under distribution $D$ or $D-$optimize $f$ under constraints $K$
-when, after observing ${\cal O}(L^c)$ set-value pairs from $D$ where $c > 0$ is
-an absolute constant, we can return a set $S$ such that $S \in K$ and $f(S)
-\geq \alpha f(S^*)$ with high probability and $\alpha$ an absolute constant.
-
-In the case of \emph{passive} observations of set-value pairs under a
-distribution $D$ for a function $f$, recent research has focused on whether we
-can efficiently and approximately \emph{learn} $f$. This was formalized in the
-PMAC model from \cite{balcan2011learning}. When thinking about passive
-optimization, it is necessary to understand the link between being able to
- $D-PMAC$ learn $f$ and being able to $D-$optimize $f$.
+The natural “diminishing return” property of submodular functions makes them
+suitable to model many real-life scenarios. However, even when they are
+suitable as a model, it is not realistic to assume that they are entirely
+known. For example, most submodular functions depend on unknown parameters that
+have to be learned from observations.
+
+A recent line of work has focused on learning submodular functions. The
+PMAC-learnable framework formalizes what it means to learn a submodular
+function in an “overall” sense. For specific examples of submodular functions,
+results are known on how well the parameters of the function can be learned
+from observations.
+
+When the experimenter wishes to optimize the unknown submodular function, the
+well-known approximation algorithms cannot be readily applied because they rely
+on a value query oracle, which is much stronger than what the present scenario
+suggests (at best, random independent samples). The results on learning
+submodular functions also seem to be insufficient in that they are myopic to
+the ultimate use which will be made of the submodular function (optimizing).
+The experimenter experimenter only cares about learning the function to the
+extent that what he knows is sufficient to optimize it. As such, the main
+question underlying the present work is the following:
+
+\begin{center}
+    \emph{What is a good model of learning for submodular functions when the
+    ultimate goal is to be able to optimize them?}
+\end{center}
+
+We make the following contributions:
+\begin{itemize}
+    \item We show how the PMAC learning framework does not provide a satisfying
+    framework for optimizing under uncertainty. In particular, we show that
+    there are functions which are PMAC-learnable but not optimizable and
+    vice-versa.
+    \item For specific class of submodular functions, we show how to optimize
+    them under uncertainty.
+\end{itemize}
+
+\subsection{Related Work}
+
+\paragraph{Maximizing submodular functions} Vast literature on maximizing
+submodular functions under many kind of constraints. All the algorithms rely on
+some kind of oracle (e.g. value query or demand query) which is stronger than
+what we have: access to polynomially many observations coming from
+a distribution. In particular, an algorithm cannot make arbitrary (and
+certainly not adaptive) queries to the oracle.
+
+\paragraph{PMAC-learning} Interesting line of work, but oblivious to the
+ultimate goal.
+
+\paragraph{Bandits} Interesting in that it ties together learning and
+optimizing. However, it assumes that arbitrary queries can be made to an oracle
+allowing the experimenter to explore more efficiently (even though the standard
+regret benchmark means that he will incur some cost for exploring vs
+exploiting.
diff --git a/paper/sections/negative.tex b/paper/sections/negative.tex
index 2eaeb47..555aa1d 100644
--- a/paper/sections/negative.tex
+++ b/paper/sections/negative.tex
@@ -1,3 +1,33 @@
+\subsection{Model}
+
+Let $\Omega$ be the universe of elements and $f$ a function defined on subsets
+of $\Omega$: $f : S \in 2^{[\Omega]} \mapsto f(S) \in \mathbb{R}$. Let $K$ be a
+collection of sets of $2^{[\Omega]}$, which we call \emph{constraints}. Let
+$S^*_K$ be any solution to $\max_{S \in K} f(S)$, which we will also denote by
+$S^*$ when there is no ambiguity. Let $L$ be the problem size, which is often
+(but not always) equal to $|\Omega|$.
+
+In general, we say we can efficiently optimize a function $f$ under constraint
+$K$ when we have a polynomial-time algorithm making adaptive value queries to
+$f$,which returns a set $S$ such that $S \in K$ and $f(S) \geq \alpha f(S^*)$
+with high probability and $\alpha$ an absolute constant.
+
+Here, we consider the scenario where we cannot make adaptive value queries, and
+in fact, where we cannot make queries at all! Instead, we suppose that we
+observe a polynomial number of set-value pairs $(S, f(S))$ where $S$ is taken
+from a known distribution $D$. We say we can efficiently \emph{passively
+optimize} $f$ under distribution $D$ or $D-$optimize $f$ under constraints $K$
+when, after observing ${\cal O}(L^c)$ set-value pairs from $D$ where $c > 0$ is
+an absolute constant, we can return a set $S$ such that $S \in K$ and $f(S)
+\geq \alpha f(S^*)$ with high probability and $\alpha$ an absolute constant.
+
+In the case of \emph{passive} observations of set-value pairs under a
+distribution $D$ for a function $f$, recent research has focused on whether we
+can efficiently and approximately \emph{learn} $f$. This was formalized in the
+PMAC model from \cite{balcan2011learning}. When thinking about passive
+optimization, it is necessary to understand the link between being able to
+ $D-PMAC$ learn $f$ and being able to $D-$optimize $f$.
+
 \subsection{Learnable $\nRightarrow$ Optimizable}
 
 Consider the following function:
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