related

1 files changed, 3 insertions, 3 deletions

diff --git a/intro.tex b/intro.tex
index ce66c1d..f69620f 100644
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+++ b/intro.tex
@@ -39,10 +39,10 @@ subject to a budget constraint $\sum_{i\in S}c_i\leq B$, where $B$ is \E's budge
 \smallskip
 The objective function, which is the key, is formally obtained by optimizing  the information gain in  $\beta$ when the latter is learned  through linear regression, and is related to  the so-called $D$-optimality criterion~\cite{pukelsheim2006optimal,atkinson2007optimum}. 
 \item
-We present a polynomial time, truthful mechanism for \SEDP{}, yielding a constant factor ($\approx 12.98$) approximation. In contrast to this, we show that no truthful, budget-feasible algorithms are possible for \SEDP{}  within a factor 2 approximation. 
+We present a polynomial time, truthful mechanism for \SEDP{}, yielding a constant factor ($\approx 12.98$) approximation to the optimal value of \eqref{obj}. In contrast to this, we show that no truthful, budget-feasible algorithms are possible for \SEDP{}  within a factor 2 approximation. 
 
 \smallskip
-We note that the objective \eqref{obj} is submodular. Using this fact, applying previous results for budget feasible mechanisms under general submodular objectives would yield either a truthful deterministic mechanism that requires exponential time,  or a poly-time algorithm that is not deterministic~\cite{singer-mechanisms,chen}.
+We note that the objective \eqref{obj} is submodular. Using this fact, applying previous results for budget feasible mechanisms under general submodular objectives~\cite{singer-mechanisms,chen} would yield either a deterministic, truthful, constant-approximation mechanism that requires exponential time,  or a non-deterministic, (universally) truthful, poly-time mechanism that yields a constant approximation ratio only \emph{in expectation} (\emph{i.e.}, its approximation guarantee for a given instance may in fact be unbounded).
 \end{itemize}
 
 
@@ -66,7 +66,7 @@ From a technical perspective, we present a convex relaxation of \eqref{obj}, and
 %Our approach to mechanisms for experimental design --- by 
 % optimizing  the information gain in parameters like $\beta$ which are estimated through the data analysis process --- is general.  We give examples of this approach beyond linear regression to a general class that includes logistic regression and learning binary functions, and show that the corresponding budgeted mechanism design problem is also expressed through a submodular optimization.  Hence,  prior work \cite{chen,singer-mechanisms} immediately applies, and gives randomized, universally truthful, polynomial time, constant factor approximation mechanisms for problems in this class. Getting deterministic, truthful, polynomial time mechanisms with a constant approximation factor for this class or specific problems in it, like we did for \EDP, remains an open problem.
 
-In what follows, we describe related work in Section~\ref{sec:related}. We briefly review  experimental design and budget feasible mechanisms in Section~\ref{sec:peel} and define \SEDP\ formally. In Section~\ref{sec:main} we present our mechanism for \SEDP\ and prove our main results. A generalization of our framework to machine learning tasks beyond linear regression is presented in Section~\ref{sec:ext}. 
+In what follows, we describe related work in Section~\ref{sec:related}. We briefly review  experimental design and budget feasible mechanisms in Section~\ref{sec:peel} and define \SEDP\ formally. In Section~\ref{sec:main} we present our mechanism for \SEDP\ and state our main results,  which are proved in Section~\ref{sec:proofs}. A generalization of our framework to machine learning tasks beyond linear regression is presented in Section~\ref{sec:ext}. 
 
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