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1 files changed, 3 insertions, 3 deletions

diff --git a/intro.tex b/intro.tex
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@@ -61,10 +61,10 @@ We note that the objective \eqref{obj} is submodular. Using this fact, applying
 
 %From a technical perspective, we present a convex relaxation of \eqref{obj}, and show that its optimal value is within a constant factor from the so-called multi-linear relaxation of  \eqref{obj}, which in turn can be related to \eqref{obj} through pipage rounding. We establish the constant factor to the multi-linear relaxation by bounding  the partial derivatives of these two functions; we achieve the latter by exploiting convexity properties of matrix functions over the convex cone of positive semidefinite matrices.
 
-From a technical perspective, we propose a convex optimization problem and establish that its optimal value within a constant factor from the optimal value of \EDP.  
+From a technical perspective, we propose a convex optimization problem and establish that its optimal value is within a constant factor from the optimal value of \EDP.  
  In particular, we show our relaxed objective is within a constant factor from the so-called multi-linear extension of  \eqref{obj}, which in turn can be related to \eqref{obj} through pipage rounding. We establish the constant factor to the multi-linear extension by bounding  the partial derivatives of these two functions; we achieve the latter by exploiting convexity properties of matrix functions over the convex cone of positive semidefinite matrices.
 
-Our convex relaxation of \EDP{} involves maximizing a self-concordant function subject to linear constraints. Its optimal value can be computed with arbitrary accuracy in polynomial time using the so-called barrier method. However, the outcome of this computation may not necessarily be monotone, a property needed in designing a truthful mechanism. Nevetheless, we construct an algorithm that solves the above convex relaxation and is ``almost'' monotone; we achieve this by applying the barrier method on a set perturbed constraints, over which our objective is ``sufficiently'' concave. In turn, we also show that this algorithm can be employed to design a $\delta$-truthful mechanism for \EDP{}.
+Our convex relaxation of \EDP{} involves maximizing a self-concordant function subject to linear constraints. Its optimal value can be computed with arbitrary accuracy in polynomial time using the so-called barrier method. However, the outcome of this computation may not necessarily be monotone, a property needed in designing a truthful mechanism. Nevetheless, we construct an algorithm that solves the above convex relaxation and is ``almost'' monotone; we achieve this by applying the barrier method on a set perturbed constraints, over which our objective is ``sufficiently'' concave. In turn, we show how to employ this algorithm to design a poly-time, $\delta$-truthful, constant-approximation mechanism for \EDP{}.
 
 
 %This allows us to adopt the approach followed by prior work in budget feasible mechanisms by Chen \emph{et al.}~\cite{chen} and Singer~\cite{singer-influence}.   %{\bf FIX the last sentence}
@@ -73,7 +73,7 @@ Our convex relaxation of \EDP{} involves maximizing a self-concordant function s
 %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. We present our convex relaxation to \EDP{} in Section~\ref{sec:approximation} and, finally, show how it can be used to construct our mechanism in Section~\ref{sec:main}; all proofs of our technical results are provided in the appendix. %we present our mechanism for \SEDP\ and state 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. We present our convex relaxation to \EDP{} in Section~\ref{sec:approximation} and use it to construct our mechanism in Section~\ref{sec:main}. We conclude in Section~\ref{sec:concl}. All proofs of our technical results are provided in the appendix. %we present our mechanism for \SEDP\ and state our main results. %A generalization of our framework to machine learning tasks beyond linear regression is presented in Section~\ref{sec:ext}. 
 
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