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We have proposed a convex relaxation for \EDP, and showed that it can be used to design a $\delta$-truthful, constant approximation mechanism that runs in polynomial time. Our objective function, commonly known as the Bayes $D$-optimality criterion, is motivated by linear regression, and in particular captures the information gain when experiments are used to learn a linear model. %in \reals^d.
A natural question to ask is to what extent the results we present here
generalize to other machine learning tasks beyond linear regression. We outline
a path in pursuing such generalizations in Appendix~\ref{sec:ext}. In
particular, although the information gain is not generally a submodular
function, we show that for a wide class of models, in which experiments
outcomes are perturbed by independent noise, the information gain indeed
exhibits submodularity. Several important learning tasks fall under this
category, including generalized linear regression, logistic regression,
\emph{etc.} In light of this, it would be interesting to investigate whether
our convex relaxation approach generalizes to other learning tasks in this
broader class.
The literature on experimental design includes several other optimality
criteria~\cite{pukelsheim2006optimal,atkinson2007optimum}. Our convex
relaxation \eqref{eq:our-relaxation} involved swapping the $\log\det$
scalarization with the expectation appearing in the multi-linear extension
\eqref{eq:multi-linear}. The same swap is known to yield concave objectives for
several other optimality criteria, even when the latter are not submodular
(see, \emph{e.g.}, \citeN{boyd2004convex}). Exploiting the convexity of such
relaxations to design budget feasible mechanisms is an additional open problem
of interest.
%Many can be seen as scalarizations (\emph{i.e.}, scalar mappings) of the the matrix $(X_S^TX_T)^{-1}$---the $\log\det$ being one of them. Studying such alternative objectives, even within the linear regression setting we study here, is also an interesting related problem. Crucially, o
%To be written. Will contain
%(a) list of extensions with a forward pointer to Appendix
%(b) some concluding remark that we initiated the area, the opt criteria is not a priori clear, etc.
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