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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 from 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 pursueing 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 experiments outcomes are perturbed by independent noise, the information does indeed exhibit 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}. 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, 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 necessarily submodular (see, \emph{e.g.}, \citeN{boyd2004convex}). Exploiting the convexity of such relaxations to design budget feasible mechanisms, is an additonal open problem of interest.
%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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