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%We initiate the study of mechanisms for \emph{experimental design}.
In the classical {\em experimental design} setting,
an experimenter \E\
%with a budget $B$
has access to a population of $n$ potential experiment subjects $i\in \{1,\ldots,n\}$, each associated with a vector of features $x_i\in\reals^d$.
%as well as a cost $c_i>0$.
Conducting an experiment with subject $i$ reveals an unknown value $y_i\in \reals$ to \E. \E\ typically assumes some
hypothetical relationship between $x_i$'s and $y_i$'s, \emph{e.g.}, $y_i \approx \T{\beta} x_i$, and estimates
$\beta$ from experiments, \emph{e.g.}, through linear regression.
%conducting the experiments and obtaining the measurements $y_i$ allows
%\E\ can estimate $\beta$.
As a proxy for various practical constraints, \E{} may select only a subset of subjects on which to conduct the experiment.
%\E 's goal is to select which experiments to conduct, subject to her budget constraint.
%, to obtain the best estimate possible for $\beta$.
We initiate the study of budgeted mechanisms for experimental design. In this setting, \E{} has a budget $B$.
Each subject $i$ declares an associated cost $c_i >0$ to be part of the experiment, and must be paid at least her cost. In particular, the {\em Experimental Design Problem} (\SEDP) is to find a set $S$ of subjects for the experiment that maximizes $V(S) = \log\det(I_d+\sum_{i\in S}x_i\T{x_i})$ under the constraint $\sum_{i\in S}c_i\leq B$; our objective function corresponds to the information gain in parameter $\beta$ that is learned through linear regression methods, and is related to the so-called $D$-optimality criterion. Further, the subjects are \emph{strategic} and may lie about their costs. Thus, we need to design a
mechanism for \SEDP{} with suitable properties.
We present a deterministic, polynomial time, $\delta$-truthful, budget feasible mechanism for \SEDP{}.
By applying previous work on budget feasible mechanisms with submodular objective, one could {\em only} have derived either an exponential time deterministic mechanism or a randomized polynomial time mechanism. Our mechanism yields a constant factor ($\approx 12.68$) approximation, and we show that no truthful, budget-feasible algorithms are possible within a factor $2$ approximation. We also show how to generalize our approach to a wide class of learning problems.
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