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\title{Budget Feasible Mechanisms\\ for Experimental Design}
\author{
    Thibaut Horel\inst{1}
    \and
    Stratis Ioannidis\inst{2}
    \and
    S. Muthukrishnan\inst{3}
}
\institute{École Normale Supérieure, \email{thibaut.horel@normalesup.org}
    \and
    Technicolor, \email{stratis.ioannidis@technicolor.com}
    \and
    Rutgers University, \email{muthu@cs.rutgers.edu}
}
\begin{document}
\maketitle
\vspace{2em}

In the classical {\em experimental design} setting, an  experimenter  \E\ has
access to a population of $n$ potential experiment subjects $i\in
\{1,\ldots,n\}$,  each associated with a vector of features $x_i\in\reals^d$.
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.  As a proxy
for various practical constraints, \E{} may select only a subset of subjects on
which to conduct the experiment. 

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, budget feasible mechanism scheme, that is approximately
truthful and yields a 12.98 factor  approximation to \EDP.
% By applying previous work on budget feasible mechanisms with
% a submodular objective, one could {\em only} have derived either an exponential
% time deterministic mechanism or a randomized  polynomial time mechanism.
We also establish that no truthful, budget-feasible mechanism is possible
within a factor $2$ approximation, and show how to generalize our approach to
a wide class of learning problems, beyond linear regression.
\end{document}
