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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 assume 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.
%conducting the experiments and obtaining the measurements $y_i$ allows
%\E\ can estimate $\beta$.
\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 mechanisms for experimental design. In this setting,
subjects are \emph{strategic} and may lie about their costs. In particular, we formulate the {\em Experimental Design Problem} (\EDP) as finding a set $S$ of subjects that maximize $V(S) = \log\det(I_d+\sum_{i\in S}x_i\T{x_i})$ under the constraint $\sum_{i\in S}c_i\leq B$; our objective function corresponds to the information gain in $\beta$ when it is learned through linear regression methods, and is related to the so-called $D$-optimality criterion. We present the first known
deterministic, polynomial time, truthful, budget feasible mechanism for \EDP{}.
Our mechanism yields a constant factor ($\approx 19.68$) approximation, and we show that no truthful, budget-feasible algorithms are possible within a factor 2 approximation.
Our approach here generally applies to a wider class of learning problems and
obtains polynomial time universally truthful (\emph{i.e.}, randomized) budget feasible mechanism, also within a constant factor approximation.
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