In the classic setting of experimental design \cite{pukelsheim2006optimal,atkinson2007optimum},
 an {\em experimenter}  \E\ has access to a population of $n$ potential experiment subjects. 
Each subject $i\in  \{1,\ldots,n\}$ is associated with a set of parameters (or features) $x_i\in \reals^d$, 
known to the experimenter. 
\E\ wishes to perform an experiment that measures a certain inherent property of the subjects: the outcome for a subject $i$ is denoted $y_i$, which is unknown to \E\ before the experiment is performed.
Typically, \E\ has a hypothesis of the relationship between $x_i$'s and $y_i$'s, such as, say linear, \emph{i.e.},  $y_i \approx  \T{\beta} x_i$; conducting the experiments and obtaining the measurements $y_i$ lets \E\  estimate  $\beta$. %The goal of experimental design amounts to determining which subjects to experiment upon to produce the best possible such estimate.
 
The above experimental design scenario  has many applications, from medical testing to marketing research and others. 
There is a rich literature about estimation procedures, as well as for means of quantifying the quality of the produced estimate  \cite{pukelsheim2006optimal}.  There is also an extensive theory on how to select subjects 
if \E\ can conduct only a limited number of experiments, so the estimation process returns $\beta$
that approximates the true parameter of the underlying population \cite{ginebra2007measure,le1996comparison,chaloner1995bayesian,boyd2004convex}. 

We depart from this classical setup by viewing experimental design in a strategic setting, and by studying mechanism design issues. 
In our setting, experiments cannot be manipulated and hence measurements are considered reliable. %\footnote{Thus, the experiments of our interest are statistically significant ones where each experiment provides a reliable outcome.}  
However, there
is a cost $c_i$ associated with experimenting on
subject $i$ which varies from subject to subject. This may be viewed as the  
cost subject $i$ incurs when tested and for which she needs to be reimbursed; or, it might be viewed as the incentive for $i$
to participate in the experiment; or, it might be the inherent value of the data.   This economic aspect has always been inherent in experimental design: experimenters often work within strict budgets and design creative incentives. However, we are not aware of principled study of this setting from a strategic point of view. When subjects are strategic, they may have an incentive to misreport their cost and the choice of experiments and payments need to be more sophisticated.


Our contributions are as follows.
\begin{itemize}
\item
We formulate the problem of experimental design subject to a given budget, in the presence of strategic agents who may lie about their costs. In particular, we focus on linear regression. This is naturally viewed  as a budget feasible mechanism design problem, in which the objective function %is sophisticated and 
is  related to the covariance of the $x_i$'s. In particular, we formulate the  {\em Experimental Design Problem} (\EDP) as follows: the experimenter \E\ wishes to find set $S$ of subjects to maximize 
\begin{align}V(S) = \log\det\Big(I_d+\sum_{i\in S}x_i\T{x_i}\Big) \label{obj}\end{align}
subject to a budget constraint $\sum_{i\in S}c_i\leq B$, where $B$ is \E's budget. %, and other {\em strategic constraints} we don't list here.
The objective function, which is the key,  is obtained by optimizing  the information gain in  $\beta$ when it is learned  through linear regression methods, and is related to the so-called $D$-optimality criterion. 
\item
The above objective is submodular. 
There are several recent results in budget feasible 
mechanisms~\cite{singer-mechanisms,chen,singer-influence,bei2012budget,dobz2011-mechanisms}, and some apply to  the submodular optimization in 
\EDP.
There is a randomized, 7.91-approximate polynomial time mechanism for maximizing a general submodular function that is universally truthful, \emph{i.e.}, it is sampled from a distribution among truthful mechanisms. Also, there is a $8.34$-approximate exponential time deterministic mechanism. 
There are however no known deterministic, truthful, polynomial time mechanisms for general submodular functions.
For specific combinatorial problems such as \textsc{Knapsack} or \textsc{Coverage}, there exist 
deterministic, truthful, polynomial constant-approximation algorithms~\cite{singer-mechanisms,chen, singer-influence}, but they do not work for the linear-algebraic  objective function in \EDP.
%{\bf S+T: could we verify that the above sentence is correct in its implication?}

We present the first known, polynomial time truthful mechanism for \EDP{}. Our mechanism is a constant factor ($\approx 12.98$) approximation for \EDP{}. In contrast to this, we show that no truthful, budget-feasible algorithms are possible for \EDP{}  within a factor 2 approximation. From a technical perspective, we present a convex relaxation of \eqref{obj}, and show that it is within a constant factor from the so-called multi-linear relaxation of  \eqref{obj}. This allows us to adopt the approach followed by prior work in budget feasible mechanisms by Chen \emph{et al.}~\cite{chen} and Singer~\cite{singer-influence}.   %{\bf FIX the last sentence}

\item
Our approach to mechanisms for experimental design --- by 
 optimizing  the information gain in parameters like $\beta$ which are estimated through the data analysis process --- is general.  We give examples of this approach beyond linear regression to a general class that includes logistic regression and learning binary functions, and show that the corresponding budgeted mechanism design problem is also expressed through a submodular optimization.  Hence,  prior work \cite{chen,singer-mechanisms} immediately applies, and gives randomized, universally truthful, polynomial time, constant factor approximation mechanisms for problems in this class. Getting deterministic, truthful, polynomial time mechanisms with a constant approximation factor for this class or specific problems in it, like we did for \EDP, remains an open problem.
\end{itemize}

In what follows, we describe related work in Section~\ref{sec:related}. We briefly review  experimental design and budget feasible mechanisms in Section~\ref{sec:peel} and define \EDP\ formally. In Section~\ref{sec:main} we present our mechanism for \EDP\ and prove our main results. The present applications of our general framework are presented in Section~\ref{sec:ext}. 

\junk{

\stratis{maximizing other ``optimality criteria'' (E-optimality, G-optimality, etc.) Improved upper lower bound for $R\neq I$. General entropy gain optimization}

\begin{itemize}
    \item already existing field of experiment design: survey-like setup, what
    are the best points to include in your experiment? Measure of the
    usefulness of the data: variance-reduction or entropy-reduction.
    \item nowadays, there is also a big focus on purchasing data: paid surveys,
    mechanical turk, etc. that add economic aspects to the problem of
    experiment design
    \item recent advances (Singer, Chen) in the field of budgeted mechanisms
    \item we study ridge regression, very widely used in statistical learning,
    and treat it as a problem of budgeted experiment design
    \item we make the following contributions: ...
    \item extension to a more general setup which includes a wider class of
    machine learning problems
\end{itemize}

}

\input{related}