From 574e1f7e48f238efada8bff73ba26fd6dd50aaac Mon Sep 17 00:00:00 2001 From: Stratis Ioannidis Date: Mon, 11 Feb 2013 16:36:11 -0800 Subject: related --- intro.tex | 8 +++++--- 1 file changed, 5 insertions(+), 3 deletions(-) (limited to 'intro.tex') diff --git a/intro.tex b/intro.tex index f69620f..39e6b52 100644 --- a/intro.tex +++ b/intro.tex @@ -34,15 +34,17 @@ We initiate the study of experimental design problem in presence of budgets and %is related to the covariance of the $x_i$'s. In particular, we formulate the {\em Strategic Experimental Design Problem} (\SEDP) 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, as well as the usual constraints of truthfulness and individual rationality. %, and other {\em strategic constraints} we don't list here. +subject to a budget constraint $\sum_{i\in S}c_i\leq B$, where $B$ is \E's budget, as well as the usual constraints of truthfulness and individual rationality. +This is naturally viewed as a \emph{budget feasible mechanism design} problem, as introduced by \citeN{singer-mechanisms}. +%, and other {\em strategic constraints} we don't list here. \smallskip The objective function, which is the key, is formally obtained by optimizing the information gain in $\beta$ when the latter is learned through linear regression, and is related to the so-called $D$-optimality criterion~\cite{pukelsheim2006optimal,atkinson2007optimum}. \item -We present a polynomial time, truthful mechanism for \SEDP{}, yielding a constant factor ($\approx 12.98$) approximation to the optimal value of \eqref{obj}. In contrast to this, we show that no truthful, budget-feasible algorithms are possible for \SEDP{} within a factor 2 approximation. +We present a polynomial time, truthful mechanism for \SEDP{}, yielding a constant factor ($\approx 12.98$) approximation to the optimal value of \eqref{obj}. In contrast to this, we show that no truthful, budget-feasible mechanisms are possible for \SEDP{} within a factor 2 approximation. \smallskip -We note that the objective \eqref{obj} is submodular. Using this fact, applying previous results for budget feasible mechanisms under general submodular objectives~\cite{singer-mechanisms,chen} would yield either a deterministic, truthful, constant-approximation mechanism that requires exponential time, or a non-deterministic, (universally) truthful, poly-time mechanism that yields a constant approximation ratio only \emph{in expectation} (\emph{i.e.}, its approximation guarantee for a given instance may in fact be unbounded). +We note that the objective \eqref{obj} is submodular. Using this fact, applying previous results on budget feasible mechanism design under general submodular objectives~\cite{singer-mechanisms,chen} would yield either a deterministic, truthful, constant-approximation mechanism that requires exponential time, or a non-deterministic, (universally) truthful, poly-time mechanism that yields a constant approximation ratio only \emph{in expectation} (\emph{i.e.}, its approximation guarantee for a given instance may in fact be unbounded). \end{itemize} -- cgit v1.2.3-70-g09d2