Approximation Algorithms for D-optimal Design

Experimental design is a classical statistics problem and its aim is to estimate an unknown $m$-dimensional vector $\beta$ from linear measurements where a Gaussian noise is introduced in each measurement. For the combinatorial experimental design problem, the goal is to pick $k$ out of the given $n$ experiments so as to make the most accurate estimate of the unknown parameters, denoted as $\hat{\beta}$. In this paper, we will study one of the most robust measures of error estimation - $D$-optimality criterion, which corresponds to minimizing the volume of the confidence ellipsoid for the estimation error $\beta-\hat{\beta}$. The problem gives rise to two natural variants depending on whether repetitions of experiments are allowed or not. We first propose an approximation algorithm with a $\frac1e$-approximation for the $D$-optimal design problem with and without repetitions, giving the first constant factor approximation for the problem. We then analyze another sampling approximation algorithm and prove that it is $(1-\epsilon)$-approximation if $k\geq \frac{4m}{\epsilon}+\frac{12}{\epsilon^2}\log(\frac{1}{\epsilon})$ for any $\epsilon \in (0,1)$. Finally, for $D$-optimal design with repetitions, we study a different algorithm proposed by literature and show that it can improve this asymptotic approximation ratio.