Approximation Algorithms for D-optimal Design

Experimental design is a classical statistics problem and its aim is to estimate an unknown m-dimensional vector β 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 hatβ. 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 β-hatβ. 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-ε)-approximation if k≥ frac4mε+frac12ε2log(frac1ε) for any ε ∈ (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.