Approximation Algorithms for D-optimal Design
Mohit Singh, Weijun Xie
TL;DR
This work addresses the combinatorial $D$-optimal design problem, introducing both without- and with-repetition variants. It develops a constant-factor randomized algorithm with a polynomial-time deterministic derandomization that achieves at least a $1/e$ of the optimum, and a sampling-based approach that attains a $(1-\varepsilon)$-approximation when the sample size $k$ scales as $k\ge \tfrac{4m}{\varepsilon}+\tfrac{12}{\varepsilon^2}\log(1/\varepsilon)$. For the repetition setting, the authors analyze a related algorithm and provide improved asymptotic guarantees along with a deterministic implementation. Overall, the paper advances both constant-factor and near-optimal approximation guarantees for D-optimal design and connects to matrix sparsification and related combinatorial optimization techniques.
Abstract
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 $k\geq \frac{4m}ε+\frac{12}{ε^2}\log(\frac{1}ε)$ for any $ε\in (0,1)$. 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.