We revisit the classical problem of optimal experimental design (OED) under a new mathematical model grounded in a geometric motivation. Specifically, we introduce models based on elementary symmetric polynomials; these polynomials capture "partial volumes" and offer a graded interpolation between the widely used A-optimal design and D-optimal design models, obtaining each of them as special cases. We analyze properties of our models, and derive both greedy and convex-relaxation algorithms for computing the associated designs. Our analysis establishes approximation guarantees on these algorithms, while our empirical results substantiate our claims and demonstrate a curious phenomenon concerning our greedy method. Finally, as a byproduct, we obtain new results on the theory of elementary symmetric polynomials that may be of independent interest.

We revisit the classical problem of optimal experimental design (OED) under a new mathematical model grounded in a geometric motivation. Specifically, we introduce models based on elementary symmetric polynomials; these polynomials capture "partial volumes" and offer a graded interpolation between the widely used A-optimal design and D-optimal design models, obtaining each of them as special cases. We analyze properties of our models, and derive both greedy and convex-relaxation algorithms for computing the associated designs. Our analysis establishes approximation guarantees on these algorithms, while our empirical results substantiate our claims and demonstrate a curious phenomenon concerning our greedy method. Finally, as a byproduct, we obtain new results on the theory of elementary symmetric polynomials that may be of independent interest.

In this work, we study the $\lambda$-regularized $A$-optimal design problem and introduce the $\lambda$-regularized proportional volume sampling algorithm, generalized from [Nikolov, Singh, and Tantipongpipat, 2019], for this problem with the approximation guarantee that extends upon the previous work. In this problem, we are given vectors $v_1,\ldots,v_n\in\mathbb{R}^d$ in $d$ dimensions, a budget $k\leq n$, and the regularizer parameter $\lambda\geq0$, and the goal is to find a subset $S\subseteq [n]$ of size $k$ that minimizes the trace of $\left(\sum_{i\in S}v_iv_i^\top + \lambda I_d\right)^{-1}$ where $I_d$ is the $d\times d$ identity matrix. The problem is motivated from optimal design in ridge regression, where one tries to minimize the expected squared error of the ridge regression predictor from the true coefficient in the underlying linear model. We introduce $\lambda$-regularized proportional volume sampling and give its polynomial-time implementation to solve this problem. We show its $(1+\frac{\epsilon}{\sqrt{1+\lambda'}})$-approximation for $k=\Omega\left(\frac d\epsilon+\frac{\log 1/\epsilon}{\epsilon^2}\right)$ where $\lambda'$ is proportional to $\lambda$, extending the previous bound in [Nikolov, Singh, and Tantipongpipat, 2019] to the case $\lambda>0$ and obtaining asymptotic optimality as $\lambda\rightarrow \infty$.

We revisit the classical problem of optimal experimental design (OED) under a new mathematical model grounded in a geometric motivation. Specifically, we introduce models based on elementary symmetric polynomials; these polynomials capture "partial volumes" and offer a graded interpolation between the widely used A-optimal and D-optimal design models, obtaining each of them as special cases. We analyze properties of our models, and derive both greedy and convex-relaxation algorithms for computing the associated designs. Our analysis establishes approximation guarantees on these algorithms, while our empirical results substantiate our claims and demonstrate a curious phenomenon concerning our greedy algorithm. Finally, as a byproduct, we obtain new results on the theory of elementary symmetric polynomials that may be of independent interest.