  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 $\lambda\rightarrow \infty$.
