$λ$-Regularized A-Optimal Design and its Approximation by $λ$-Regularized Proportional Volume Sampling
This work provides an incremental improvement in algorithmic design for optimal experimental selection in ridge regression, benefiting researchers in statistics and machine learning.
The paper tackles the λ-regularized A-optimal design problem for ridge regression by introducing a λ-regularized proportional volume sampling algorithm, achieving a (1+ε/√(1+λ'))-approximation for sufficiently large k, which extends previous bounds to λ>0 and shows asymptotic optimality as λ→∞.
In this work, we study the $λ$-regularized $A$-optimal design problem and introduce the $λ$-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 $λ\geq0$, and the goal is to find a subset $S\subseteq [n]$ of size $k$ that minimizes the trace of $\left(\sum_{i\in S}v_iv_i^\top + λI_d\right)^{-1}$ 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 $λ$-regularized proportional volume sampling and give its polynomial-time implementation to solve this problem. We show its $(1+\fracε{\sqrt{1+λ'}})$-approximation for $k=Ω\left(\frac dε+\frac{\log 1/ε}{ε^2}\right)$ where $λ'$ is proportional to $λ$, extending the previous bound in [Nikolov, Singh, and Tantipongpipat, 2019] to the case $λ>0$ and obtaining asymptotic optimality as $λ\rightarrow \infty$.