English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this work, we study the $\lambda$-regularized $A$-optimal design problemand introduce the $\lambda$-regularized proportional volume sampling algorithm,generalized from [Nikolov, Singh, and Tantipongpipat, 2019], for this problemwith the approximation guarantee that extends upon the previous work. In thisproblem, 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 minimizesthe 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 optimaldesign in ridge regression, where one tries to minimize the expected squarederror of the ridge regression predictor from the true coefficient in theunderlying linear model. We introduce $\lambda$-regularized proportional volumesampling and give its polynomial-time implementation to solve this problem. Weshow 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 $\lambda>0$ andobtaining asymptotic optimality as $\lambda\rightarrow \infty$.

$\lambda$-Regularized A-Optimal Design and its Approximation by $\lambda$-Regularized Proportional Volume Sampling