Regression Learning with Limited Observations of Multivariate Outcomes and Features

Multivariate linear regression models are broadly used to facilitate relationships between outcomes and features. However, their effectiveness is compromised by the presence of missing observations, a ubiquitous challenge in real-world applications. Considering a scenario where learners access only limited components for both outcomes and features, we develop efficient algorithms tailored for the least squares ($L_2$) and least absolute ($L_1$) loss functions, each coupled with a ridge-like and Lasso-type penalty, respectively. Moreover, we establish rigorous error bounds for all proposed algorithms. Notably, our $L_2$ loss function algorithms are probably approximately correct (PAC), distinguishing them from their $L_1$ counterparts. Extensive numerical experiments show that our approach outperforms methods that apply existing algorithms for univariate outcome individually to each coordinate of multivariate outcomes in a naive manner. Further, utilizing the $L_1$ loss function or introducing a Lasso-type penalty can enhance predictions in the presence of outliers or high dimensional features. This research contributes valuable insights into addressing the challenges posed by incomplete data.