We consider a problem in Multi-Task Learning (MTL) where multiple linear models are jointly trained on a collection of datasets ("tasks"). A key novelty of our framework is that it allows the sparsity pattern of regression coefficients and the values of non-zero coefficients to differ across tasks while still leveraging partially shared structure. Our methods encourage models to share information across tasks through separately encouraging 1) coefficient supports, and/or 2) nonzero coefficient values to be similar. This allows models to borrow strength during variable selection even when non-zero coefficient values differ across tasks. We propose a novel mixed-integer programming formulation for our estimator. We develop custom scalable algorithms based on block coordinate descent and combinatorial local search to obtain high-quality (approximate) solutions for our estimator. Additionally, we propose a novel exact optimization algorithm to obtain globally optimal solutions. We investigate the theoretical properties of our estimators. We formally show how our estimators leverage the shared support information across tasks to achieve better variable selection performance. We evaluate the performance of our methods in simulations and two biomedical applications. Our proposed approaches appear to outperform other sparse MTL methods in variable selection and prediction accuracy. We provide the sMTL package on CRAN.

翻译：我们考虑多任务学习（MTL）中的一个问题，即在线性模型集合（“任务”）上联合训练多个模型。我们框架的一个关键创新在于，它允许回归系数的稀疏模式以及非零系数的值在不同任务间存在差异，同时仍能利用部分共享的结构。我们的方法通过分别鼓励1）系数支持集，和/或2）非零系数值相似，来促使模型在任务间共享信息。这使得即使在非零系数值在不同任务间存在差异时，模型也能在变量选择过程中借用彼此优势。我们为估计量提出了一种新颖的混合整数规划形式。我们开发了基于块坐标下降和组合局部搜索的自定义可扩展算法，以获得估计量的高质量（近似）解。此外，我们提出了一种新颖的精确优化算法，以获得全局最优解。我们研究了估计量的理论性质。我们正式证明了估计量如何利用任务间的共享支持信息来实现更好的变量选择性能。我们在模拟实验和两个生物医学应用中评估了所提方法的性能。所提出的方法在变量选择和预测精度方面似乎优于其他稀疏MTL方法。我们在CRAN上提供了sMTL包。