Large and complex datasets are often collected from several, possibly heterogeneous sources. Multitask learning methods improve efficiency by leveraging commonalities across datasets while accounting for possible differences among them. Here, we study multitask linear regression and contextual bandits under sparse heterogeneity, where the source/task-associated parameters are equal to a global parameter plus a sparse task-specific term. We propose a novel two-stage estimator called MOLAR that leverages this structure by first constructing a covariate-wise weighted median of the task-wise linear regression estimates and then shrinking the task-wise estimates towards the weighted median. Compared to task-wise least squares estimates, MOLAR improves the dependence of the estimation error on the data dimension. Extensions of MOLAR to generalized linear models and constructing confidence intervals are discussed in the paper. We then apply MOLAR to develop methods for sparsely heterogeneous multitask contextual bandits, obtaining improved regret guarantees over single-task bandit methods. We further show that our methods are minimax optimal by providing a number of lower bounds. Finally, we support the efficiency of our methods by performing experiments on both synthetic data and the PISA dataset on student educational outcomes from heterogeneous countries.

翻译：大型复杂数据集通常从多个可能异质的来源收集。多任务学习方法通过利用数据集间的共性，同时考虑其间的潜在差异来提高效率。本文研究稀疏异质性下的多任务线性回归与上下文赌博机问题，其中源/任务相关参数等于一个全局参数加上一个稀疏的任务特定项。我们提出了一种名为MOLAR的新型两阶段估计器，该估计器通过首先构建任务级线性回归估计的协变量加权中位数，然后将任务级估计向该加权中位数收缩来利用此结构。与任务级最小二乘估计相比，MOLAR改善了估计误差对数据维度的依赖关系。本文讨论了MOLAR向广义线性模型的扩展以及置信区间的构建。随后，我们将MOLAR应用于开发稀疏异质多任务上下文赌博机的方法，获得了优于单任务赌博机方法的遗憾保证。我们进一步通过提供若干下界证明我们的方法具有极小极大最优性。最后，我们通过在合成数据以及来自不同国家的学生教育成果PISA数据集上进行实验，验证了所提方法的有效性。