Large and complex datasets are often collected from several, possibly heterogeneous sources. Collaborative learning methods improve efficiency by leveraging commonalities across datasets while accounting for possible differences among them. Here we study collaborative linear regression and contextual bandits, where each instance's associated parameters are equal to a global parameter plus a sparse instance-specific term. We propose a novel two-stage estimator called MOLAR that leverages this structure by first constructing an entry-wise median of the instances' linear regression estimates, and then shrinking the instance-specific estimates towards the median. MOLAR improves the dependence of the estimation error on the data dimension, compared to independent least squares estimates. We then apply MOLAR to develop methods for sparsely heterogeneous collaborative contextual bandits, which lead to improved regret guarantees compared to independent 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：大规模复杂数据集通常来自多个可能异质的来源。协同学习方法通过利用数据集间的共性同时考虑其潜在差异来提高效率。本文研究协同线性回归与上下文Bandits问题，其中每个实例的关联参数等于全局参数加上一个稀疏的实例特定项。我们提出一种名为MOLAR的新型两阶段估计器：首先构建各实例线性回归估计的逐元素中位数，然后将实例特定估计向该中位数收缩。与独立最小二乘估计相比，MOLAR改善了估计误差对数据维度的依赖关系。随后，我们将MOLAR应用于稀疏异质协同上下文Bandits方法的设计，相较于独立Bandits方法获得了更优的遗憾界保证。进一步通过给出多个下界证明所提方法达到极小化最优。最后，我们在合成数据及异质国家学生教育成果的PISA数据集上进行实验，验证了方法的有效性。