稀疏异质性下的最优多任务线性回归与上下文赌博机
Optimal Multitask Linear Regression and Contextual Bandits under Sparse Heterogeneity
- Univ. of Pennsylvania（宾夕法尼亚大学）
- Arizona State University（亚利桑那州立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对稀疏异质性下的多任务线性回归与上下文赌博机问题，提出两阶段估计器MOLAR，改善估计误差对维度的依赖，获更优遗憾保证且达极小极大最优，实验验证了其效率。
AI中文摘要：
大型复杂数据集通常来自多个可能具有异质性的来源。多任务学习方法通过利用不同数据集间的共性同时考量其间的潜在差异，来提升效率。本文研究稀疏异质性场景下的多任务线性回归与上下文赌博机问题，该场景中与来源/任务相关的参数等于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，它利用该结构，首先构建任务级线性回归估计的协变量级加权中位数，随后将任务级估计向加权中位数收缩。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。本文还讨论了MOLAR向广义线性模型的扩展以及置信区间的构建方法。我们进一步将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。通过给出多个下界，我们进一步证明了所提方法是极小极大最优的。最后，我们在合成数据以及来自异质国家的学生教育成果PISA数据集上开展实验，验证了方法的有效性。
英文摘要：
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.
