In this work, we formulate a new multi-task active learning setting in which the learner's goal is to solve multiple matrix completion problems simultaneously. At each round, the learner can choose from which matrix it receives a sample from an entry drawn uniformly at random. Our main practical motivation is market segmentation, where the matrices represent different regions with different preferences of the customers. The challenge in this setting is that each of the matrices can be of a different size and also of a different rank which is unknown. We provide and analyze a new algorithm, MAlocate that is able to adapt to the unknown ranks of the different matrices. We then give a lower-bound showing that our strategy is minimax-optimal and demonstrate its performance with synthetic experiments.

翻译：本文提出了一种新的多任务主动学习场景，其中学习者的目标是同时解决多个矩阵补全问题。在每一轮中，学习者可以选择从哪个矩阵中随机均匀抽取一个条目的样本。该研究的主要实际动机来自市场细分领域，其中不同矩阵代表具有不同客户偏好的不同区域。本场景的挑战在于每个矩阵的规模可能不同，且其未知的秩数也存在差异。我们提出并分析了一种名为MAlocate的新算法，该算法能够自适应地适应不同矩阵的未知秩数。随后我们给出了下界证明，表明该策略在极小化极大意义下是最优的，并通过合成实验验证了其性能。