Active multiple matrix completion with adaptive confidence sets

arXiv stat.ML / 5/5/2026

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Key Points

The paper proposes a new multi-task active learning framework focused on solving multiple matrix completion problems at once.
In each round, the learner selects which matrix to sample from, with the sampled entry drawn uniformly at random from that matrix.
The method targets a market segmentation use case where each matrix corresponds to a region and may reflect different (unknown) customer preference patterns.
The key technical challenge is that each matrix may have different dimensions and unknown ranks, and the authors introduce the MAlocate algorithm to adapt to these unknown ranks.
The paper includes theoretical results, including a minimax lower bound proving the strategy’s optimality, and validates the approach via synthetic experiments.

Abstract

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.