Active multiple matrix completion with adaptive confidence sets

TL;DR

This paper introduces a multi-task active learning framework for matrix completion, addressing unknown ranks and varying matrix sizes, with an adaptive algorithm proven to be minimax-optimal.

Contribution

The paper proposes MAlocate, an adaptive algorithm for multi-task matrix completion that handles unknown ranks and sizes, with theoretical optimality guarantees.

Findings

MAlocate adapts to unknown matrix ranks effectively.

The algorithm is proven to be minimax-optimal.

Synthetic experiments demonstrate the algorithm's strong performance.

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.