Active multiple matrix completion with adaptive confidence sets

2026-05-04

AI Summary

This work addresses the problem of jointly completing multiple matrices with unknown dimensions and ranks in multi-task active learning settings—such as modeling heterogeneous regional customer preferences in market segmentation—and proposes a novel multi-matrix active completion framework. The framework adaptively selects which matrix to sample from at each round and learns from uniformly randomly observed entries. Its core contribution is the MAlocate algorithm, which seamlessly integrates active learning, matrix completion, and multi-task optimization while automatically adapting to the unknown rank of each individual matrix. Theoretical analysis establishes a minimax lower bound, demonstrating the algorithm’s optimality. Extensive synthetic experiments confirm its superior effectiveness and performance over existing approaches.

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.

Problem

• matrix completion
• active learning
• multi-task learning
• unknown rank
• market segmentation

Innovation

• active learning
• matrix completion
• multi-task learning
• adaptive confidence sets
• minimax optimality