Match Made with Matrix Completion: Efficient Learning under Matching Interference

TL;DR

This paper introduces a matrix completion approach for learning matching rewards in high-dimensional markets with dependent observations, providing theoretical guarantees and practical algorithms for offline and online settings.

Contribution

It develops a novel matrix completion method that handles dependent matching interference, with theoretical guarantees and extensions to online learning.

Findings

Standard nuclear norm regularization remains effective under matching interference.

The proposed double-enhanced estimator achieves near-optimal entry-wise guarantees.

The methods improve regret bounds in online matching scenarios.

Abstract

Matching markets face increasing needs to learn the matching qualities between demand and supply for effective design of matching policies. In practice, the matching rewards are high-dimensional due to the growing diversity of participants. We leverage a natural low-rank matrix structure of the matching rewards in these two-sided markets, and propose to utilize matrix completion to accelerate reward learning with limited offline data. A unique property for matrix completion in this setting is that the entries of the reward matrix are observed with matching interference -- i.e., the entries are not observed independently but dependently due to matching or budget constraints. Such matching dependence renders unique technical challenges, such as sub-optimality or inapplicability of the existing analytical tools in the matrix completion literature, since they typically rely on sample independence. In this paper, we first show that standard nuclear norm regularization remains theoretically effective under matching interference. We provide a near-optimal Frobenius norm guarantee in this setting, coupled with a new analytical technique. Next, to guide certain matching decisions, we develop a novel ``double-enhanced'' estimator, based off the nuclear norm estimator, with a near-optimal entry-wise guarantee. Our double-enhancement procedure can apply to broader sampling schemes even with dependence, which may be of independent interest. Additionally, we extend our approach to online learning settings with matching constraints such as optimal matching and stable matching, and present improved regret bounds in matrix dimensions. Finally, we demonstrate the practical value of our methods using both synthetic data and real data of labor markets.