Match Made with Matrix Completion: Efficient Learning under Matching Interference

TL;DR

This work studies efficient learning of high-dimensional two-sided matching rewards by exploiting a natural low-rank structure in the reward matrix Θ*. It introduces offline matrix completion using nuclear norm regularization under matching interference, proving near-optimal Frobenius-norm guarantees and establishing a minimax lower bound. To obtain practical downstream guarantees, the paper adds a double-enhancement step that delivers an entry-wise ∞-norm bound, applicable to broader sampling schemes. It then extends these offline insights to online settings for both optimal and stable matching, offering regret bounds that scale favorably with matrix dimensions in data-poor regimes. Empirical results on synthetic and real labor-market data validate substantial improvements in both offline learning accuracy and online matching performance, underscoring the value of matrix-completion-based approaches in complex matching markets.

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.

Figures (6)

• Figure 1: Toy example of one-to-one matching for $3$ worker types and $4$ job types. (a) presents one matching of this labor market. (b) shows the matrix representation of the matching problem, where each entry denotes the reward of matching the corresponding worker and job pair; the matched pairs in (a) are circled in red.
• Figure 2: Toy example of different sampling schemes for $N=3$ worker types and $K=4$ job types. One red triangle indicates that the corresponding entry is sampled once. (a) shows an independent sampling scheme, where all entries are sampled at random with replacement. (b) shows an independent row sampling scheme, where the column is selected at random for each row independently. (c) shows a dependent sampling scheme with matching interference in our matching setting.
• Figure 3: Relative estimation errors of the matching reward matrix $\mathbf{\Theta}^*$ in Frobenius norm (left) and infinity norm (right) averaged over 50 trials. Error bars represent 95% confidence intervals. We consider $N=100$ worker types, $K=100$ job types, and the matrix rank $r=3$. Sample size on the x-axis refers to the number of matchings $n$. "Low-Rank" represents the nuclear norm estimator, and "Naive" represents the entry-wise sample average estimator.
• Figure 4: Regret for optimal matching (left) and maximum per-worker regret for stable matching (right) averaged over 50 trials. Error bars represent 95% confidence intervals. We consider $N=100$ worker types, $K=100$ job types, and the matrix rank $r=3$. 'CombLRB' and 'CompLRB' represent our combinatorial low-rank bandit algorithm and competing low-rank bandit algorithm respectively.
• Figure 5: Relative estimation errors of the matching reward matrix $\mathbf{\Theta}^*$ in Frobenius norm (left) and infinity norm (right) averaged over 50 trials. Error bars represent 95% confidence intervals. We consider $N=50$ worker types, and $K=50$ job types. Sample size on the x-axis refers to the number of matchings $n$. 'Low-Rank' represents our nuclear norm regularization approach.

Theorems & Definitions (40)

• Remark 1
• Definition 1
• Remark 2
• Theorem 1
• Proposition 1
• Theorem 2
• Remark 3
• Theorem 3
• Remark 4
• Theorem 4