Proportional Allocation: Simple, Distributed, and Diverse Matching with High Entropy
Shipra Agrawal, Morteza Zadimoghaddam, Vahab Mirrokni
Abstract
Inspired by many applications of bipartite matching in online advertising and machine learning, we study a simple and natural iterative proportional allocation algorithm: Maintain a priority score $\priority_a$ for each node $a\in \mathds{A}$ on one side of the bipartition, initialized as $\priority_a=1$. Iteratively allocate the nodes $i\in \impressions$ on the other side to eligible nodes in $\mathds{A}$ in proportion of their priority scores. After each round, for each node $a\in \mathds{A}$, decrease or increase the score $\priority_a$ based on whether it is over- or under- allocated. Our first result is that this simple, distributed algorithm converges to a $(1-\epsilon)$-approximate fractional $b$-matching solution in $O({\log n\over \epsilon^2} )$ rounds. We also extend the proportional allocation algorithm and convergence results to the maximum weighted matching problem, and show that the algorithm can be naturally tuned to produce maximum matching with
BibTeX
@InProceedings{pmlr-v80-agrawal18b,
title = {Proportional Allocation: Simple, Distributed, and Diverse Matching with High Entropy},
author = {Agrawal, Shipra and Zadimoghaddam, Morteza and Mirrokni, Vahab},
booktitle = {Proceedings of the 35th International Conference on Machine Learning},
pages = {99--108},
year = {2018},
editor = {Dy, Jennifer and Krause, Andreas},
volume = {80},
series = {Proceedings of Machine Learning Research},
month = {10--15 Jul},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v80/agrawal18b/agrawal18b.pdf},
url = {https://proceedings.mlr.press/v80/agrawal18b.html},
abstract = {Inspired by many applications of bipartite matching in online advertising and machine learning, we study a simple and natural iterative proportional allocation algorithm: Maintain a priority score $\priority_a$ for each node $a\in \mathds{A}$ on one side of the bipartition, initialized as $\priority_a=1$. Iteratively allocate the nodes $i\in \impressions$ on the other side to eligible nodes in $\mathds{A}$ in proportion of their priority scores. After each round, for each node $a\in \mathds{A}$, decrease or increase the score $\priority_a$ based on whether it is over- or under- allocated. Our first result is that this simple, distributed algorithm converges to a $(1-\epsilon)$-approximate fractional $b$-matching solution in $O({\log n\over \epsilon^2} )$ rounds. We also extend the proportional allocation algorithm and convergence results to the maximum weighted matching problem, and show that the algorithm can be naturally tuned to produce maximum matching with