Proportional Allocation: Simple, Distributed, and Diverse Matching with High Entropy

Shipra Agrawal, Morteza Zadimoghaddam, Vahab Mirrokni
Proceedings of the 35th International Conference on Machine Learning, PMLR 80:99-108, 2018.

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 high entropy. High entropy, in turn, implies additional desirable properties of this matching, e.g., it satisfies certain diversity and fairness (aka anonymity) properties that are desirable in a variety of applications in online advertising and machine learning.

Cite this Paper


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 = {http://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 high entropy. High entropy, in turn, implies additional desirable properties of this matching, e.g., it satisfies certain diversity and fairness (aka anonymity) properties that are desirable in a variety of applications in online advertising and machine learning.} }
Endnote
%0 Conference Paper %T Proportional Allocation: Simple, Distributed, and Diverse Matching with High Entropy %A Shipra Agrawal %A Morteza Zadimoghaddam %A Vahab Mirrokni %B Proceedings of the 35th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2018 %E Jennifer Dy %E Andreas Krause %F pmlr-v80-agrawal18b %I PMLR %P 99--108 %U http://proceedings.mlr.press/v80/agrawal18b.html %V 80 %X 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 high entropy. High entropy, in turn, implies additional desirable properties of this matching, e.g., it satisfies certain diversity and fairness (aka anonymity) properties that are desirable in a variety of applications in online advertising and machine learning.
APA
Agrawal, S., Zadimoghaddam, M. & Mirrokni, V.. (2018). Proportional Allocation: Simple, Distributed, and Diverse Matching with High Entropy. Proceedings of the 35th International Conference on Machine Learning, in Proceedings of Machine Learning Research 80:99-108 Available from http://proceedings.mlr.press/v80/agrawal18b.html.

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