On the Strategyproofness of the Geometric Median

El-Mahdi El-Mhamdi, Sadegh Farhadkhani, Rachid Guerraoui, Lê-Nguyên Hoang
Proceedings of The 26th International Conference on Artificial Intelligence and Statistics, PMLR 206:2603-2640, 2023.

Abstract

The geometric median, an instrumental component of the secure machine learning toolbox, is known to be effective when robustly aggregating models (or gradients), gathered from potentially malicious (or strategic) users. What is less known is the extent to which the geometric median incentivizes dishonest behaviors. This paper addresses this fundamental question by quantifying its strategyproofness. While we observe that the geometric median is not even approximately strategyproof, we prove that it is asymptotically $\alpha$-strategyproof: when the number of users is large enough, a user that misbehaves can gain at most a multiplicative factor $\alpha$, which we compute as a function of the distribution followed by the users. We then generalize our results to the case where users actually care more about specific dimensions, determining how this impacts $\alpha$. We also show how the skewed geometric medians can be used to improve strategyproofness.

Cite this Paper

BibTeX
@InProceedings{pmlr-v206-el-mhamdi23a, title = {On the Strategyproofness of the Geometric Median}, author = {El-Mhamdi, El-Mahdi and Farhadkhani, Sadegh and Guerraoui, Rachid and Hoang, L\^e-Nguy\^en}, booktitle = {Proceedings of The 26th International Conference on Artificial Intelligence and Statistics}, pages = {2603--2640}, year = {2023}, editor = {Ruiz, Francisco and Dy, Jennifer and van de Meent, Jan-Willem}, volume = {206}, series = {Proceedings of Machine Learning Research}, month = {25--27 Apr}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v206/el-mhamdi23a/el-mhamdi23a.pdf}, url = {https://proceedings.mlr.press/v206/el-mhamdi23a.html}, abstract = {The geometric median, an instrumental component of the secure machine learning toolbox, is known to be effective when robustly aggregating models (or gradients), gathered from potentially malicious (or strategic) users. What is less known is the extent to which the geometric median incentivizes dishonest behaviors. This paper addresses this fundamental question by quantifying its strategyproofness. While we observe that the geometric median is not even approximately strategyproof, we prove that it is asymptotically $\alpha$-strategyproof: when the number of users is large enough, a user that misbehaves can gain at most a multiplicative factor $\alpha$, which we compute as a function of the distribution followed by the users. We then generalize our results to the case where users actually care more about specific dimensions, determining how this impacts $\alpha$. We also show how the skewed geometric medians can be used to improve strategyproofness.} }
Endnote
%0 Conference Paper %T On the Strategyproofness of the Geometric Median %A El-Mahdi El-Mhamdi %A Sadegh Farhadkhani %A Rachid Guerraoui %A Lê-Nguyên Hoang %B Proceedings of The 26th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2023 %E Francisco Ruiz %E Jennifer Dy %E Jan-Willem van de Meent %F pmlr-v206-el-mhamdi23a %I PMLR %P 2603--2640 %U https://proceedings.mlr.press/v206/el-mhamdi23a.html %V 206 %X The geometric median, an instrumental component of the secure machine learning toolbox, is known to be effective when robustly aggregating models (or gradients), gathered from potentially malicious (or strategic) users. What is less known is the extent to which the geometric median incentivizes dishonest behaviors. This paper addresses this fundamental question by quantifying its strategyproofness. While we observe that the geometric median is not even approximately strategyproof, we prove that it is asymptotically $\alpha$-strategyproof: when the number of users is large enough, a user that misbehaves can gain at most a multiplicative factor $\alpha$, which we compute as a function of the distribution followed by the users. We then generalize our results to the case where users actually care more about specific dimensions, determining how this impacts $\alpha$. We also show how the skewed geometric medians can be used to improve strategyproofness.
APA
El-Mhamdi, E., Farhadkhani, S., Guerraoui, R. & Hoang, L.. (2023). On the Strategyproofness of the Geometric Median. Proceedings of The 26th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 206:2603-2640 Available from https://proceedings.mlr.press/v206/el-mhamdi23a.html.

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