# On the Strategyproofness of the Geometric Median

arXiv: https://arxiv.org/abs/2106.02394 · HTML: https://mlsanity.com/paper/2106.02394/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Jun 2021.

## Built on by (most cited first, 9 of 9)

- [Byzantine Machine Learning Made Easy by Resilient Averaging of Momentums](https://mlsanity.com/paper/2205.12173/) (2022)
- [Moral Machine or Tyranny of the Majority?](https://mlsanity.com/paper/2305.17319/) (2023)
- [An Equivalence Between Data Poisoning and Byzantine Gradient Attacks](https://mlsanity.com/paper/2202.08578/) (2022)
- [Byzantine-Robust Federated Learning: Impact of Client Subsampling and Local Updates](https://mlsanity.com/paper/2402.12780/) (2024)
- [Can AI Model the Complexities of Human Moral Decision-Making? A Qualitative Study of Kidney Allocation Decisions](https://mlsanity.com/paper/2503.00940/) (2025)
- [Private Geometric Median](https://mlsanity.com/paper/2406.07407/) (2024)
- [The poison of dimensionality](https://mlsanity.com/paper/2409.17328/) (2024)
- [Data Quality in Edge Machine Learning: A State-of-the-Art Survey](https://mlsanity.com/paper/2406.02600/) (2024)
- [LiD-FL: Towards List-Decodable Federated Learning](https://mlsanity.com/paper/2408.04963/) (2024)

## Abstract (arXiv)

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 α-strategyproof: when the number of users is large enough, a user that misbehaves can gain at most a multiplicative factor α, 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 α. We also show how the skewed geometric medians can be used to improve strategyproofness.