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.

翻译：几何中位数作为安全机器学习工具箱中的关键组件，在鲁棒聚合来自潜在恶意（或策略性）用户的模型（或梯度）时显示出有效性，但其对不诚实行为的激励程度仍鲜为人知。本文通过量化几何中位数的策略可信性来回答这一基本问题。虽然我们观察到几何中位数甚至不满足近似策略可信性，但证明了其具有渐近α-策略可信性：当用户数量足够大时，行为异常的用户最多可获得乘数因子α的收益，该因子可通过用户服从的分布计算得出。我们进一步将结果推广至用户更关注特定维度的场景，并确定其如何影响α值。同时展示了倾斜几何中位数如何用于提升策略可信性。