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The geometric median, an instrumental component of the secure machinelearning toolbox, is known to be effective when robustly aggregating models (orgradients), gathered from potentially malicious (or strategic) users. What isless known is the extent to which the geometric median incentivizes dishonestbehaviors. This paper addresses this fundamental question by quantifying itsstrategyproofness. While we observe that the geometric median is not evenapproximately strategyproof, we prove that it is asymptotically$\alpha$-strategyproof: when the number of users is large enough, a user thatmisbehaves can gain at most a multiplicative factor $\alpha$, which we computeas a function of the distribution followed by the users. We then generalize ourresults to the case where users actually care more about specific dimensions,determining how this impacts $\alpha$. We also show how the skewed geometricmedians can be used to improve strategyproofness.

On the Strategyproofness of the Geometric Median