  This paper proposes two split-and-conquer (SC) learning estimators for finite
mixture models that are tolerant to Byzantine failures. In SC learning,
individual machines obtain local estimates, which are then transmitted to a
central server for aggregation. During this communication, the server may
receive malicious or incorrect information from some local machines, a scenario
known as Byzantine failures. While SC learning approaches have been devised to
mitigate Byzantine failures in statistical models with Euclidean parameters,
developing Byzantine-tolerant methods for finite mixture models with
non-Euclidean parameters requires a distinct strategy. Our proposed
distance-based methods are hyperparameter tuning free, unlike existing methods,
and are resilient to Byzantine failures while achieving high statistical
efficiency. We validate the effectiveness of our methods both theoretically and
empirically via experiments on simulated and real data from machine learning
applications for digit recognition. The code for the experiment can be found at
https://github.com/SarahQiong/RobustSCGMM.
