Improved Robust CM Estimation Method for Distributed Data under Lipschitz Condition

In this paper, we propose improved robust estimated methods for the "Byzantine failure" problem in homogeneous and heterogeneous datasets under the framework of centralized distributed system. Firstly, we use the Lipschitz condition to improve the classic coordinate-wise median method, and propose the Lipschitz-CM method, which is suitable for homogeneous datasets and prove its convergence rate form. The results of numerical experiments of simulated dataset and real dataset also show that the Lipschitz-CM method has better robustness and effectiveness compared with the existing geometric median method and coordinate-wise median method. For the case of heterogeneous datasets, we improve the Lipschitz-CM method by bucketing and propose the Bucketing Lipschitz-CM method, we prove that the convergence rate of the Bucketing Lipschitz-CM method has the same form as the Lipschitz-CM method, which means that bucketing does not bring additional computational complexity. Through the numerical experiments and real dataset, it is verified that the Bucketing Lipschitz-CM method has a better performance on the heterogeneous datasets.