Noisy SIGNSGD Is More Differentially Private Than You (Might) Think

The prevalent distributed machine learning paradigm faces two critical challenges: communication efficiency and data privacy. SIGNSGD provides a simple-to-implement approach with improved communication efficiency by requiring workers to share only the signs of the gradients. However, it fails to converge in the presence of data heterogeneity, and a simple fix is to add Gaussian noise before taking the signs, which leads to the Noisy SIGNSGD algorithm that enjoys competitive performance while significantly reducing the communication overhead. Existing results suggest that Noisy SIGNSGD with additive Gaussian noise has the same privacy guarantee as classic DP-SGD due to the post-processing property of differential privacy, and logistic noise may be a good alternative to Gaussian noise when combined with the sign-based compressor. Nonetheless, discarding the magnitudes in Noisy SIGNSGD leads to information loss, which may intuitively amplify privacy. In this paper, we make this intuition rigorous and quantify the privacy amplification of the sign-based compressor. Particularly, we analytically show that Gaussian noise leads to a smaller estimation error than logistic noise when combined with the sign-based compressor and may be more suitable for distributed learning with heterogeneous data. Then, we further establish the convergence of Noisy SIGNSGD. Finally, extensive experiments are conducted to validate the theoretical results.