  While differential privacy and gradient compression are separately
well-researched topics in machine learning, the study of interaction between
these two topics is still relatively new. We perform a detailed empirical study
on how the Gaussian mechanism for differential privacy and gradient compression
jointly impact test accuracy in deep learning. The existing literature in
gradient compression mostly evaluates compression in the absence of
differential privacy guarantees, and demonstrate that sufficiently high
compression rates reduce accuracy. Similarly, existing literature in
differential privacy evaluates privacy mechanisms in the absence of
compression, and demonstrates that sufficiently strong privacy guarantees
reduce accuracy. In this work, we observe while gradient compression generally
has a negative impact on test accuracy in non-private training, it can
sometimes improve test accuracy in differentially private training.
Specifically, we observe that when employing aggressive sparsification or rank
reduction to the gradients, test accuracy is less affected by the Gaussian
noise added for differential privacy. These observations are explained through
an analysis how differential privacy and compression effects the bias and
variance in estimating the average gradient. We follow this study with a
recommendation on how to improve test accuracy under the context of
differentially private deep learning and gradient compression. We evaluate this
proposal and find that it can reduce the negative impact of noise added by
differential privacy mechanisms on test accuracy by up to 24.6%, and reduce the
negative impact of gradient sparsification on test accuracy by up to 15.1%.
