Implicit Bias in Noisy-SGD: With Applications to Differentially Private
  Training
Tom Sander Maxime Sylvestre Alain Durmus
Training Deep Neural Networks (DNNs) with small batches using Stochastic
Gradient Descent (SGD) yields superior test performance compared to larger
batches. The specific noise structure inherent to SGD is known to be
responsible for this implicit bias. DP-SGD, used to ensure differential privacy
(DP) in DNNs' training, adds Gaussian noise to the clipped gradients.
Surprisingly, large-batch training still results in a significant decrease in
performance, which poses an important challenge because strong DP guarantees
necessitate the use of massive batches. We first show that the phenomenon
extends to Noisy-SGD (DP-SGD without clipping), suggesting that the
stochasticity (and not the clipping) is the cause of this implicit bias, even
with additional isotropic Gaussian noise. We theoretically analyse the
solutions obtained with continuous versions of Noisy-SGD for the Linear Least
Square and Diagonal Linear Network settings, and reveal that the implicit bias
is indeed amplified by the additional noise. Thus, the performance issues of
large-batch DP-SGD training are rooted in the same underlying principles as
SGD, offering hope for potential improvements in large batch training
strategies.