Optimizer Dynamics at the Edge of Stability with Differential Privacy

TL;DR

This paper investigates how differential privacy affects neural network training dynamics, revealing that privacy-preserving modifications alter stability patterns but some characteristic behaviors persist.

Contribution

It provides a detailed analysis of how differential privacy impacts optimizer stability and sharpness dynamics during training.

Findings

DP reduces sharpness and alters stability thresholds

Patterns of Edge of Stability persist under DP conditions

Large privacy budgets can approach classical stability limits

Abstract

Deep learning models can reveal sensitive information about individual training examples, and while differential privacy (DP) provides guarantees restricting such leakage, it also alters optimization dynamics in poorly understood ways. We study the training dynamics of neural networks under DP by comparing Gradient Descent (GD), and Adam to their privacy-preserving variants. Prior work shows that these optimizers exhibit distinct stability dynamics: full-batch methods train at the Edge of Stability (EoS), while mini-batch and adaptive methods exhibit analogous edge-of-stability behavior. At these regimes, the training loss and the sharpness--the maximum eigenvalue of the training loss Hessian--exhibit certain characteristic behavior. In DP training, per-example gradient clipping and Gaussian noise modify the update rule, and it is unclear whether these stability patterns persist. We analyze how clipping and noise change sharpness and loss evolution and show that while DP generally reduces the sharpness and can prevent optimizers from fully reaching the classical stability thresholds, patterns from EoS and analogous adaptive methods stability regimes persist, with the largest learning rates and largest privacy budgets approaching, and sometimes exceeding, these thresholds. These findings highlight the unpredictability introduced by DP in neural network optimization.