Understanding Gradient Clipping in Private SGD: A Geometric Perspective

TL;DR

This paper analyzes how gradient clipping affects convergence in private SGD from a geometric perspective, revealing its bias and proposing a correction method to improve privacy-preserving training.

Contribution

It provides a theoretical quantification of clipping bias in private SGD and introduces a novel technique to correct this bias for asymmetric gradient distributions.

Findings

Gradient clipping can hinder SGD convergence due to bias.

Gradient distributions along private SGD trajectories tend to be symmetric.

A new perturbation-based method can correct clipping bias effectively.

Abstract

Deep learning models are increasingly popular in many machine learning applications where the training data may contain sensitive information. To provide formal and rigorous privacy guarantee, many learning systems now incorporate differential privacy by training their models with (differentially) private SGD. A key step in each private SGD update is gradient clipping that shrinks the gradient of an individual example whenever its L2 norm exceeds some threshold. We first demonstrate how gradient clipping can prevent SGD from converging to stationary point. We then provide a theoretical analysis that fully quantifies the clipping bias on convergence with a disparity measure between the gradient distribution and a geometrically symmetric distribution. Our empirical evaluation further suggests that the gradient distributions along the trajectory of private SGD indeed exhibit symmetric structure that favors convergence. Together, our results provide an explanation why private SGD with gradient clipping remains effective in practice despite its potential clipping bias. Finally, we develop a new perturbation-based technique that can provably correct the clipping bias even for instances with highly asymmetric gradient distributions.