Understanding Gradient Clipping in Private SGD: A Geometric Perspective (2006.15429v2)

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.

• The paper shows that gradient clipping introduces bias in DP-SGD, potentially hindering convergence by affecting stationary point attainment.
• The paper demonstrates through empirical analysis that symmetric gradient distributions help align clipped gradients with true gradients.
• The paper proposes a perturbation technique that adds noise before clipping, effectively mitigating bias while preserving privacy guarantees.

Understanding Gradient Clipping in Private Stochastic Gradient Descent: A Geometric Perspective

This paper explores the impact of gradient clipping in differentially private stochastic gradient descent (DP-SGD) from a geometric standpoint. Gradient clipping is an essential mechanism in DP-SGD that manages the $\ell_2$ norm of individual gradients by enforcing a threshold, ensuring a bounded sensitivity which is crucial for maintaining differential privacy. However, this operation introduces bias that can affect convergence to optimal solutions.

Key Findings

1. Impact on Stationary Points and Convergence: The research identifies that gradient clipping can prevent convergence to stationary points and provides a theoretical analysis quantifying clipping bias by using a disparity measure that resembles the Wasserstein distance. This measure is between the gradient distribution and an ideal symmetric distribution. Their analysis demonstrates how gradient clipping remains effective in practice by aligning clipped gradients with true gradients, provided the gradient distributions exhibit symmetric structures.
2. Empirical Evaluations: The paper conducts extensive empirical tests on DP-SGD applied to datasets, revealing that gradient distributions tend to show symmetric patterns during training. This supports the assertion that the symmetric structures favor convergent behavior despite aggressive clipping.
3. Perturbation-Based Technique: The authors propose a new perturbation approach designed to remedy clipping bias in scenarios with high asymmetric gradient distributions. This technique involves adding noise before clipping, mitigating bias without compromising privacy guarantees.

Theoretical Implications

The examination of gradient clipping through geometric configurations offers a deeper understanding of the convergence properties in machine learning models trained under differential privacy constraints. It elucidates how symmetric gradient distributions help in retaining the efficacy of clipped gradients, contributing to the convergence assurance of significant optimization tasks. This perspective can inform the design of more robust DP algorithms, emphasizing configurations that align well with symmetric gradient distributions.

Practical Implications

Gradient clipping ensures that DP-SGD maintains formal privacy guarantees while training on sensitive datasets. The insights from this paper may aid practitioners in setting appropriate clipping thresholds, understanding the influence of gradient distributions in different types of neural network architectures, and leveraging perturbation techniques to enhance the reliability of DP-SGD regardless of gradient distribution forms.

Future Directions

The results suggest looking into methods of symmetrizing gradient distributions further, which could include adapting learning models explicitly towards symmetrical configurations. Additionally, more empirical studies are warranted across various model types to assess the generality and limitations of these findings, particularly those models that naturally exhibit less symmetric gradient distributions, like certain LSTM architectures. Understanding and optimizing gradient distributions beyond symmetric frameworks can help refine DP-SGD algorithms, ultimately improving the balance between privacy and performance.