Gradient Descent with Linearly Correlated Noise: Theory and Applications to Differential Privacy

TL;DR

This paper analyzes gradient descent algorithms affected by linearly correlated noise, motivated by differential privacy methods, providing tighter theoretical insights and practical matrix factorizations for privacy-preserving optimization.

Contribution

It introduces a simplified model for linearly correlated noise in gradient descent, offering tighter analysis and new matrix factorizations for differential privacy applications.

Findings

Tighter theoretical bounds for gradient descent with correlated noise

Development of new matrix factorizations improving differential privacy

Empirical validation showing improved privacy-utility trade-offs

Abstract

We study gradient descent under linearly correlated noise. Our work is motivated by recent practical methods for optimization with differential privacy (DP), such as DP-FTRL, which achieve strong performance in settings where privacy amplification techniques are infeasible (such as in federated learning). These methods inject privacy noise through a matrix factorization mechanism, making the noise linearly correlated over iterations. We propose a simplified setting that distills key facets of these methods and isolates the impact of linearly correlated noise. We analyze the behavior of gradient descent in this setting, for both convex and non-convex functions. Our analysis is demonstrably tighter than prior work and recovers multiple important special cases exactly (including anticorrelated perturbed gradient descent). We use our results to develop new, effective matrix factorizations for differentially private optimization, and highlight the benefits of these factorizations theoretically and empirically.