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

Published: 21 Sept 2023, Last Modified: 02 Nov 2023NeurIPS 2023 posterEveryoneRevisionsBibTeX
Keywords: optimization, machine learning, differential privacy
TL;DR: We study the convergence of gradient descent with linearly correlated noise, and use our insights to design improved methods for differentially private optimization.
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.
Supplementary Material: pdf
Submission Number: 1952