Improved Differential Privacy for SGD via Optimal Private Linear Operators on Adaptive Streams

Motivated by recent applications requiring differential privacy over adaptive streams, we investigate the question of optimal instantiations of the matrix mechanism in this setting. We prove fundamental theoretical results on the applicability of matrix factorizations to adaptive streams, and provide a parameter-free fixed-point algorithm for computing optimal factorizations. We instantiate this framework with respect to concrete matrices which arise naturally in machine learning, and train user-level differentially private models with the resulting optimal mechanisms, yielding significant improvements in a notable problem in federated learning with user-level differential privacy.