Privacy Amplification by Iteration for ADMM with (Strongly) Convex Objective Functions

We examine a private ADMM variant for (strongly) convex objectives which is a primal-dual iterative method. Each iteration has a user with a private function used to update the primal variable, masked by Gaussian noise for local privacy, without directly adding noise to the dual variable. Privacy amplification by iteration explores if noises from later iterations can enhance the privacy guarantee when releasing final variables after the last iteration. Cyffers et al. [ICML 2023] explored privacy amplification by iteration for the proximal ADMM variant, where a user's entire private function is accessed and noise is added to the primal variable. In contrast, we examine a private ADMM variant requiring just one gradient access to a user's function, but both primal and dual variables must be passed between successive iterations. To apply Balle et al.'s [NeurIPS 2019] coupling framework to the gradient ADMM variant, we tackle technical challenges with novel ideas. First, we address the non-expansive mapping issue in ADMM iterations by using a customized norm. Second, because the dual variables are not masked with any noise directly, their privacy guarantees are achieved by treating two consecutive noisy ADMM iterations as a Markov operator. Our main result is that the privacy guarantee for the gradient ADMM variant can be amplified proportionally to the number of iterations. For strongly convex objective functions, this amplification exponentially increases with the number of iterations. These amplification results align with the previously studied special case of stochastic gradient descent.