Differentially Private Learning Needs Hidden State (Or Much Faster Convergence)

Differential privacy analysis of randomized learning algorithms typically relies on composition theorems, where the implicit assumption is that the internal state of the iterative algorithm is revealed to the adversary. However, by assuming hidden states for DP algorithms (when only the last-iterate is observable), recent works prove a converging privacy bound for noisy gradient descent (on strongly convex smooth loss function) that is significantly smaller than composition bounds after $O(1/\text{step-size})$ epochs. In this paper, we extend this hidden-state analysis to the noisy mini-batch stochastic gradient descent algorithms on strongly-convex smooth loss functions. We prove converging R\'enyi DP bounds under various mini-batch sampling schemes, such as "shuffle and partition" (which are used in practical implementations of DP-SGD) and "sampling without replacement". We prove that, in these settings, our privacy bound is much smaller than the composition bound for training with a large number of iterations (which is the case for learning from high-dimensional data). Our converging privacy analysis, thus, shows that differentially private learning, with a tight bound, needs hidden state privacy analysis or a fast convergence. To complement our theoretical results, we run experiment on training classification models on MNIST, FMNIST and CIFAR-10 datasets, and observe a better accuracy given fixed privacy budgets, under the hidden-state analysis.