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Prior work on differential privacy analysis of randomized SGD algorithmsrelies on composition theorems, where the implicit (unrealistic) assumption isthat the internal state of the iterative algorithm is revealed to theadversary. As a result, the R\'enyi DP bounds derived by such composition-basedanalyses linearly grow with the number of training epochs. When the internalstate of the algorithm is hidden, we prove a converging privacy bound for noisystochastic gradient descent (on strongly convex smooth loss functions). We showhow to take advantage of privacy amplification by sub-sampling and randomizedpost-processing, and prove the dynamics of privacy bound for "shuffle andpartition" and "sample without replacement" stochastic mini-batch gradientdescent schemes. We prove that, in these settings, our privacy bound convergesexponentially fast and is substantially smaller than the composition bounds,notably after a few number of training epochs. Thus, unless the DP algorithmconverges fast, our privacy analysis shows that hidden state analysis cansignificantly amplify differential privacy.

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