Convergent Privacy Loss of Noisy-SGD without Convexity and Smoothness

Eli Chien, Pan Li

Keywords: privacy, differential privacy, hidden state SGD, nonconvex, nonsmooth

TL;DR: We show the convergent privacy loss of hidden state noisy SGD even without convexity and smoothness assumptions.

Abstract:

We study the Differential Privacy (DP) guarantee of hidden-state Noisy-SGD algorithms over a bounded domain. Standard privacy analysis for Noisy-SGD assumes all internal states are revealed, which leads to a divergent R'enyi DP bound with respect to the number of iterations. Ye & Shokri (2022) and Altschuler & Talwar (2022) proved convergent bounds for smooth (strongly) convex losses, and raise open questions about whether these assumptions can be relaxed. We provide positive answers by proving convergent R'enyi DP bound for non-convex non-smooth losses, where we show that requiring losses to have H"older continuous gradient is sufficient. We also provide a strictly better privacy bound compared to state-of-the-art results for smooth strongly convex losses. Our analysis relies on the improvement of shifted divergence analysis in multiple aspects, including forward Wasserstein distance tracking, identifying the optimal shifts allocation, and the H"older reduction lemma. Our results further elucidate the benefit of hidden-state analysis for DP and its applicability.

Primary Area: alignment, fairness, safety, privacy, and societal considerations

Code Of Ethics: I acknowledge that I and all co-authors of this work have read and commit to adhering to the ICLR Code of Ethics.

Submission Guidelines: I certify that this submission complies with the submission instructions as described on https://iclr.cc/Conferences/2025/AuthorGuide.

Anonymous Url: I certify that there is no URL (e.g., github page) that could be used to find authors’ identity.

No Acknowledgement Section: I certify that there is no acknowledgement section in this submission for double blind review.

Submission Number: 3180