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.

Machine unlearning has raised significant interest with the adoption of laws ensuring the ``right to be forgotten''. Researchers have provided a probabilistic notion of approximate unlearning under a similar definition of Differential Privacy (DP), where privacy is defined as statistical indistinguishability to retraining from scratch. We propose Langevin unlearning, an unlearning framework based on noisy gradient descent with privacy guarantees for approximate unlearning problems. Langevin unlearning unifies the DP learning process and the privacy-certified unlearning process with many algorithmic benefits. These include approximate certified unlearning for non-convex problems, complexity saving compared to retraining, sequential and batch unlearning for multiple unlearning requests. We verify the practicality of Langevin unlearning by studying its privacy-utility-complexity trade-off via experiments on benchmark datasets, and also demonstrate its superiority against gradient-decent-plus-output-perturbation based approximate unlearning.