# Mixing Times and Privacy Analysis for the Projected Langevin Algorithm under a Modulus of Continuity

arXiv: https://arxiv.org/abs/2501.04134 · HTML: https://mlsanity.com/paper/2501.04134/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 33 references that are in the map.
- First posted on arXiv 7 Jan 2025.

## Builds on (most cited first, 21 of 21)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014)
- [R'enyi Differential Privacy of the Sampled Gaussian Mechanism](https://mlsanity.com/paper/1908.10530/) (2019)
- [Stability of Stochastic Gradient Descent on Nonsmooth Convex Losses](https://mlsanity.com/paper/2006.06914/) (2020)
- [High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1605.01559/) (2016)
- [Fine-Grained Analysis of Stability and Generalization for Stochastic Gradient Descent](https://mlsanity.com/paper/2006.08157/) (2020)
- [Privacy Amplification by Iteration](https://mlsanity.com/paper/1808.06651/) (2018)
- [Efficient Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets Moreau](https://mlsanity.com/paper/1612.07471/) (2016)
- [Differential Privacy Dynamics of Langevin Diffusion and Noisy Gradient Descent](https://mlsanity.com/paper/2102.05855/) (2021)
- [Privacy of Noisy Stochastic Gradient Descent: More Iterations without More Privacy Loss](https://mlsanity.com/paper/2205.13710/) (2022)
- [Sampling from a log-concave distribution with Projected Langevin Monte Carlo](https://mlsanity.com/paper/1507.02564/) (2015)
- [Differentially Private Learning Needs Hidden State (Or Much Faster Convergence)](https://mlsanity.com/paper/2203.05363/) (2022)
- [Proximal Markov chain Monte Carlo algorithms](https://mlsanity.com/paper/1306.0187/) (2013)
- [Proximal Langevin Algorithm: Rapid Convergence Under Isoperimetry](https://mlsanity.com/paper/1911.01469/) (2019)
- [Langevin Monte Carlo without smoothness](https://mlsanity.com/paper/1905.13285/) (2019)
- [The Langevin Monte Carlo algorithm in the non-smooth log-concave case](https://mlsanity.com/paper/2101.10695/) (2021)
- [Resolving the Mixing Time of the Langevin Algorithm to its Stationary Distribution for Log-Concave Sampling](https://mlsanity.com/paper/2210.08448/) (2022)
- [Fast Convergence of Phi-Divergence Along the Unadjusted Langevin Algorithm and Proximal Sampler](https://mlsanity.com/paper/2410.10699/) (2024)
- [Privacy Loss of Noisy Stochastic Gradient Descent Might Converge Even for Non-Convex Losses](https://mlsanity.com/paper/2305.09903/) (2023)
- [Convergent Privacy Loss of Noisy-SGD without Convexity and Smoothness](https://mlsanity.com/paper/2410.01068/) (2024)
- [Privacy of the last iterate in cyclically-sampled DP-SGD on nonconvex composite losses](https://mlsanity.com/paper/2407.05237/) (2024)

## Abstract (arXiv)

We study the mixing time of the projected Langevin algorithm (LA) and the privacy curve of noisy Stochastic Gradient Descent (SGD), beyond nonexpansive iterations. Specifically, we derive new mixing time bounds for the projected LA which are, in some important cases, dimension-free and poly-logarithmic on the accuracy, closely matching the existing results in the smooth convex case. Additionally, we establish new upper bounds for the privacy curve of the subsampled noisy SGD algorithm. These bounds show a crucial dependency on the regularity of gradients, and are useful for a wide range of convex losses beyond the smooth case. Our analysis relies on a suitable extension of the Privacy Amplification by Iteration (PABI) framework (Feldman et al., 2018; Altschuler and Talwar, 2022, 2023) to noisy iterations whose gradient map is not necessarily nonexpansive. This extension is achieved by designing an optimization problem which accounts for the best possible R'enyi divergence bound obtained by an application of PABI, where the tractability of the problem is crucially related to the modulus of continuity of the associated gradient mapping. We show that, in several interesting cases -- namely the nonsmooth convex, weakly smooth and (strongly) dissipative -- such optimization problem can be solved exactly and explicitly, yielding the tightest possible PABI-based bounds.