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

TL;DR

This paper provides new mixing time bounds for the projected Langevin algorithm and privacy bounds for noisy SGD, extending analysis to nonexpansive gradient maps with various regularity conditions, and introduces a novel extension of the PABI framework.

Contribution

It extends the PABI framework to noisy iterations with nonexpansive gradient maps, deriving tight bounds for mixing times and privacy curves under broader conditions.

Findings

Dimension-free, poly-logarithmic mixing time bounds for projected Langevin algorithm.

New upper bounds for privacy curves of subsampled noisy SGD.

Explicit solutions for the optimization problem in various regularity cases.

Abstract

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.