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

Mario Bravo, Juan P. Flores-Mella, Cristóbal Guzmán

TL;DR

This paper addresses the joint challenges of sampling via the Projected Langevin Algorithm and private optimization by analyzing mixing times and privacy guarantees beyond nonexpansive iterations. It extends Privacy Amplification by Iteration to general gradient mappings through a modulus of continuity, yielding explicit $R_{oldsymbol{α}}$ bounds for convex Lipschitz, $(p,M)$-weakly smooth, and strongly dissipative potentials. A central technical contribution is the shifts-optimization problem, which admits a unique closed-form solution when the modulus is φ(δ)=√{c δ^2 + h}, enabling dimension-free or polylogarithmic mixing-time results and a capped privacy curve for noisy SGD with an extra term $V$ reflecting gradient regularity. The framework unifies privacy and sampling analyses under a common PABI approach and extends applicability to nondifferentiable and weakly smooth regimes, with implications for differential privacy in constrained convex settings and efficient sampling. Overall, the results provide tight, regime-specific bounds that improve understanding of last-iterate privacy and mixing behavior for projected/noisy iterative methods.

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 Rényi 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.

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

TL;DR

This paper addresses the joint challenges of sampling via the Projected Langevin Algorithm and private optimization by analyzing mixing times and privacy guarantees beyond nonexpansive iterations. It extends Privacy Amplification by Iteration to general gradient mappings through a modulus of continuity, yielding explicit bounds for convex Lipschitz, -weakly smooth, and strongly dissipative potentials. A central technical contribution is the shifts-optimization problem, which admits a unique closed-form solution when the modulus is φ(δ)=√{c δ^2 + h}, enabling dimension-free or polylogarithmic mixing-time results and a capped privacy curve for noisy SGD with an extra term reflecting gradient regularity. The framework unifies privacy and sampling analyses under a common PABI approach and extends applicability to nondifferentiable and weakly smooth regimes, with implications for differential privacy in constrained convex settings and efficient sampling. Overall, the results provide tight, regime-specific bounds that improve understanding of last-iterate privacy and mixing behavior for projected/noisy iterative methods.

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 Rényi 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.
Paper Structure (25 sections, 27 theorems, 118 equations, 2 figures, 1 table)

This paper contains 25 sections, 27 theorems, 118 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our Results
  3. Related Work
  4. Organization of the Paper
  5. Preliminaries
  6. Vector Spaces and Convex Functions
  7. Information Theory and Probability Divergences
  8. Privacy Amplification by Iteration Under a Modulus of Continuity
  9. The Shifts Optimization Problem
  10. Solving the Shifts Optimization Problem
  11. Consequences of Theorem \ref{['cor:minimized_E']}
  12. Convex and Lipschitz Potentials, Convex and (p,M)-Weakly Smooth Potentials
  13. Strongly Dissipative Potential
  14. Mixing Time Bounds for the Projected Langevin Algorithm
  15. Convex and Weakly Smooth Case
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $\mathop{\mathrm{\mathcal{X}}}\nolimits\subseteq\mathbb{R}^{d}$ be a convex, compact set with diameter $D>0$ and suppose that $f:\mathop{\mathrm{\mathcal{X}}}\nolimits\to\mathbb{R}^{}$ is a convex and $(p,M)$-weakly smooth function, with $0\leq p\leq 1$. There exists a constant $\Theta$ such tha

Figures (2)

  • Figure 1: Different values for the bound $2 \overline{T}+V(D,M,\overline{T},\eta,p)$ in logarithmic scale, for $\eta \in [n^{-1}, n^{-1/5}]$, $n=1000$, $L=1$, $M=2$, $D=1$.
  • Figure 2: $E(u_1,3)$ with $D=1$, $\eta\equiv 1$, $L=1$, $\sigma_0=1$, $\sigma_1=0.1$, $\sigma_2=1$.

Theorems & Definitions (54)

  • Theorem 1.1: Abridged version of Theorem \ref{['thm:mixing_for_weakly_smooth']}
  • Theorem 1.2: Abridged version of Theorem \ref{['thm:mixing2']}
  • Theorem 1.3: Abridged version of Theorem \ref{['thm:privacy_analysis_nsgd']}
  • Definition 2.1: Modulus of continuity
  • Definition 2.2: Rényi Divergence
  • Definition 2.3: Coupling and $\infty$-Wasserstein Distance
  • Definition 2.4: Shifted Rényi Divergence
  • Lemma 2.5: Shift-reduction
  • Lemma 3.1: Coupling under modulus of continuity
  • proof
  • ...and 44 more