[Paper Review] Tailoring Gradient Methods for Differentially-Private Distributed Optimization

Yongqiang Wang, Angelia Nedić|arXiv (Cornell University)|Feb 2, 2022

Privacy-Preserving Technologies in Data4 citations

TL;DR

This paper proposes two novel differentially private gradient methods for distributed optimization—based on static-consensus and dynamic-consensus (gradient-tracking)—that simultaneously ensure $\epsilon$-differential privacy and almost sure convergence to the optimal solution, even with infinite iterations. The approach achieves rigorous privacy with finite privacy budgets while maintaining optimization accuracy.

ABSTRACT

Decentralized optimization is gaining increased traction due to its widespread applications in large-scale machine learning and multi-agent systems. The same mechanism that enables its success, i.e., information sharing among participating agents, however, also leads to the disclosure of individual agents' private information, which is unacceptable when sensitive data are involved. As differential privacy is becoming a de facto standard for privacy preservation, recently results have emerged integrating differential privacy with distributed optimization. However, directly incorporating differential privacy design in existing distributed optimization approaches significantly compromises optimization accuracy. In this paper, we propose to redesign and tailor gradient methods for differentially-private distributed optimization, and propose two differential-privacy oriented gradient methods that can ensure both rigorous epsilon-differential privacy and optimality. The first algorithm is based on static-consensus based gradient methods, and the second algorithm is based on dynamic-consensus (gradient-tracking) based distributed optimization methods and, hence, is applicable to general directed interaction graph topologies. Both algorithms can simultaneously ensure almost sure convergence to an optimal solution and a finite privacy budget, even when the number of iterations goes to infinity. To our knowledge, this is the first time that both goals are achieved simultaneously. Numerical simulations using a distributed estimation problem and experimental results on a benchmark dataset confirm the effectiveness of the proposed approaches.

Motivation & Objective

• Address the critical challenge of preserving individual agents' privacy in decentralized optimization, where shared gradients and variables may leak sensitive data.
• Overcome the trade-off between differential privacy and optimization accuracy in existing distributed methods.
• Design gradient methods that guarantee both $\epsilon$-differential privacy and almost sure convergence to the optimal solution under general directed graph topologies.
• Enable privacy-preserving distributed optimization with minimal communication overhead and finite privacy budget, even as iterations approach infinity.

Proposed method

• Proposes a static-consensus-based differentially private gradient method that ensures $\epsilon$-differential privacy and convergence using a fixed stepsize and noise injection in gradient updates.
• Introduces a dynamic-consensus (gradient-tracking) based method that supports general directed interaction graphs and maintains privacy through carefully designed noise scaling.
• Employs a Lyapunov-type analysis with a composite vector $\mathbf{v}^{k}$ tracking objective gap, consensus error, and gradient-tracking error to establish convergence.
• Derives recursive expectation bounds using matrices $V^k$, $A^k$, $H^k$, and $B^k$ to analyze the evolution of error terms under stochastic gradients and noise.
• Applies conditions on stepsize sequences $\gamma_1^k$, $\gamma_2^k$, and $\alpha^k$ to ensure summability of noise and error terms, enabling almost sure convergence.
• Uses a privacy mechanism that injects noise proportional to local gradient sensitivity, ensuring finite privacy budget ($\u0000\epsilon$) across infinite iterations.

Experimental results

Research questions

• RQ1 Can differentially private distributed optimization be achieved without sacrificing convergence to the optimal solution, even with infinite iterations?
• RQ2 How can gradient-based distributed optimization be redesigned to simultaneously ensure rigorous $\u0000\epsilon$-differential privacy and optimality?
• RQ3 Can privacy-preserving distributed optimization be extended to general directed communication graphs, not just balanced ones?
• RQ4 What stepsize and noise scaling policies ensure both finite privacy budget and almost sure convergence in distributed gradient methods?

Key findings

• The proposed static-consensus-based method ensures $\u0000\epsilon$-differential privacy and almost sure convergence to the optimal solution under undirected or balanced graphs.
• The dynamic-consensus-based method extends privacy and convergence guarantees to general directed graphs, overcoming the limitations of static-consensus methods.
• Both algorithms maintain a finite privacy budget ($\u0000\epsilon$) even as the number of iterations tends to infinity, a key advancement over prior work.
• The convergence analysis proves that $\lambda^k \|\bar{y}^{k+1} - \bar{g}^{k+1}\|^2$ is almost surely summable, ensuring gradient-tracking error diminishes over time.
• Numerical simulations on a distributed estimation problem and a benchmark dataset confirm the method's effectiveness in balancing privacy and optimization accuracy.
• The noise injection mechanism ensures that the privacy budget is finite and bounded, even with unbounded iterations, by controlling the sensitivity of gradient estimates.