[论文解读] Tailoring Gradient Methods for Differentially-Private Distributed Optimization

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

Privacy-Preserving Technologies in Data被引用 4

一句话总结

本文提出两种新颖的分布式优化差分隐私梯度方法——基于静态一致性与动态一致性（梯度跟踪）——在无限迭代下同时确保 $\epsilon$-差分隐私与几乎必然收敛至最优解。该方法在有限隐私预算下实现严格隐私保护，同时保持优化精度。

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.

研究动机与目标

• 解决去中心化优化中个体代理隐私保护的关键挑战，其中共享的梯度和变量可能泄露敏感数据。
• 克服现有分布式方法中差分隐私与优化精度之间的权衡。
• 设计一种保证 $\epsilon$-差分隐私与在一般有向图拓扑下几乎必然收敛至最优解的梯度方法。
• 实现在最小通信开销与有限隐私预算下，即使迭代次数趋于无穷，仍可实现隐私保护的分布式优化。

提出的方法

• 提出一种基于静态一致性的差分隐私梯度方法，通过固定步长与梯度更新中的噪声注入，确保 $\epsilon$-差分隐私与收敛性。
• 引入一种基于动态一致性（梯度跟踪）的方法，支持一般有向交互图，并通过精心设计的噪声缩放策略维持隐私。
• 采用李雅普诺夫型分析，利用复合向量 $\mathbf{v}^{k}$ 同时追踪目标差距、一致性误差与梯度跟踪误差，以建立收敛性。
• 通过矩阵 $V^k$、$A^k$、$H^k$ 与 $B^k$ 推导递归期望界，分析随机梯度与噪声下误差项的演化。
• 对步长序列 $\gamma_1^k$、$\gamma_2^k$ 与 $\alpha^k$ 施加条件，以确保噪声与误差项的可 summability，从而实现几乎必然收敛。
• 采用一种隐私机制，在梯度估计中注入与局部梯度敏感性成比例的噪声，确保在无限迭代下隐私预算仍为有限值 ($\epsilon$)。

实验结果

研究问题

• RQ1 能否在无限迭代下实现差分隐私的分布式优化，且不牺牲对最优解的收敛性？
• RQ2 如何重新设计基于梯度的分布式优化方法，以同时确保严格的 $\epsilon$-差分隐私与最优性？
• RQ3 能否将隐私保护的分布式优化扩展至一般有向通信图，而不仅限于平衡图？
• RQ4 何种步长与噪声缩放策略可确保分布式梯度方法中隐私预算有限且几乎必然收敛？

主要发现

• 所提出的基于静态一致性的方法在无向图或平衡图下，可确保 $\epsilon$-差分隐私与几乎必然收敛至最优解。
• 基于动态一致性的方法将隐私与收敛性保证扩展至一般有向图，克服了静态一致性方法的局限性。
• 两种算法在迭代次数趋于无穷时，仍保持有限的隐私预算 ($\epsilon$)，这是对先前工作的关键改进。
• 收敛性分析证明 $\lambda^k \|\bar{y}^{k+1} - \bar{g}^{k+1}\|^2$ 几乎必然可求和，确保梯度跟踪误差随时间衰减。
• 在分布式估计问题与基准数据集上的数值仿真验证了该方法在隐私与优化精度之间平衡的有效性。
• 噪声注入机制通过控制梯度估计的敏感性，确保即使在迭代次数无界时，隐私预算仍为有限且有界。