This paper considers distributed optimization (DO) where multiple agents cooperate to minimize a global objective function, expressed as a sum of local objectives, subject to some constraints. In DO, each agent iteratively solves a local optimization model constructed by its own data and communicates some information (e.g., a local solution) with its neighbors until a global solution is obtained. Even though locally stored data are not shared with other agents, it is still possible to reconstruct the data from the information communicated among agents, which could limit the practical usage of DO in applications with sensitive data. To address this issue, we propose a privacy-preserving DO algorithm for constrained convex optimization models, which provides a statistical guarantee of data privacy, known as differential privacy, and a sequence of iterates that converges to an optimal solution in expectation. The proposed algorithm generalizes a linearized alternating direction method of multipliers by introducing a multiple local updates technique to reduce communication costs and incorporating an objective perturbation method in the local optimization models to compute and communicate randomized feasible local solutions that cannot be utilized to reconstruct the local data, thus preserving data privacy. Under the existence of convex constraints, we show that, while both algorithms provide the same level of data privacy, the objective perturbation used in the proposed algorithm can provide better solutions than does the widely adopted output perturbation method that randomizes the local solutions by adding some noise. We present the details of privacy and convergence analyses and numerically demonstrate the effectiveness of the proposed algorithm by applying it in two different applications, namely, distributed control of power flow and federated learning, where data privacy is of concern.

翻译：本文研究分布式优化（DO）问题，其中多个智能体协同最小化一个全局目标函数（表示为局部目标函数之和），并满足若干约束条件。在DO中，每个智能体通过自身数据迭代求解局部优化模型，并与邻居智能体交换某些信息（如局部解），直至获得全局解。尽管本地存储的数据不与其他智能体共享，但通过智能体间通信的信息仍可能重建数据，这限制了DO在敏感数据应用中的实际使用。为解决此问题，我们针对带约束的凸优化模型提出了一种隐私保护DO算法，该算法能提供数据隐私的统计保证（即差分隐私），并生成期望上收敛到最优解的迭代序列。所提算法对线性化交替方向乘子法进行推广，引入多局部更新技术以降低通信成本，并在局部优化模型中融入目标扰动方法，用于计算并通信随机化的可行局部解（这些解无法用于重建局部数据），从而保护数据隐私。在凸约束存在的情况下，我们证明，尽管两种算法提供相同级别的数据隐私保护，但所提算法采用的目标扰动方法相比广泛采用的输出扰动方法（通过添加噪声对局部解进行随机化）能提供更优解。我们详细阐述了隐私性和收敛性分析，并通过两个关注数据隐私的应用场景（即电力潮流分布式控制与联邦学习）数值验证了所提算法的有效性。