We deal with a general distributed constrained online learning problem with privacy over time-varying networks, where a class of nondecomposable objectives are considered. Under this setting, each node only controls a part of the global decision, and the goal of all nodes is to collaboratively minimize the global cost over a time horizon $T$ while guarantees the security of the transmitted information. For such problems, we first design a novel generic algorithm framework, named as DPSDA, of differentially private distributed online learning using the Laplace mechanism and the stochastic variants of dual averaging method. Note that in the dual updates, all nodes of DPSDA employ the noise-corrupted gradients for more generality. Then, we propose two algorithms, named as DPSDA-C and DPSDA-PS, under this framework. In DPSDA-C, the nodes implement a circulation-based communication in the primal updates so as to alleviate the disagreements over time-varying undirected networks. In addition, for the extension to time-varying directed ones, the nodes implement the broadcast-based push-sum dynamics in DPSDA-PS, which can achieve average consensus over arbitrary directed networks. Theoretical results show that both algorithms attain an expected regret upper bound in $\mathcal{O}( \sqrt{T} )$ when the objective function is convex, which matches the best utility achievable by cutting-edge algorithms. Finally, numerical experiment results on both synthetic and real-world datasets verify the effectiveness of our algorithms.

翻译：我们研究了时变网络下具有隐私约束的通用分布式在线学习问题，其中考虑了一类非分解目标函数。在该设定中，每个节点仅控制全局决策的一部分，所有节点的目标是在时间范围$T$内协同最小化全局代价，同时保证传输信息的安全性。针对此类问题，我们首先设计了一种新颖的通用算法框架DPSDA，该框架采用拉普拉斯机制和随机变体对偶平均方法实现差分隐私分布式在线学习。值得注意的是，在对偶更新中，DPSDA所有节点均使用噪声扰动梯度以获得更广泛的适用性。基于此框架，我们提出了两种算法：DPSDA-C和DPSDA-PS。在DPSDA-C中，节点在原始更新中采用基于循环的通信方式，以缓解时变无向网络中的分歧。此外，针对时变有向网络的扩展，DPSDA-PS中的节点采用基于广播的推求和动态，可在任意有向网络上实现平均共识。理论结果表明：当目标函数为凸函数时，两种算法的期望遗憾上界均为$\mathcal{O}( \sqrt{T} )$，这与前沿算法可实现的最优效用相匹配。最后，基于合成数据集和真实数据集的数值实验结果验证了我们算法的有效性。