In this paper, we consider an online distributed composite optimization problem over a time-varying multi-agent network that consists of multiple interacting nodes, where the objective function of each node consists of two parts: a loss function that changes over time and a regularization function. This problem naturally arises in many real-world applications ranging from wireless sensor networks to signal processing. We propose a class of online distributed optimization algorithms that are based on approximate mirror descent, which utilize the Bregman divergence as distance-measuring function that includes the Euclidean distances as a special case. We consider two standard information feedback models when designing the algorithms, that is, full-information feedback and bandit feedback. For the full-information feedback model, the first algorithm attains an average regularized regret of order $\mathcal{O}(1/\sqrt{T})$ with the total number of rounds $T$. The second algorithm, which only requires the information of the values of the loss function at two predicted points instead of the gradient information, achieves the same average regularized regret as that of the first algorithm. Simulation results of a distributed online regularized linear regression problem are provided to illustrate the performance of the proposed algorithms.

中文翻译：

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用于在线复合优化的分布式镜像下降

在本文中，我们考虑在由多个交互节点组成的时变多智能体网络上的在线分布式复合优化问题，其中每个节点的目标函数由两部分组成：随时间变化的损失函数和正则化功能。这个问题在从无线传感器网络到信号处理的许多实际应用中自然会出现。我们提出了一类基于近似镜像下降的在线分布式优化算法，该算法利用 Bregman 散度作为距离测量函数，其中包括欧几里得距离作为一个特例。我们在设计算法时考虑了两种标准的信息反馈模型，即全信息反馈和老虎机反馈。对于全信息反馈模型，第一个算法在总轮数为 $T$ 的情况下获得 $\mathcal{O}(1/\sqrt{T})$ 订单的平均正则化遗憾。第二种算法只需要两个预测点的损失函数值的信息而不是梯度信息，实现了与第一种算法相同的平均正则化遗憾。提供了分布式在线正则化线性回归问题的仿真结果来说明所提出算法的性能。