We consider an $n$ agents distributed optimization problem with imperfect information characterized in a parametric sense, where the unknown parameter can be solved by a distinct distributed parameter learning problem. Though each agent only has access to its local parameter learning and computational problem, they mean to collaboratively minimize the average of their local cost functions. To address the special optimization problem, we propose a coupled distributed stochastic approximation algorithm, in which every agent updates the current beliefs of its unknown parameter and decision variable by stochastic approximation method; and then averages the beliefs and decision variables of its neighbors over network in consensus protocol. Our interest lies in the convergence analysis of this algorithm. We quantitatively characterize the factors that affect the algorithm performance, and prove that the mean-squared error of the decision variable is bounded by $\mathcal{O}(\frac{1}{nk})+\mathcal{O}\left(\frac{1}{\sqrt{n}(1-\rho_w)}\right)\frac{1}{k^{1.5}}+\mathcal{O}\big(\frac{1}{(1-\rho_w)^2} \big)\frac{1}{k^2}$, where $k$ is the iteration count and $(1-\rho_w)$ is the spectral gap of the network weighted adjacency matrix. It reveals that the network connectivity characterized by $(1-\rho_w)$ only influences the high order of convergence rate, while the domain rate still acts the same as the centralized algorithm. In addition, we analyze that the transient iteration needed for reaching its dominant rate $\mathcal{O}(\frac{1}{nk})$ is $\mathcal{O}(\frac{n}{(1-\rho_w)^2})$. Numerical experiments are carried out to demonstrate the theoretical results by taking different CPUs as agents, which is more applicable to real-world distributed scenarios.

翻译：我们考虑一个含$n$个智能体的分布式优化问题，该问题存在参数意义上的不完全信息，其中未知参数可通过一个独立的分布式参数学习问题求解。尽管每个智能体仅能访问其本地的参数学习和计算问题，但它们旨在协作最小化本地代价函数的平均值。针对这一特殊优化问题，我们提出了一种耦合分布式随机近似算法：每个智能体通过随机近似方法更新其未知参数和决策变量的当前信念，然后通过共识协议在网络中对其邻域内智能体的信念和决策变量进行平均。我们关注该算法的收敛性分析，定量刻画了影响算法性能的因素，并证明决策变量的均方误差以$\mathcal{O}(\frac{1}{nk})+\mathcal{O}\left(\frac{1}{\sqrt{n}(1-\rho_w)}\right)\frac{1}{k^{1.5}}+\mathcal{O}\big(\frac{1}{(1-\rho_w)^2} \big)\frac{1}{k^2}$为界，其中$k$为迭代次数，$(1-\rho_w)$为网络加权邻接矩阵的谱隙。这表明由$(1-\rho_w)$表征的网络连通性仅影响收敛速率的高阶项，而主导速率仍与集中式算法相同。此外，我们分析达到主导速率$\mathcal{O}(\frac{1}{nk})$所需的暂态迭代次数为$\mathcal{O}(\frac{n}{(1-\rho_w)^2})$。通过采用不同CPU作为智能体进行数值实验，验证了理论结果，这更适用于真实分布式场景。