We consider a distributed stochastic optimization problem in networks with finite number of nodes. Each node adjusts its action to optimize the global utility of the network, which is defined as the sum of local utilities of all nodes. While Gradient descent method is a common technique to solve such optimization problem, the computation of the gradient may require much information exchange. In this paper, we consider that each node can only have a noisy numerical observation of its local utility, of which the closed-form expression is not available. This assumption is quite realistic, especially when the system is either too complex or constantly changing. Nodes may exchange partially the observation of their local utilities to estimate the global utility at each timeslot. We propose a distributed algorithm based on stochastic perturbation, under the assumption that each node has only part of the local utilities of the other nodes. We use stochastic approximation tools to prove that our algorithm converges almost surely to the optimum, given that the objective function is smooth and strictly concave. The convergence rate is also derived, under the additional assumption of strongly concave objective function. It is shown that the convergence rate scales as $O\left ({K^{-0.5}}\right )$ after a sufficient number of iterations $K>K_{0}$ , which is the optimal rate order in terms of $K$ for our problem. Although the proposed algorithm can be applied to general optimization problems, we perform simulations for a typical power control problem in wireless networks and present numerical results to corroborate our claims.

中文翻译：

---

低信息交换网络中的分布式随机优化

我们考虑节点数量有限的网络中的分布式随机优化问题。每个节点调整其操作以优化网络的全局效用，该全局效用定义为所有节点的本地效用之和。虽然梯度下降法是解决此类优化问题的常用技术，但梯度的计算可能需要大量信息交换。在本文中，我们认为每个节点只能对其本地效用进行嘈杂的数值观测，而封闭形式的表达式则不可用。这种假设是很现实的，尤其是在系统过于复杂或不断变化的情况下。节点可以部分交换对其本地效用的观察，以估计每个时隙的全局效用。我们提出了一种基于随机扰动的分布式算法，假设每个节点仅具有其他节点的局部实用程序的一部分。考虑到目标函数是光滑且严格凹的，我们使用随机逼近工具来证明我们的算法几乎可以肯定地收敛到最优值。在强凹目标函数的附加假设下，还可以得出收敛速度。结果表明，收敛速度与 $ O \ left（{K ^ {-0.5}} \ right）$ 经过足够的迭代次数 $ K> K_ {0} $ ，这是根据 $ K $ 对于我们的问题。尽管所提出的算法可以应用于一般的优化问题，但是我们对无线网络中的典型功率控制问题进行了仿真，并给出了数值结果以证实我们的主张。