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首页> 外文期刊>Journal of the Franklin Institute >Primal-dual stochastic distributed algorithm for constrained convex optimization
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Primal-dual stochastic distributed algorithm for constrained convex optimization

机译：约束凸优化的本对偶随机分布算法

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This paper investigates distributed convex optimization problems over an undirected and connected network, where each node's variable lies in a private constrained convex set, and overall nodes aim at collectively minimizing the sum of all local objective functions. Motivated by a variety of applications in machine learning problems with large-scale training sets distributed to multiple autonomous nodes, each local objective function is further designed as the average of moderate number of local instantaneous functions. Each local objective function and constrained set cannot be shared with others. A primal-dual stochastic algorithm is presented to address the distributed convex optimization problems, where each node updates its state by resorting to unbiased stochastic averaging gradients and projects on its private constrained set. At each iteration, for each node the gradient of one local instantaneous function selected randomly is evaluated and the average of the most recent stochastic gradients is used to approximate the true local gradient. In the constrained case, we show that with strong-convexity of the local instantaneous function and Lipschitz continuity of its gradient, the algorithm converges to the global optimization solution almost surely. In the unconstrained case, an explicit linear convergence rate of the algorithm is provided. Numerical experiments are presented to demonstrate correctness of the theoretical results. (C) 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
机译：本文研究了无向和连接网络上的分布式凸优化问题，其中每个节点的变量都位于一个私有约束凸集中，而整个节点的目标是共同最小化所有局部目标函数的总和。机器学习问题中的各种应用都将大规模的训练集分布到多个自治节点，因此，每个局部目标函数被进一步设计为中等数量的局部瞬时函数的平均值。每个局部目标函数和约束集不能与其他对象共享。提出了一种原始-对偶随机算法来解决分布式凸优化问题，在该算法中，每个节点通过使用无偏随机平均梯度来更新其状态，并投影在其私有约束集上。在每次迭代中，对于每个节点，将评估随机选择的一个局部瞬时函数的梯度，并使用最新随机梯度的平均值来近似真实的局部梯度。在受约束的情况下，我们证明了该算法具有局部瞬时函数的强凸性和其梯度的Lipschitz连续性，几乎可以肯定地收敛于全局优化解。在无约束的情况下，提供了算法的显式线性收敛速率。数值实验表明了理论结果的正确性。 （C）2019富兰克林研究所。由Elsevier Ltd.出版。保留所有权利。

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