A Primal-Dual SGD Algorithm for Distributed Nonconvex Optimization

• Published in: IEEE/CAA journal of automatica sinica Vol. 9; no. 5; pp. 812 - 833
• Main Authors: Yi, Xinlei, Zhang, Shengjun, Yang, Tao, Chai, Tianyou, Johansson, Karl Henrik
• Format: Journal Article
• Language: English
• Published: Piscataway Chinese Association of Automation (CAA) 01.05.2022; The Institute of Electrical and Electronics Engineers, Inc. (IEEE); Division of Decision and Control Systems,School of Electrical Engineering and Computer Science,KTH Royal Institute of Technology,and also affiliated with the Digital Futures,Stockholm 10044,Sweden%Department of Electrical Engineering,University of North Texas,Denton,TX 76203 USA%State Key Laboratory of Synthetical Automation for Process Industries,Northeastern University,Shenyang 110819,China
• Subjects: Algorithms; Collaborative work; Communication networks; Convergence; Cost function; Data exchange; Deep learning; Distributed nonconvex optimization; Information exchange; linear speedup; Machine learning; Machine learning algorithms; Optimization; Parallel processing; Polyak-Lojasiewicz (P-L) condition; Polyak-Łojasiewicz (P-Ł) condition; primal-dual algorithm; stochastic gradient descent; Distributed nonconvex optimization; Polyak-?ojasiewicz (P-?) condition; primal-dual algorithm; linear speedup; stochastic gradient descent
• ISSN: 2329-9266; 2329-9274
• DOI: 10.1109/JAS.2022.105554

The distributed nonconvex optimization problem of minimizing a global cost function formed by a sum of n local cost functions by using local information exchange is considered. This problem is an important component of many machine learning techniques with data parallelism, such as deep learning and federated learning. We propose a distributed primal-dual stochastic gradient descent (SGD) algorithm, suitable for arbitrarily connected communication networks and any smooth (possibly nonconvex) cost functions. We show that the proposed algorithm achieves the linear speedup convergence rate \mathcal{O}(1/\sqrt{nT}) for general nonconvex cost functions and the linear speedup convergence rate \mathcal{O}(1/(nT)) when the global cost function satisfies the Polyak-Łojasiewicz (P-Ł) condition, where T is the total number of iterations. We also show that the output of the proposed algorithm with constant parameters linearly converges to a neighborhood of a global optimum. We demonstrate through numerical experiments the efficiency of our algorithm in comparison with the baseline centralized SGD and recently proposed distributed SGD algorithms.