S-ADDOPT: Decentralized stochastic first-order optimization over directed graphs

• URL: http://arxiv.org/abs/2005.07785v3
• Date: Wed, 22 Jul 2020 19:15:28 GMT
• Title: S-ADDOPT: Decentralized stochastic first-order optimization over directed graphs
• Authors: Muhammad I. Qureshi, Ran Xin, Soummya Kar, and Usman A. Khan
• Abstract summary: Decentralized convex optimization is proposed to minimize a sum of smooth and strongly cost functions when the functions are distributed over a directed network nodes. In particular, we propose thetextbftextttS-ADDOPT algorithm that assumes a first-order oracle at each node. For decaying step-sizes$mathcalO (1/k)$, we show thattextbfttS-ADDOPT reaches the exact solution subly at$mathcalO (1/k)$ and its convergence is networkally-independent
• Score: 16.96562173221624
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this report, we study decentralized stochastic optimization to minimize a sum of smooth and strongly convex cost functions when the functions are distributed over a directed network of nodes. In contrast to the existing work, we use gradient tracking to improve certain aspects of the resulting algorithm. In particular, we propose the~\textbf{\texttt{S-ADDOPT}} algorithm that assumes a stochastic first-order oracle at each node and show that for a constant step-size~$\alpha$, each node converges linearly inside an error ball around the optimal solution, the size of which is controlled by~$\alpha$. For decaying step-sizes~$\mathcal{O}(1/k)$, we show that~\textbf{\texttt{S-ADDOPT}} reaches the exact solution sublinearly at~$\mathcal{O}(1/k)$ and its convergence is asymptotically network-independent. Thus the asymptotic behavior of~\textbf{\texttt{S-ADDOPT}} is comparable to the centralized stochastic gradient descent. Numerical experiments over both strongly convex and non-convex problems illustrate the convergence behavior and the performance comparison of the proposed algorithm.

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