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arXiv · 1805.06045

Optimal Distributed Optimization on Slowly Time-Varying Graphs

Alexander Rogozin·César A. Uribe·Alexander Gasnikov·Nikolay Malkovsky·Angelia Nedić
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Abstract

We study optimal distributed first-order optimization algorithms when the network (i.e., communication constraints between the agents) changes with time. This problem is motivated by scenarios where agents experience network malfunctions. We provide a sufficient condition that guarantees a convergence rate with optimal (up lo logarithmic terms) dependencies on the network and function parameters if the network changes are constrained to a small percentage $α$ of the total number of iterations. We call such networks slowly time-varying networks. Moreover, we show that Nesterov's method has an iteration complexity of $Ω\big( \big(\sqrt{κ_Φ\cdot \barχ} + α\log(κ_Φ\cdot \barχ)\big) \log(1 / \varepsilon)\big)$ for decentralized algorithms, where $κ_Φ$ is condition number of the objective function, and $\barχ$ is a worst case bound on the condition number of the sequence of communication graphs. Additionally, we provide an explicit upper bound on $α$ in terms of the condition number of the objective function and network topologies.

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Alexander Rogozin, César A. Uribe, Alexander Gasnikov, Nikolay Malkovsky, Angelia Nedić. 2019-11-28. Optimal Distributed Optimization on Slowly Time-Varying Graphs. https://doi.org/10.1109/tcns.2019.2949439

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