An improved convergence analysis for decentralized online stochastic non-convex optimization

• URL: http://arxiv.org/abs/2008.04195v2
• Date: Tue, 29 Dec 2020 02:20:10 GMT
• Title: An improved convergence analysis for decentralized online stochastic non-convex optimization
• Authors: Ran Xin, Usman A. Khan, and Soummya Kar
• Abstract summary: In this paper, we show that a technique called GT-Loakjasiewics (GT-Loakjasiewics) satisfies the existing condition GT-Loakjasiewics (GT-Loakjasiewics) satisfies the current best convergence rates. The results are not only immediately applicable but also the currently known best convergence rates.
• Score: 17.386715847732468
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we study decentralized online stochastic non-convex optimization over a network of nodes. Integrating a technique called gradient tracking in decentralized stochastic gradient descent, we show that the resulting algorithm, GT-DSGD, enjoys certain desirable characteristics towards minimizing a sum of smooth non-convex functions. In particular, for general smooth non-convex functions, we establish non-asymptotic characterizations of GT-DSGD and derive the conditions under which it achieves network-independent performances that match the centralized minibatch SGD. In contrast, the existing results suggest that GT-DSGD is always network-dependent and is therefore strictly worse than the centralized minibatch SGD. When the global non-convex function additionally satisfies the Polyak-Lojasiewics (PL) condition, we establish the linear convergence of GT-DSGD up to a steady-state error with appropriate constant step-sizes. Moreover, under stochastic approximation step-sizes, we establish, for the first time, the optimal global sublinear convergence rate on almost every sample path, in addition to the asymptotically optimal sublinear rate in expectation. Since strongly convex functions are a special case of the functions satisfying the PL condition, our results are not only immediately applicable but also improve the currently known best convergence rates and their dependence on problem parameters.

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