An Improved Convergence Analysis for Decentralized Online Stochastic Non-Convex Optimization

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 min...

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Bibliographic Details
Published in:IEEE transactions on signal processing Vol. 69; pp. 1842 - 1858
Main Authors: Xin, Ran, Khan, Usman A., Kar, Soummya
Format: Journal Article
Language:English
Published: New York IEEE 2021
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Algorithms
Asymptotic properties
Computational geometry
Convergence
Convex analysis
Convexity
Decentralized optimization
Linear programming
multi-agent systems
non-convex problems
Optimization
Signal processing algorithms
Steady-state
stochastic gradient methods
Stochastic processes
Transient analysis
ISSN:1053-587X, 1941-0476
Online Access:Get full text
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Summary: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-Łojasiewics (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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ISSN:1053-587X
1941-0476
DOI:10.1109/TSP.2021.3062553