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A simpler approach to accelerated optimization: iterative averaging meets optimism

Pooria Joulani · Anant Raj · András György · Csaba Szepesvari

Keywords: [ Convex Optimization ] [ Online Learning / Bandits ] [ Online Learning, Active Learning, and Bandits ]


Recently there have been several attempts to extend Nesterov's accelerated algorithm to smooth stochastic and variance-reduced optimization. In this paper, we show that there is a simpler approach to acceleration: applying optimistic online learning algorithms and querying the gradient oracle at the online average of the intermediate optimization iterates. In particular, we tighten a recent result of Cutkosky (2019) to demonstrate theoretically that online iterate averaging results in a reduced optimization gap, independently of the algorithm involved. We show that carefully combining this technique with existing generic optimistic online learning algorithms yields the optimal accelerated rates for optimizing strongly-convex and non-strongly-convex, possibly composite objectives, with deterministic as well as stochastic first-order oracles. We further extend this idea to variance-reduced optimization. Finally, we also provide ``universal'' algorithms that achieve the optimal rate for smooth and non-smooth composite objectives simultaneously without further tuning, generalizing the results of Kavis et al. (2019) and solving a number of their open problems.

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