Lower Bounds for Smooth Nonconvex Finite-Sum Optimization

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Dongruo Zhou, Quanquan Gu ;
Proceedings of the 36th International Conference on Machine Learning, PMLR 97:7574-7583, 2019.

Abstract

Smooth finite-sum optimization has been widely studied in both convex and nonconvex settings. However, existing lower bounds for finite-sum optimization are mostly limited to the setting where each component function is (strongly) convex, while the lower bounds for nonconvex finite-sum optimization remain largely unsolved. In this paper, we study the lower bounds for smooth nonconvex finite-sum optimization, where the objective function is the average of $n$ nonconvex component functions. We prove tight lower bounds for the complexity of finding $\epsilon$-suboptimal point and $\epsilon$-approximate stationary point in different settings, for a wide regime of the smallest eigenvalue of the Hessian of the objective function (or each component function). Given our lower bounds, we can show that existing algorithms including {KatyushaX} \citep{allen2018katyushax}, {Natasha} \citep{allen2017natasha} and {StagewiseKatyusha} \citep{yang2018does} have achieved optimal {Incremental First-order Oracle} (IFO) complexity (i.e., number of IFO calls) up to logarithm factors for nonconvex finite-sum optimization. We also point out potential ways to further improve these complexity results, in terms of making stronger assumptions or by a different convergence analysis.

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