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In this paper, we propose a novel accelerated gradient method called ANITAfor solving the fundamental finite-sum optimization problems. Concretely, weconsider both general convex and strongly convex settings: i) For generalconvex finite-sum problems, ANITA improves previous state-of-the-art resultgiven by Varag (Lan et al., 2019). In particular, for large-scale problems orthe convergence error is not very small, i.e., $n \geq \frac{1}{\epsilon^2}$,ANITA obtains the \emph{first} optimal result $O(n)$, matching the lower bound$\Omega(n)$ provided by Woodworth and Srebro (2016), while previous results are$O(n \log \frac{1}{\epsilon})$ of Varag (Lan et al., 2019) and$O(\frac{n}{\sqrt{\epsilon}})$ of Katyusha (Allen-Zhu, 2017). ii) For stronglyconvex finite-sum problems, we also show that ANITA can achieve the optimalconvergence rate $O\big((n+\sqrt{\frac{nL}{\mu}})\log\frac{1}{\epsilon}\big)$matching the lower bound$\Omega\big((n+\sqrt{\frac{nL}{\mu}})\log\frac{1}{\epsilon}\big)$ provided byLan and Zhou (2015). Besides, ANITA enjoys a simpler loopless algorithmicstructure unlike previous accelerated algorithms such as Varag (Lan et al.,2019) and Katyusha (Allen-Zhu, 2017) where they use double-loop structures.Moreover, we provide a novel \emph{dynamic multi-stage convergence analysis},which is the key technical part for improving previous results to the optimalrates. We believe that our new theoretical rates and novel convergence analysisfor the fundamental finite-sum problem will directly lead to key improvementsfor many other related problems, such as distributed/federated/decentralizedoptimization problems (e.g., Li and Richt\'arik, 2021). Finally, the numericalexperiments show that ANITA converges faster than the previous state-of-the-artVarag (Lan et al., 2019), validating our theoretical results and confirming thepractical superiority of ANITA.

ANITA: An Optimal Loopless Accelerated Variance-Reduced Gradient Method