Tight Lower Complexity Bounds for Strongly Convex Finite-Sum Optimization

TL;DR

This paper establishes tight lower complexity bounds for randomized incremental gradient methods in strongly convex finite-sum optimization, matching the best known upper bounds under various conditions.

Contribution

It derives the first tight lower bounds for these methods, improving understanding of their fundamental complexity limits in different strongly convex settings.

Findings

Lower bounds match upper bounds of Katyusha and VRADA for strongly convex and smooth functions.

Lower bounds match upper bounds of SDCA without duality and KatyushaX for average smooth functions.

Results clarify the optimality of existing algorithms in finite-sum optimization.

Abstract

Finite-sum optimization plays an important role in the area of machine learning, and hence has triggered a surge of interest in recent years. To address this optimization problem, various randomized incremental gradient methods have been proposed with guaranteed upper and lower complexity bounds for their convergence. Nonetheless, these lower bounds rely on certain conditions: deterministic optimization algorithm, or fixed probability distribution for the selection of component functions. Meanwhile, some lower bounds even do not match the upper bounds of the best known methods in certain cases. To break these limitations, we derive tight lower complexity bounds of randomized incremental gradient methods, including SAG, SAGA, SVRG, and SARAH, for two typical cases of finite-sum optimization. Specifically, our results tightly match the upper complexity of Katyusha or VRADA when each component function is strongly convex and smooth, and tightly match the upper complexity of SDCA without duality and of KatyushaX when the finite-sum function is strongly convex and the component functions are average smooth.