Tight Lower Complexity Bounds for Strongly Convex Finite-Sum Optimization
This work addresses a theoretical gap in optimization for machine learning by providing tight lower bounds that resolve inconsistencies in prior results, though it is incremental as it builds on existing methods.
The paper derived tight lower complexity bounds for randomized incremental gradient methods in strongly convex finite-sum optimization, matching the upper bounds of state-of-the-art methods like Katyusha and SDCA in specific cases.
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.