A General Analysis Framework of Lower Complexity Bounds for Finite-Sum Optimization

This paper studies the lower bound complexity for the optimization problem whose objective function is the average of n individual smooth convex functions. We consider the algorithm which gets access to gradient and proximal oracle for each individual component. For the strongly-convex case, we prove such an algorithm can not reach an varepsilon-suboptimal point in fewer than Omega((n+√kappa n)log(1/varepsilon)) iterations, where kappa is the condition number of the objective function. This lower bound is tighter than previous results and perfectly matches the upper bound of the existing proximal incremental first-order oracle algorithm Point-SAGA. We develop a novel construction to show the above result, which partitions the tridiagonal matrix of classical examples into n groups. This construction is friendly to the analysis of proximal oracle and also could be used to general convex and average smooth cases naturally.