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

Abstract

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 $\Omega((n+\sqrt{\kappa n})\log(1/\varepsilon))$ 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.