# 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 ε-suboptimal point in fewer than Ω((n+√(κ n))(1/ε)) iterations, where κ 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.

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