A General Analysis Framework of Lower Complexity Bounds for Finite-Sum Optimization
This provides a foundational theoretical limit for optimization algorithms in machine learning, addressing a key bottleneck in large-scale data processing.
The paper tackles the problem of determining the lower bound complexity for finite-sum optimization with smooth convex functions, proving a tighter lower bound of Ω((n+√(κn))log(1/ε)) iterations for strongly-convex cases that matches the upper bound of Point-SAGA.
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 $Ω((n+\sqrt{κn})\log(1/\varepsilon))$ 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.