Lower Complexity Bounds of Finite-Sum Optimization Problems: The Results and Construction

Abstract

In this paper, we study the lower complexity bounds for finite-sum optimization problems, where the objective is the average of $n$ individual component functions. We consider Proximal Incremental First-order (PIFO) algorithms which have access to the gradient and proximal oracles for each component function. To incorporate loopless methods, we also allow PIFO algorithms to obtain the full gradient infrequently. We develop a novel approach to constructing the hard instances, which partitions the tridiagonal matrix of classical examples into $n$ groups. This construction is friendly to the analysis of PIFO algorithms. Based on this construction, we establish the lower complexity bounds for finite-sum minimax optimization problems when the objective is convex-concave or nonconvex-strongly-concave and the class of component functions is $L$-average smooth. Most of these bounds are nearly matched by existing upper bounds up to log factors. We can also derive similar lower bounds for finite-sum minimization problems as previous work under both smoothness and average smoothness assumptions. Our lower bounds imply that proximal oracles for smooth functions are not much more powerful than gradient oracles.

We fix some typos

References in corpus (3)

• Efficient Algorithms for Smooth Minimax Optimization
• Stochastic Variance Reduction for Variational Inequality Methods
• Lower Bounds for Smooth Nonconvex Finite-Sum Optimization

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• Nonconvex-Nonconcave Min-Max Optimization with a Small Maximization Domain
• Near Optimal Stochastic Algorithms for Finite-Sum Unbalanced Convex-Concave Minimax Optimization
• Finding Second-Order Stationary Points in Nonconvex-Strongly-Concave Minimax Optimization
• Method with Batching for Stochastic Finite-Sum Variational Inequalities in Non-Euclidean Setting
• Optimal Analysis of Method with Batching for Monotone Stochastic Finite-Sum Variational Inequalities