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

• URL: http://arxiv.org/abs/2103.08280v1
• Date: Mon, 15 Mar 2021 11:20:31 GMT
• Title: Lower Complexity Bounds of Finite-Sum Optimization Problems: The Results and Construction
• Authors: Yuze Han, Guangzeng Xie, Zhihua Zhang
• Abstract summary: We consider Proximal Incremental First-order (PIFO) algorithms which have access to gradient and proximal oracle for each individual component. We develop a novel approach for constructing adversarial problems, which partitions the tridiagonal matrix of classical examples into $n$ groups.
• Score: 18.65143269806133
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The contribution of this paper includes two aspects. First, we study the lower bound complexity for the minimax optimization problem whose objective function is the average of $n$ individual smooth component functions. We consider Proximal Incremental First-order (PIFO) algorithms which have access to gradient and proximal oracle for each individual component. We develop a novel approach for constructing adversarial problems, which partitions the tridiagonal matrix of classical examples into $n$ groups. This construction is friendly to the analysis of incremental gradient and proximal oracle. With this approach, we demonstrate the lower bounds of first-order algorithms for finding an $\varepsilon$-suboptimal point and an $\varepsilon$-stationary point in different settings. Second, we also derive the lower bounds of minimization optimization with PIFO algorithms from our approach, which can cover the results in \citep{woodworth2016tight} and improve the results in \citep{zhou2019lower}.

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