A Note on Complexity for Two Classes of Structured Non-Smooth Non-Convex Compositional Optimization
Year: 2024
Type: preprint
Abstract: This note studies numerical methods for solving compositional optimization problems, where the inner function is smooth, and the outer function is Lipschitz continuous, non-smooth, and non-convex but exhibits one of two special structures that enable the design of efficient first-order methods. In the first structure, the outer function allows for an easily solvable proximal mapping. We demonstrate that, in this case, a smoothing compositional gradient method can find a $(δ,ε)$-stationary point--specifically defined for compositional optimization--in $O(1/(δε^2))$ iterations. In the second structure, the outer function is expressed as a difference-of-convex function, where each convex component is simple enough to allow an efficiently solvable proximal linear subproblem. In this case, we show that a prox-linear method can find a nearly $ε$-critical point in $O(1/ε^2)$ iterations.
Source: arXiv (Cornell University)
Authors: Yao, Yao, Lin, Qihang, Yang, Tianbao
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4404649432