An Inexact Halpern Iteration with Application to Distributionally Robust Optimization

Ling Liang (), Zusen Xu (), Kim-Chuan Toh () and Jia-Jie Zhu ()
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Ling Liang: College Park
Zusen Xu: Weierstrass Institute for Applied Analysis and Stochastics
Kim-Chuan Toh: National University of Singapore
Jia-Jie Zhu: Weierstrass Institute for Applied Analysis and Stochastics

Journal of Optimization Theory and Applications, 2025, vol. 206, issue 3, No 2, 41 pages

Abstract: Abstract The Halpern iteration for solving monotone inclusion problems has gained increasing interests in recent years due to its simple form and appealing convergence properties. In this paper, we investigate the inexact variants of the scheme in both deterministic and stochastic settings. We conduct extensive convergence analysis and show that by choosing the inexactness tolerances appropriately, the inexact schemes admit an $$O(k^{-1})$$ O ( k - 1 ) convergence rate in terms of the (expected) residue norm. Our results relax the state-of-the-art inexactness conditions employed in the literature while sharing the same competitive convergence properties. We then demonstrate how the proposed methods can be applied for solving two classes of data-driven Wasserstein distributionally robust optimization problems that admit convex-concave min-max optimization reformulations. We highlight its capability of performing inexact computations for distributionally robust learning with stochastic first-order methods and for general nonlinear convex-concave loss functions, which are competitive in the literature.

Keywords: Halpern iteration; Monotone inclusion problem; Convex-concave min-max optimization; Data-driven Wasserstein distributionally robust optimization; 90C25; 90C15; 90C17 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10957-025-02744-y

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