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 $O(k^{-1})$ 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.

通过选择适当的不精确度容限，我们的结果放宽了文献中采用的不精确条件，同时具有相同的竞争性收敛性能，从而研究了解决单调包含问题的Halpern迭代的不精确变体，并进行广泛的收敛性分析。我们还演示了如何将该方法应用于两类数据驱动的Wasserstein分布鲁棒优化问题的求解，这些问题可以转换为凸凹极小极大优化问题。我们强调了该方法在具有随机一阶方法的分布鲁棒学习中执行不精确计算的能力。

不精确的Halpern迭代在分布鲁棒优化中的应用