论文标题
随机不进行的增强Lagrangian方法，用于非convex期望受约束优化
Stochastic Inexact Augmented Lagrangian Method for Nonconvex Expectation Constrained Optimization
论文作者
论文摘要
许多现实世界中的问题不仅具有复杂的非Convex功能约束，而且使用大量数据点。这激发了在有限和期望的约束问题上设计有效的随机方法的设计。在本文中，我们设计和分析了随机不进行的增强拉格朗日方法（stoc-iAlm），以解决涉及非凸复合材料（即平滑+非平滑度）物镜和非凸平平滑功能约束的问题。我们采用标准IALM框架，并使用基于动量的方差降低近端随机梯度方法（PSTORM）和后处理步骤来设计子例程。在某些规律性条件（在现有作品中也假定），要达到预期的$ \ varepsilon $ -KKT点，我们建立了$ o（\ varepsilon^{ - 5}）$的甲骨文复杂性结果，该结果比最适合的$ o（\ varepsilon^{ - 6}）$更好。关于公平性问题的数值实验和使用实际数据的Neyman-Pearson分类问题表明，我们所提出的方法的表现优于以前最著名的复杂性结果的现有方法。
Many real-world problems not only have complicated nonconvex functional constraints but also use a large number of data points. This motivates the design of efficient stochastic methods on finite-sum or expectation constrained problems. In this paper, we design and analyze stochastic inexact augmented Lagrangian methods (Stoc-iALM) to solve problems involving a nonconvex composite (i.e. smooth+nonsmooth) objective and nonconvex smooth functional constraints. We adopt the standard iALM framework and design a subroutine by using the momentum-based variance-reduced proximal stochastic gradient method (PStorm) and a postprocessing step. Under certain regularity conditions (assumed also in existing works), to reach an $\varepsilon$-KKT point in expectation, we establish an oracle complexity result of $O(\varepsilon^{-5})$, which is better than the best-known $O(\varepsilon^{-6})$ result. Numerical experiments on the fairness constrained problem and the Neyman-Pearson classification problem with real data demonstrate that our proposed method outperforms an existing method with the previously best-known complexity result.
