提出新方法解决非凸非凹极小极大问题，用于分布鲁棒优化与对抗训练。
A stochastic smoothing framework for nonconvex-nonconcave minEmax problems with applications to Wasserstein distributionally robust optimization
- 用对数均值指数平滑处理价值函数，构建随机近端梯度算法。
- 理论证明收敛至Clarke平稳点，数值实验显示性能优于现有基线。
- 适合做分布鲁棒优化、对抗训练等复杂场景的优化求解。
我们研究一类随机非光滑优化问题，其中外变量最小化一个逐点最大值的期望。此类最小化-期望-最大化（minEmax）问题出现在Wasserstein分布鲁棒优化和对抗鲁棒训练中，当底层分布非经验分布时，通常无法重构成有限维极小极大问题。本文基于对数均值指数平滑方法，提出一种随机平滑近端梯度算法。在紧性与Lipschitz型假设下，我们给出了关于Goldstein平稳性的非渐近分析，并证明该方法生成的几乎必然聚类点是原始问题的Clarke平稳点；由Clarke正则性，该点也是方向平稳点。在报童问题、鲁棒回归及对抗鲁棒学习问题上的数值实验表明，所提方法在性能上可与现有基线相媲美。
原文摘要 · Abstract (English)
We study a class of stochastic nonsmooth optimization problems in which an outer variable minimizes the expectation of a pointwise maximum. This minimization--expectation--maximization (minEmax) problem arises in Wasserstein distributionally robust optimization and adversarially robust training, and it cannot in general be reformulated as a finite-dimensional minimax problem when the underlying distribution is not empirical. We propose a stochastic smoothing proximal gradient method based on log-mean-exp smoothing of the value function. Under compactness and Lipschitz-type assumptions, we present nonasymptotic analysis in terms of Goldstein stationarity and show that every almost-sure cluster point generated by our method is a Clarke stationary point; by Clarke regularity, such a point is also directional stationary for the original problem. Numerical experiments on newsvendor, robust regression, and adversarially robust learning problems show that the proposed method is competitive with existing baselines.
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