提出几何新框架，更精准评估模型在不确定数据下的最坏风险。
Concave Certificates: Geometric Framework for Distributionally Robust Risk and Complexity Analysis
- 用凹函数上界替代传统梯度约束，提升鲁棒性分析精度。
- 理论证明可消除网络深度、宽度对风险的依赖影响。
- 设计可计算的对抗评分，支持神经网络逐层分析。
分布式鲁棒优化旨在对水车斯坦不确定性集内的最坏情况风险进行认证。现有方法通常依赖全局Lipschitz界（过于保守）或局部梯度信息（仅一阶近似）。本文提出基于增长速率函数最小凹上界的几何新框架，引入凹证书以紧致地控制非Lipschitz和非可微损失下的分布鲁棒风险。该框架拓展至复杂性分析，提出与标准统计泛化界互补的最坏情况泛化界。进一步利用该证书，证明对抗风险与经验Rademacher复杂度之间的差距可摆脱输入直径、网络宽度和深度的依赖。针对深度学习应用，提出对抗评分作为凹证书的可计算松弛，实现高效且逐层的神经网络分析。在真实世界数据上的分类与回归任务中，通过多组数值实验验证了理论结果的有效性。
原文摘要 · Abstract (English)
Distributionally Robust (DR) optimization aims to certify worst-case risk within a Wasserstein uncertainty set. Current certifications typically rely either on global Lipschitz bounds, which are often conservative, or on local gradient information, which provides only a first-order approximation. This paper introduces a novel geometric framework based on the least concave majorants of the growth rate functions. Our proposed concave certificate establishes a tight bound on DR risk that remains applicable to non-Lipschitz and non-differentiable losses. We extend this framework to complexity analysis, introducing the worst-case generalization bound that complements the standard statistical generalization bound. Furthermore, we utilize this certificate to bound the gap between adversarial and empirical Rademacher complexity, demonstrating that dependencies on input diameter, network width, and depth can be eliminated. For practical application in deep learning, we introduce the adversarial score as a tractable relaxation of the concave certificate that enables efficient and layer-wise analysis of neural networks. We validate our theoretical results in various numerical experiments on classification and regression tasks using real-world data.
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