用Wasserstein距离优化后悔值，平衡鲁棒性与收益潜力。
Wasserstein Distributionally Robust Regret Optimization
- 基于Wasserstein模糊集构建后悔优化框架，通过梯度差异选择最优解
- 在平滑条件下，最优解由一阶梯度差决定；唯一解时二阶项主导偏差
- 即使无双线性项，计算后悔值仍为NP难，但可精确求解或获得紧松弛
分布鲁棒优化（DRO）广泛用于不确定性下的决策，但其对最坏情况损失的偏好可能导致策略过于保守。为缓解此问题，我们研究了事前分布鲁棒后悔优化（DRRO），采用Wasserstein模糊集，旨在平衡鲁棒性与潜在收益。我们建立了与Wasserstein DRO平行的WDRRO理论：在光滑性和正则性假设下，WDRRO通过一阶梯度差异规则从经验风险最小化（ERM）最优解中选择。若ERM最优解唯一，则一阶敏感性消失，二阶展开主导偏离。对于凸二次型的ERM，任意半径下其与DRRO一致。随后我们研究假设失效的情形：不可微的最大仿射损失、离散参考分布及较大半径，此时WDRRO可能区别于ERM和WDRO。我们证明，即使无双线性项，计算WDRRO后悔值也为NP-hard。然而，我们提出了精确算法、具有保证的可处理凸松弛，并通过实验验证了其紧致性和损失依赖行为。
原文摘要 · Abstract (English)
Distributionally robust optimization (DRO) is widely used for decision-making under uncertainty, but its adversarial focus on worst-case loss can lead to overly conservative policies. To mitigate this, we study ex-ante Distributionally Robust Regret Optimization (DRRO) with Wasserstein ambiguity sets, designed to balance robustness with upside potential. We develop a theory of Wasserstein DRRO (WDRRO) paralleling Wasserstein DRO. Under smoothness and regularity, WDRRO selects among ERM optima by a first-order gradient-discrepancy rule. If the ERM optimizer is unique, first-order sensitivity vanishes and a second-order expansion governs deviations. For convex quadratics ERM and DRRO coincide for any radius. We then study regimes where these assumptions fail: nondifferentiable max-affine losses, discrete references, and larger radii, where WDRRO can differ from ERM and WDRO. We show that computing WDRRO regret is NP-hard even without bilinear terms. Nevertheless, we develop exact algorithms, a tractable convex relaxation with guarantees, and experiments showing tightness and loss-dependent behavior.
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