研究表明 MRO 与 SNR 在 Wasserstein 鲁棒线性回归中优于 DRO
Regret-Optimal Wasserstein-Robust Linear Regression
如果你用 DRO 做鲁棒回归，这篇指出了它什么时候会慢一截，还给了两个能达到最优速率的替代算法和实现方法。
论文研究多元最小二乘回归中源分布与目标分布相差 ε 尺寸的 2-Wasserstein 扰动时的鲁棒决策问题。在 ε 趋于 0 时，DRO 等多种简单算法都能达到由条件数 κ 刻画的实例最优速率 O(κε²)。但对于固定的 ε > 0，论文证明 DRO 只在残差有效秩有界时达到 minimax 速率，在其他场景会被维度相关因子拖慢。替代方案 minimax regret optimization（MRO）和 spectral-norm regularization（SNR）被证明能达到 minimax 速率，其中 MRO 有精确凸重构、SNR 有加速近端算法，数值实验也区分了两者的表现。
Regret-Optimal Wasserstein-Robust Linear Regression
Distributionally robust optimization (DRO) minimizes worst-case risk under distributional uncertainty, but its decisions can be suboptimal in terms of their excess risk, or regret. We study this gap for multivariate least-squares regression, assuming that the observed source law and unknown target law differ by an adversarial 2-Wasserstein perturbation of size $\varepsilon$. We first identify the asymptotic complexity of robust decision-making in the $\varepsilon \to 0$ limit, characterized by a problem-specific condition number $κ$. Here, a wide spectrum of simple algorithms, including DRO, achieve the instance-optimal rate of $O(κ\varepsilon^2)$. For fixed $\varepsilon > 0$, we prove that DRO still achieves the minimax rate if the problem has bounded residual effective rank. On the other hand, we identify significant failure modes where DRO is provably suboptimal by dimension-dependent factors. Fortunately, we prove that two alternative procedures, minimax regret optimization (MRO) and spectral-norm regularization (SNR), attain the minimax rate. We derive an exact convex reformulation of MRO and an accelerated proximal algorithm for SNR, along with high-probability finite-sample guarantees. We support these theoretical results with numerical experiments separating the performance of MRO and SNR from DRO.