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type- ∞ \infty ∞ Wasserstein distance

给定两个分布 P 1 , P 2 \mathbb{P}_1, \mathbb{P}_2 P1​,P2​，它们之间的 type- ∞ \infty ∞ Wasserstein distance 定义为：
d ∞ ( P 1 , P 2 ) : = inf ⁡ Q ∈ P ( P 1 , P 2 , W ) Q -ess sup ∥ w ~ 1 − w ~ 2 ∥ d_{\infty}\left(\mathbb{P}_1, \mathbb{P}_2\right) := \mathop{\inf}\limits_{\mathbb{Q}\in\mathcal{P}(\mathbb{P}_1, \mathbb{P}_2,\mathcal{W})} \mathbb{Q}\text{-ess sup}\lVert \tilde{\boldsymbol{w}}_1 - \tilde{\boldsymbol{w}}_2 \rVert d∞​(P1​,P2​):=Q∈P(P1​,P2​,W)inf​Q-ess sup∥w~1​−w~2​∥ 其中 w ~ 1 ∼ P 1 , w ~ 2 ∼ P 2 \tilde{\boldsymbol{w}}_1\sim\mathbb{P}_1, \tilde{\boldsymbol{w}}_2\sim\mathbb{P}_2 w~1​∼P1​,w~2​∼P2​, P ( P 1 , P 2 , W ) \mathcal{P}(\mathbb{P}_1, \mathbb{P}_2,\mathcal{W}) P(P1​,P2​,W) 表示 W × W \mathcal{W}\times\mathcal{W} W×W 上所有以 P 1 , P 2 \mathbb{P}_1, \mathbb{P}_2 P1​,P2​ 为边缘的分布。

Q -ess sup ∥ ⋅ ∥ \mathbb{Q}\text{-ess sup} \lVert\cdot\rVert Q-ess sup∥⋅∥ 表示相对于联合分布 Q \mathbb{Q} Q 的范数 ∥ ⋅ ∥ \lVert\cdot\rVert ∥⋅∥ 的本质上界（essential supremum）：
Q -ess sup ∥ w ~ 1 − w ~ 2 ∥ : = inf ⁡ { t : Q [ ∥ w ~ 1 − w ~ 2 ∥ > t ] = 0 } \mathbb{Q}\text{-ess sup}\lVert \tilde{\boldsymbol{w}}_1 - \tilde{\boldsymbol{w}}_2 \rVert := \inf \left\{ t : \mathbb{Q}\left[ \lVert \tilde{\boldsymbol{w}}_1 - \tilde{\boldsymbol{w}}_2 \rVert > t \right]=0 \right\} Q-ess sup∥w~1​−w~2​∥:=inf{t:Q[∥w~1​−w~2​∥>t]=0}

type- ∞ \infty ∞ Wasserstein ball

F ∞ ( θ ) : = { P ∈ P ( W ) ∣ d ∞ ( P , P ^ ) ≤ θ } \mathcal{F}_{\infty}(\theta) := \left\{ \mathbb{P}\in\mathcal{P}(\mathcal{W}) \big| d_{\infty}(\mathbb{P},\hat{\mathbb{P}})\leq\theta \right\} F∞​(θ):={P∈P(W)∣∣​d∞​(P,P^)≤θ} 其中 P ^ = 1 N ∑ i ∈ [ N ] δ w ^ i \hat{\mathbb{P}} =\frac{1}{N}\mathop{\sum}\limits_{i\in[N]}\delta_{\hat{\boldsymbol{w}}_i} P^=N1​i∈[N]∑​δw^i​​。

一个等价形式

Bertsimas et al. (2018) 证明：
F ∞ ( θ ) = { P ∈ P ( W ) ∣ P = 1 N ∑ i ∈ [ N ] P i , P i ∈ F i ( θ ) , ∀ i ∈ [ N ] } \mathcal{F}_{\infty}(\theta) = \left\{ \mathbb{P}\in\mathcal{P}(\mathcal{W}) \big| \mathbb{P}= \frac{1}{N}\mathop{\sum}\limits_{i\in[N]} \mathbb{P}_i, \mathbb{P}_i\in\mathcal{F}_i(\theta), \forall i\in[N] \right\} F∞​(θ)=⎩⎨⎧​P∈P(W)∣∣​P=N1​i∈[N]∑​Pi​,Pi​∈Fi​(θ),∀i∈[N]⎭⎬⎫​ 其中 F i ( θ ) = { P ∈ P ( W ) ∣ P [ ∥ w ~ − w ^ i ∥ ≤ θ ] = 1 } \mathcal{F}_i(\theta) = \left\{ \mathbb{P}\in\mathcal{P}(\mathcal{W}) \big| \mathbb{P}[\lVert\tilde{\boldsymbol{w}}-\hat{\boldsymbol{w}}_i\rVert\leq\theta]=1 \right\} Fi​(θ)={P∈P(W)∣∣​P[∥w~−w^i​∥≤θ]=1}

type- ρ \rho ρ Wasserstein distance

对任何 ρ ∈ [ 1 , ∞ ) \rho\in[1,\infty) ρ∈[1,∞)，type- ρ \rho ρ Wasserstein distance 定义为
d ρ ( P 1 , P 2 ) : = inf ⁡ Q ∈ P ( P 1 , P 2 , W ) E Q [ ∥ w ~ 1 − w ~ 2 ∥ ρ ] ρ d_{\rho}\left(\mathbb{P}_1, \mathbb{P}_2\right) := \mathop{\inf}\limits_{\mathbb{Q}\in\mathcal{P}(\mathbb{P}_1, \mathbb{P}_2,\mathcal{W})} \sqrt[\rho]{ \mathbb{E}_{\mathbb{Q}} \left[ \lVert \tilde{\boldsymbol{w}}_1 - \tilde{\boldsymbol{w}}_2 \rVert^{\rho} \right]} dρ​(P1​,P2​):=Q∈P(P1​,P2​,W)inf​ρEQ​[∥w~1​−w~2​∥ρ] ​

Givens et al. 1984 证明：当 ρ \rho ρ 趋于无穷时，type- ρ \rho ρ Wasserstein distance 收敛到 type- ∞ \infty ∞ Wasserstein distance.

参考文献

[1] Bertsimas, Dimitris, Shtern Shimrit, Sturt Bradley. 2018. A data-driven approach for multi-stage linear optimization. Available at Optimization Online.
[2] Givens, Clark R, Rae Michael Shortt, et al. 1984. A class of wasserstein metrics for probability distributions. The Michigan Mathematical Journal 31(2) 231–240.
[3] Zhi Chen, Weijun Xie. Regret in the Newsvendor Model Revisited: A Data-Driven Perspective.

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