Wasserstein Ambiguity Set (Wasserstein 模糊集)

Definition. A Wasserstein ambiguity set (Wasserstein 模糊集) is a ball of probability distributions centred on a nominal / empirical measure, with the radius measured in the p-Wasserstein metric from optimal transport. It is the most popular data-driven construction in distributionally robust optimisation (DRO, concepts/Distributionally Robust Optimization) because the radius admits a finite-sample statistical interpretation.

Definition

For two probability measures P_1,P_2 on a Polish space (\Xi,\mathcal F) and a transportation cost c:\Xi\times\Xi\to\mathbb R_+, the p-Wasserstein distance is

W_p(P_1,P_2) := \left(\inf_{\pi\in\Pi(P_1,P_2)}\ \int_{\Xi\times\Xi} c(\xi_1,\xi_2)^p\ d\pi(\xi_1,\xi_2)\right)^{1/p}, where \Pi(P_1,P_2) is the set of couplings with marginals P_1,P_2. The Wasserstein ambiguity set is \mathcal P_\epsilon := \bigl\{\,P\in\mathcal M(\Xi,\mathcal F)\ :\ W_p(P,P_0)\le \epsilon\,\bigr\},

with P_0 a nominal or empirical distribution and \epsilon\ge 0 the radius.

Why Wasserstein (and not \phi-divergence)

The Wasserstein ball contains all distributions within a certain optimal-transport cost, including distributions with different supports. \phi-divergence balls (KL, \chi^2, Hellinger) require absolute continuity, so they cannot include distributions that put mass where the nominal distribution has zero mass. In quant-finance this matters because the empirical distribution is a sum of point masses on N observed scenarios; a \phi-divergence ball cannot hedge against new realisations outside the observed support — a Wasserstein ball can.

Mohajerin Esfahani & Kuhn (2018) show that a Wasserstein-DRO solution is equivalent to a robust optimisation problem with an uncertainty set of size proportional to \epsilon, so the geometry of the solution is preserved.

Finite-sample calibration of \epsilon (the key deployable result)

For a sample \{\hat\xi_i\}_{i=1}^N\sim P^\ast and the empirical measure \hat P_N, Mohajerin Esfahani & Kuhn prove that for \epsilon_N(\beta):= \left(\frac{\log(c/\beta)}{N}\right)^{1/d} (under a sub-Gaussian moment condition and \Xi\subseteq[0,1]^d),

\mathbb P^N\Bigl\{\,P^\ast\in\hat{\mathcal P}_{\epsilon_N(\beta)}\,\Bigr\}\ \ge\ 1-\beta.

The probability that the true data-generating distribution lies inside the Wasserstein ball is at least 1-\beta. The \epsilon_N(\beta) formula is rate-optimal in N and dimension d up to logarithmic factors.

flowchart TD
    A["Sample (xi_1,...,xi_N)"] --> B["Empirical P_N"]
    B --> C["Choose confidence 1-beta"]
    C --> D["epsilon_N(beta) = (log(c/beta)/N)^(1/d)"]
    D --> E["Wasserstein ball P_eps"]
    E --> F["Solve DRO over P_eps"]
    F --> G["Robust x* with finite-sample guarantee"]

In production this means a quant team can directly pick a confidence level (say 95% or 99%) and let the radius be a deterministic function of sample size — no cross-validation over \epsilon required. The cost is the dimension-1/d curse: in d=50 (a typical equity covariance problem), \epsilon decays only as N^{-1/50}, which is the rate of convergence of empirical measures in Wasserstein distance.

Reformulation of Wasserstein-DRO

For a convex cost h(x,\xi), Mohajerin Esfahani & Kuhn (2018) show that the infinite-dimensional

\min_{x\in X}\ \sup_{P:\,W_p(P,\hat P_N)\le\epsilon}\ \mathbb E_P[h(x,\tilde\xi)] admits a finite convex reformulation: \min_{x\in X,\ \lambda\ge 0}\ \lambda\epsilon + \frac{1}{N}\sum_{i=1}^N \sup_{\xi\in\Xi}\bigl[h(x,\xi)-\lambda\,c(\xi,\hat\xi_i)^p\bigr]

The inner sup is over a (potentially infinite) support \Xi; for linear / quadratic h and standard c (e.g. \ell_1, \ell_2, \ell_\infty norm), the sup reduces to a tractable conic problem. The dual variable \lambda plays the role of a regularisation strength — as \lambda\to\infty the worst-case over the Wasserstein ball collapses to the empirical mean, recovering standard ERM.

Application flavours in the quant-finance literature

Connection to statistical learning (DRO = regularisation)

When the support \Xi is bounded and the cost is \ell_2^p, the Wasserstein-DRO problem reduces to distributionally robust regularised ERM (Shafieezadeh-Abadeh, Esfahani & Kuhn 2019):

\min_{x\in X}\ \sup_{P:\,W_p(P,\hat P_N)\le\epsilon}\ \mathbb E_P[\ell(x,\tilde\xi)]\ \equiv\ \min_{x\in X}\ \frac{1}{N}\sum_{i=1}^n \ell(x,\hat\xi_i) + \epsilon\,|x|,

with the regularisation strength controlled by \epsilon. This means LASSO, ridge, SVM, regularised logistic regression are all implicitly Wasserstein-DRO solutions at a particular choice of \epsilon. The DRO view gives a statistical interpretation of the regularisation strength: \epsilon is a confidence radius, not a free knob.

When the Wasserstein construction is not appropriate

• Heavy-tailed / infinite-variance distributions: the sub-Gaussian assumption behind \epsilon_N(\beta) fails; the bound is loose.
• The nominal P_0 is itself misspecified (e.g., Gaussian when truth is multi-modal). A Wasserstein ball around a Gaussian will not contain multi-modal truth even at large \epsilon.
• The dimension d is very large: N^{-1/d} convergence is glacial; a moment-based set is more conservative but more stable.
• The transportation cost c is itself unknown: see Blanchet et al. (2019) for data-driven c.

Cross-references

• concepts/Distributionally Robust Optimization — parent framework
• concepts/Robust Optimization — limit when \epsilon\to\infty
• concepts/Coherent Risk Measures — the same inner sup operator, viewed as a risk measure
• summaries/201908-distributionally-robust-optimization-review/201908-distributionally-robust-optimization-review — the survey that catalogues Wasserstein DRO comprehensively (§5.1.1)

See also

• Villani, C. (2009) Optimal Transport: Old and New — the mathematical reference.
• Mohajerin Esfahani, P. & Kuhn, D. (2018) "Data-driven distributionally robust optimization using the Wasserstein metric" Math. Programming 181.
• Blanchet, J., Kang, Y. & Murthy, K. (2019) "Data-driven optimal transport cost selection for distributionally robust optimization" — data-driven c via metric learning.