Quantifying Distributional Model Risk via Optimal Transport

By Jose H. Blanchet, Karthyek Murthy. 2016

516 citations recorded by the source corpus.

This paper deals with the problem of quantifying the impact of model misspecification when computing general expected values of interest. The methodology that we propose is applicable in great generality, in particular, we provide examples involving path dependent expectations of stochastic processes. Our approach consists in computing bounds for the expectation of interest regardless of the probability measure used, as long as the measure lies within a prescribed tolerance measured in terms of a flexible class of distances from a suitable baseline model. These distances, based on optimal transportation between probability measures, include Wasserstein's distances as particular cases. The proposed methodology is well-suited for risk analysis, as we demonstrate with a number of applications. We also discuss how to estimate the tolerance region non-parametrically using Skorokhod-type embeddings in some of these applications.

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Citations in the corpus

Certifying Some Distributional Robustness with Principled Adversarial Training (2017, methodology): “The following duality result [8, 24] gives the equality (2) for the relaxation and an analogous result for the problem (1).”
Generalizing to Unseen Domains via Adversarial Data Augmentation (2018, background): “The following result is a minor adaptation of [2, Theorem 1]; to ease notation, let us deﬁne the robust surrogate loss Lemma 1.”
Distributionally Robust Neural Networks for Group Shifts: On the Importance of Regularization for Worst-Case Generalization (2019, background): “Prior work in DRO typically deﬁnes the uncertainty set Q as a divergence ball around the training distribution over ( x, y ) (Ben-Tal et al., 2013; Lam & Zhou, 2015; Duchi et al., 2016; Miyato et al., 2018; Esfahani & Kuhn, 2018; Bertsimas et al., 2018; Blanchet & Murthy, 2019).”
Robust distortion risk measures (2022, unknown): “Worst-(best-) case values under probability distance constraints have been considered by Glasserman and Xu (2014) and Lam (2016), who use the Kullback–Leibler divergence, as well as by Blanchet and Murthy (2019), who utilize distances stemming from mass transportation.”
A Collaborative Learning Framework via Federated Meta-Learning (2020, methodology): “The duality result below in [20] based on Kantorovich’s duality, a widely used approach to solve the Wasserstein distance based DRO problem in optimal transport, provides us an efﬁcient way to solve (13) through a robust surrogate loss: Deﬁne the robust surrogate loss as l λ ( θ , ( x 0 , y 0 ))…”
Lipschitz Networks and Distributional Robustness (2018, background): “…the uncertainty set is a transportation cost-or Wasserstein-ball containing µ (deﬁned formally in § 2), under certain technical conditions, (1) has the dual form (Blanchet et al., 2016, 2017; Gao and Kleywegt, 2016; Shaﬁeezadeh-Abadeh et al., 2017) risk ℓ ( f ; P ) = risk ℓ ( f ; µ ) + L ( f ) .”
Regret in the Newsvendor Model with Demand and Yield Randomness (2021, background): “…close to a reference distribution with respect to certain distance measures, among which the most popular are ϕ -divergences (Ben-Tal et al. 2013, Jiang and Guan 2018, Wang et al. 2016) and the Wasserstein distance (Blanchet and Murthy 2019, Gao and Kleywegt 2016, Mohajerin Esfahani and Kuhn 2018).”
Learning Models with Uniform Performance via Distributionally Robust Optimization (2018, background): “An alternative to our f -divergence based sets {Q : Df (Q||P0) ≤ ρ} are Wasserstein balls [99, 72, 100, 81, 18, 17, 41, 38, 87, 65].”
Limitations of adversarial robustness: strong No Free Lunch Theorem (2018, methodology): “!) value of λ at which the inﬁmum is attained and we have used the previous computations and the handy formula which is a direct consequence of (Blanchet and Murthy, 2016, Remark 1).”
Certifiable Distributional Robustness with Principled Adversarial Training (2017, methodology): “For completeness, we provide an alternative proof to that given in Blanchet and Murthy [7] using convex analysis.”
Robust Wasserstein profile inference and applications to machine learning (2016, methodology): “Though the DRO formulation (8) involves optimizing over uncountably many probability measures, recent strong duality results for Wasserstein DRO (see, for example, Theorem 1 in [9]) ensures that the inner supremum in (8) admits a reformulation which is a simple, univariate optimization problem.”
Regularization via Mass Transportation (2017, background): “Blanchet and Murthy [10] and Gao and Kleywegt [19] consider generalized Wasserstein ambiguity sets deﬁned over Polish spaces.”
Distributionally Robust Optimization: A Review (2019, methodology): “supP∈PW(P0; ) EP [ h(x, ξ̃) ] , proved in a general form in Blanchet and Murthy [49].”
Wasserstein Distributional Robustness and Regularization in Statistical Learning (2017, background): “…of the statistical distance include φ -divergences (Ben-Tal et al. 2013, Jiang and Guan 2015, Bayraksan and Love 2015, Wang et al. 2016), Prokhorov metric (Erdo˘gan and Iyengar 2006), and Wasserstein distances (Wozabal 2014, Blanchet and Murthy 2016, Gao and Kleywegt 2016, Esfahani and Kuhn 2017).”
Statistics of Robust Optimization: A Generalized Empirical Likelihood Approach (2016, methodology): “The recent line of work on distributionally robust optimization using Wasserstein distances [81, 100, 43, 88, 17, 95] is similar in spirit to the formulation considered here.”