A novel formulation of a Wasserstein-2 metric that uses the Bures-Wasserstein (BW) metric over probability measures with finite second moments is developed, which allows the worst-case distribution to endogenously determine both how many mixture components receive mass and where their means and covariances lie within a continuous support.
Abstract
We study distributionally robust linear chance-constrained problems in which uncertainty is modeled by a Gaussian mixture model (GMM). Finite-support distributionally robust (FDR) formulations, widely used in data-driven robust optimization, robustify over empirical mixture support points and therefore primarily stress-test the fitted nominal mixture. This can be insufficient when service reliability depends on structural misspecification of the nominal mixture-support parameters. To address this limitation, we describe the ambiguity set of distributions by developing a novel formulation of a Wasserstein-2 metric that uses the Bures-Wasserstein (BW) metric over probability measures with finite second moments. Unlike FDR, which generally sets finitely many empirical support points a priori, the proposed ambiguity set allows the worst-case distribution to endogenously determine both how many mixture components receive mass and where their means and covariances lie within a continuous support. For the resulting ambiguity set, under mild regularity conditions, we prove strong duality for the inner worst-case chance-constraint problem and derive its semi-infinite reformulation. We then develop an adaptive cutting-surface algorithm, which endogenously determines the locations of mixture components receiving mass, and the mean and covariances of the Gaussian distributions at these locations. The algorithm attains any prescribed optimality gap in finitely many iterations, while a block-alternating local search identifies new components. A case study using the electric-vehicle charging-station energy-allocation problem demonstrates the framework's practical value in achieving any reliability targets. CDR also induces structural changes in energy allocations, unlike FDR, whose allocations remain close to the nominal solution.
The distribution of a normal mean-variance mixture depends on the law of its positive mixing variable. We compare six parametric mixing laws with a grid nonparametric maximum likelihood estimator under the same determinant identification constraint. The mixing mean $m=\E(Z)$ is estimated and is not fixed at one. A paired block bootstrap is used to compare multivariate holdout log scores. The models that cannot be distinguished from the model with the largest score define a finite ambiguity set. We then consider a cumulative prospect problem on a common portfolio direction. For each model in the set, the NMVM representation gives a scalar projected return and a corresponding prospect-value function of the exposure. The distributionally robust decision maximizes the lower envelope of these functions. We prove existence of a solution, give the candidate points for the piecewise smooth problem, derive a reference-gap scaling result, and construct an interval branch-and-bound certificate for the finite-scenario optimum. In an application to 30 stock returns, the mixture models give higher holdout density scores than the multivariate Gaussian model. Several parametric and semi-parametric models, however, remain in the ambiguity set. The worst-case model is therefore determined at the portfolio optimization stage rather than selected in advance from a point estimate of the holdout score.
A more flexible framework in which a predictive model determines the nominal distribution and a separate model estimates a data-dependent radius is developed, which treats calibration as a practical mechanism for reliable decision making rather than a universal guarantee of improved optimization performance.
This work proposes PRISM-ZO, a projection-robust framework that samples low-dimensional random tangent subspaces and combines symmetric finite differences with median-of-means or Huber aggregation and establishes the unbiasedness of the correctly rescaled projected direction in expectation over the random subspace.
Yinpu Ma, Cunlin Li, Shiyue Zhang· Journal of King Saud Univers...· 0 citations
Covariance steering provides an efficient framework for designing linear stochastic feedback policies, but its extension to nonlinear systems relies on a Gaussian surrogate obtained through local linearization. Because this surrogate may differ substantially from the true nonlinear state distribution, risk-sensitive quantities such as collision probability and mean-squared error may be inaccurately estimated. This work develops a distributionally robust covariance-steering framework based on the relative entropy, also known as the Kullback-Leibler divergence (KLD), to account for ambiguity in the propagated probability density function. Using a variational representation of exponential integrals, we derive computable upper bounds on risk-sensitive quantities over a KLD ambiguity set. We then formulate an upper bound on the time rate of change of the KLD between the true nonlinear distribution and a Gaussian reference surrogate. Under some assumptions, this bound is controlled by decision variables within a covariance-steering formulation. The resulting constraints are incorporated into a sequential convex programming algorithm to design stochastic guidance policies that keep the true distribution close to its Gaussian surrogate while enforcing bounds on risk-sensitive performance measures. The proposed approach is demonstrated on a challenging nonlinear spacecraft transfer between two near-rectilinear halo orbits.
Trevor N. Wolf, Jay W. McMahon· arXiv.org· 0 citations
Many statistical models involve parameter-dependent normalizing constants that are computationally intractable, creating substantial obstacles to standard Bayesian inference. Although existing likelihood-based algorithms can often circumvent these constants, their uncertainty quantification may be poorly calibrated under model misspecification. To address these challenges, we propose SME-BETEL, a semiparametric Bayesian framework that combines score matching estimating equations with Bayesian exponentially tilted empirical likelihood. The resulting posterior avoids evaluation of normalizing constants and does not require learning-rate calibration. We establish consistency and asymptotic normality of the score matching estimator, and prove a Bernstein-von Mises theorem for the SME-BETEL posterior. These results show that SME-BETEL credible sets are asymptotically calibrated to the sampling variability of the score matching estimator, yielding valid frequentist coverage under model misspecification. We further develop a new score matching criterion for mixed-domain data, extending SME-BETEL to models whose observations combine components from different sample spaces. This construction enables robust Bayesian inference for mixed-domain doubly-intractable models, including preferential sampling models with an intractable spatial normalizing integral. Simulation studies show that SME-BETEL remains competitive under correct specification and substantially improves uncertainty quantification under misspecification. An ozone-monitoring application further demonstrates the utility of the mixed-domain construction for spatial preferential modeling.
Jiong-Ran Wang, D. Pati, A. Bhattacharya· 0 citations
Data-driven distributionally robust optimization (DRO) typically treats the conditional outcome law as fixed and uses ambiguity sets to capture estimation error. This paper studies latent distributional heterogeneity, where each instance has an unobserved law but contributes only one observation, so uncertainty persists even if the mixture law is known. We propose Conformal-DRO, which uses nested conformal regions to construct an ambiguity set for the future latent law. Under exchangeability, the set covers this law with probability at least $1-\alpha$ in finite samples, without estimating underlying latent laws or their mixing mechanism. The conformal path induces a data-driven transport geometry, while $\alpha$ determines the radius. The worst-case problem reduces to a finite linear program over conformal shells and admits sparse adversarial solutions. The resulting robust value provides a finite-sample certificate for the selected decision's expected cost.
A weeklong summer workshop brought higher education faculty to campus to explore how AI and machine learning materials can be adapted for their classrooms.
