Lukas-Benedikt Fiechtner

Lukas-Benedikt Fiechtner, Jose Blanchet

We study, to our knowledge, the first tractable multistage ex-ante distributionally robust regret optimization (DRRO) formulation for stochastic control. We consider finite-horizon LQR under common stage-law ambiguity: disturbances are independent across time but share an unknown stage law whose mean and covariance lie in a Gelbrich ball around nominal parameters. Unlike the single-stage quadratic case, the nominal certainty-equivalent (CE) controller is generally not regret-optimal, because reuse of the stage law makes past disturbances informative for future decisions. Despite the general NP-hardness of DRRO, we show that over linear disturbance-feedback policies the resulting multistage DRRO-LQR problem admits an exact semidefinite programming reformulation. The optimal controller is the nominal certainty-equivalent LQR law plus a strictly causal empirical-mean correction. We also characterize worst-case distributions and show that those for the DRRO-optimal policy are nonunique. Numerical results show that, relative to the corresponding DRO controller under the same ambiguity set, DRRO is often substantially less conservative while preserving the intended regret guarantee, and that its correction coefficients empirically approach the certainty-equivalent feedforward coefficient.

Lukas-Benedikt Fiechtner, Jose Blanchet

Distributionally Robust Optimization (DRO) is a popular framework for decision-making under uncertainty, but its adversarial nature can lead to overly conservative solutions. To address this, we study ex-ante Distributionally Robust Regret Optimization (DRRO), focusing on Wasserstein-based ambiguity sets which are popular due to their links to regularization and machine learning. We provide a systematic analysis of Wasserstein DRRO, paralleling known results for Wasserstein DRO. Under smoothness and regularity conditions, we show that Wasserstein DRRO coincides with Empirical Risk Minimization (ERM) up to first-order terms, and exactly so in convex quadratic settings. We revisit the Wasserstein DRRO newsvendor problem, where the loss is the maximum of two linear functions of demand and decision. Extending [25], we show that the regret can be computed by maximizing two one-dimensional concave functions. For more general loss functions involving the maximum of multiple linear terms in multivariate random variables and decision vectors, we prove that computing the regret and thus also the DRRO policy is NP-hard. We then propose a convex relaxation for these more general Wasserstein DRRO problems and demonstrate its strong empirical performance. Finally, we provide an upper bound on the optimality gap of our relaxation and show it improves over recent alternatives.