In contextual optimization, a decision-maker observes historical samples of uncertain variables and associated concurrent covariates, without knowing their joint distribution. Given an additional covariate observation, the goal is to choose a decision that minimizes some operational costs. A prevalent issue here is covariate shift, where the marginal distribution of the new covariate differs from historical samples, leading to decision performance variations with nonparametric or parametric estimators. To address this, we propose a distributionally robust approach that uses an ambiguity set by the intersection of two Wasserstein balls, each centered on typical nonparametric or parametric distribution estimators. Computationally, we establish the tractable reformulation of this distributionally robust optimization problem. Statistically, we provide guarantees for our Wasserstein ball intersection approach under covariate shift by analyzing the measure concentration of the estimators. Furthermore, to reduce computational complexity, we employ a surrogate objective that maintains similar generalization guarantees. Through synthetic and empirical case studies on income prediction and portfolio optimization, we demonstrate the strong empirical performance of our proposed models.

The design of gaits for robot locomotion can be a daunting process which requires significant expert knowledge and engineering. This process is even more challenging for robots that do not have an accurate physical model, such as compliant or micro-scale robots. Data-driven gait optimization provides an automated alternative to analytical gait design. In this paper, we propose a novel approach to efficiently learn a wide range of locomotion tasks with walking robots. This approach formalizes locomotion as a contextual policy search task to collect data, and subsequently uses that data to learn multi-objective locomotion primitives that can be used for planning. As a proof-of-concept we consider a simulated hexapod modeled after a recently developed microrobot, and we thoroughly evaluate the performance of this microrobot on different tasks and gaits. Our results validate the proposed controller and learning scheme on single and multi-objective locomotion tasks. Moreover, the experimental simulations show that without any prior knowledge about the robot used (e.g., dynamics model), our approach is capable of learning locomotion primitives within 250 trials and subsequently using them to successfully navigate through a maze.