Achieving Robust Data-driven Contextual Decision Making in a Data Augmentation Way

Abstract: This paper focuses on the contextual optimization problem where a decision is subject to some uncertain parameters and covariates that have some predictive power on those parameters are available before the decision is made. More specifically, we focus on solving the Wasserstein-distance-based distributionally robust optimization (DRO) model for the problem, which maximizes the worst-case expected objective over an uncertainty set including all distributions closed enough to a nominal distribution with respect to the Wasserstein distance. We develop a stochastic gradient descent algorithm based on the idea of data augmentation to solve the model efficiently. The algorithm iteratively a) does a bootstrapping sample from the nominal distribution; b) perturbs the adversarially and c) updates decisions. Accordingly, the computational time of the algorithm is only determined by the number of iterations and the complexity of computing the gradient of a single sample. Except for efficiently solving the model, the algorithm provide additional advantages that the proposed algorithm can cope with any nominal distributions and therefore is extendable to solve the problem in an online setting. We also prove that the algorithm converges to the optimal solution of the DRO model at a rate of a $O(1/\sqrt{T})$, where $T$ is the number of iterations of bootstrapping. Consequently, the performance guarantee of the algorithm is that of the DRO model plus $O(1/\sqrt{T})$. Through extensive numerical experiments, we demonstrate the superior performance of the proposed algorithm to several benchmarks.