Orateur
Description
We study contextual stochastic optimization problems in which the joint distribution of uncertain parameters and side information covariates is modeled as a mixture of Gaussians. In a data-driven setting, the parameters of this distribution are unknown and must be estimated from historical data. To mitigate the adverse effects of estimation errors and improve out-of-sample performance, we propose a distributionally robust optimization framework based on the Kullback–Leibler divergence. We show that centering the ambiguity set at the empirical conditional distribution, rather than the joint distribution, is crucial for ensuring meaningful solutions. We derive a finite-sample out-of-sample performance guarantee and prove that the optimal KL divergence radius scales at the same rate as the statistical learning rate of the estimated parameters, up to constant factors. We further develop an efficient solution scheme based on exponential conic programming and demonstrate the superior empirical performance of our method compared to existing approaches across several real-world contextual optimization settings.
