Data-Driven Stochastic Optimization Using Upper Confidence Bounds: Performance Guarantees and Distributional Robustness

A data-driven approach for stochastic optimization should preferably offer a finite-sample performance guarantee as well as asymptotic optimality. However, existing approaches that enjoy both, mostly based on distributionally robust optimization (DRO), are often difficult to use as they require an ambiguity set containing the true distribution of uncertainty with confidence, which is difficult to obtain. The main contributions of this paper are two-fold. First, we propose a data-driven stochastic optimization approach that offers a finite-sample guarantee and asymptotic optimality without resorting to an ambiguity set. The core idea of our approach is to minimize an approximation to the true expected cost function derived using only a half of the sample data, and compute an upper confidence bound for the true expected cost using the other half. Second, we identify a case in which the proposed approach is tractable and accordingly design an algorithm for it. Simulation results demonstrate that our approach can be less conservative than existing DRO methods in terms of not only the finite-sample guarantee but also the true expected cost for a wide range of frequently used confidence levels.