From Data to Decisions: Distributionally Robust Optimization Is Optimal

Abstract

We study stochastic programs where the decision maker cannot observe the distribution of the exogenous uncertainties but has access to a finite set of independent samples from this distribution. In this setting, the goal is to find a procedure that transforms the data to an estimate of the expected cost function under the unknown data-generating distribution, that is, a predictor, and an optimizer of the estimated cost function that serves as a near-optimal candidate decision, that is, a prescriptor. As functions of the data, predictors and prescriptors constitute statistical estimators. We propose a meta-optimization problem to find the least conservative predictors and prescriptors subject to constraints on their out-of-sample disappointment. The out-of-sample disappointment quantifies the probability that the actual expected cost of the candidate decision under the unknown true distribution exceeds its predicted cost. Leveraging tools from large deviations theory, we prove that this meta-optimization problem admits a unique solution: The best predictor-prescriptor-pair is obtained by solving a distributionally robust optimization problem over all distributions within a given relative entropy distance from the empirical distribution of the data. This paper was accepted by Chung Piaw Teo, optimization.

Data-driven stochastic optimization for distributional ambiguity with integrated confidence region

“…Thirdly, we can bound the expected value, over all observations and all possible distributions, of the optimal objective function of the proposed stochastic optimization problem for a finite number of observations by a constant, and accordingly the asymptotic analysis yields the convergence of this constant to zero. This stands in contrast to the finite sample size results from Van Parys et al [54], for example their Theorem 8, in that they assume the asymptotic distribution P ∞ (i.e., the sample path distribution) while our results hold for all a-priori distributions, cf. Theorem 1.…”

“…Van Parys et al [54] also deal with data-driven distribution optimization and make a special kind of asymptotic optimal decision. Similar to our work, they also provide a series of finite sample size results.…”

Distributionally Robust Optimization: A review on theory and applications

“…Recently, Parys et al [171] gave a strong conclusion that is reflected in the article entitled "From data to decisions: distributionally robust optimization is optimal". They proved that, by solving a distributionally robust optimization problem over P φ−KL , the best data-driven decision can be obtained with guarantees of the best out-of-sample performance.…”

Probabilistic Guarantees in Robust Optimization

“…Distributionally robust optimization: Probabilistic guarantees and concentration results have been the fundamental building blocks in the distributionally robust optimization (DRO) literature. In the DRO setting, reformulations of ambiguous chance constraints (3) have been derived in cases where the ambiguity set P is constructed from bounds on moments of the distribution [18,24,48,45], or is defined as a ball around the empirical distribution according to phi-divergence [46], the f-divergence [35], the Wasserstein distance [23,15,21,42,32], or the relative entropy [40]. We refer to [28] and references therein for a comprehensive review.…”