2021/06/12 by Tobias Sutter, Li, Mengmeng, Daniel Kühn +2 · 2 citations
Computer Science · Decision Sciences · #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Optimization and Control (math.OC) #Risk and Portfolio Optimization #Simulation Techniques and Applications #Stochastic Gradient Optimization Techniques
paper · pdf · doi:10.48550/arxiv.2106.06741
openalex publication_date 2021/06/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study a stochastic program where the probability distribution of the uncertain problem parameters is unknown and only indirectly observed via finitely many correlated samples generated by an unknown Markov chain with d states. We propose a data-driven distributionally robust optimization model to estimate the problem's objective function and optimal solution. By leveraging results from large deviations theory, we derive statistical guarantees on the quality of these estimators. The underlying worst-case expectation problem is nonconvex and involves \mathcal O(d2) decision variables. Thus, it cannot be solved efficiently for large d. By exploiting the structure of this problem, we devise a customized Frank-Wolfe algorithm with convex direction-finding subproblems of size \mathcal O(d). We prove that this algorithm finds a stationary point efficiently under mild conditions. The efficiency of the method is predicated on a dimensionality reduction enabled by a dual reformulation. Numerical experiments indicate that our approach has better computational and statistical properties than the state-of-the-art methods.