DISTRIBUTIONALLY ROBUST GAUSSIAN PROCESS REGRESSION AND BAYESIAN INVERSE PROBLEMS

• 成果类型: Article
• 署名作者: Zhang, Xuhui; Blanchet, Jose; Marzouk, Youssef; Nguyen, Viet Anh; Wang, Sven
• 署名单位: Stanford University; Massachusetts Institute of Technology (MIT); Chinese University of Hong Kong; Humboldt University of Berlin
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/24-AAP2141
• 发表日期: 2025
• 页码: 1489-1530
• 关键词: approximations; uncertainty

摘要

We study a distributionally robust optimization formulation (i.e., a min-max game) for two representative problems in Bayesian nonparametric estimation: Gaussian process regression and, more generally, linear inverse problems. Our formulation seeks the best mean-squared error predictor in an infinite-dimensional space against an adversary who chooses the worst-case model in a Wasserstein ball around a nominal infinite-dimensional Bayesian model. The transport cost is chosen to control features such as the degree of roughness of the sample paths that the adversary is allowed to inject. We show that strong duality holds in the sense that max-min equals min-max, and that there exists a unique Nash equilibrium that can be computed by a sequence of finite-dimensional approximations. Crucially, the worst-case distribution is itself Gaussian. We explore the properties of the Nash equilibrium and the effects of hyperparameters through numerical experiments, demonstrating the versatility of our modeling framework.