Statistical Inference for Conditional Group Distributionally Robust Optimization with Cross-Entropy Loss

• URL: http://arxiv.org/abs/2507.09905v1
• Date: Mon, 14 Jul 2025 04:21:23 GMT
• Title: Statistical Inference for Conditional Group Distributionally Robust Optimization with Cross-Entropy Loss
• Authors: Zijian Guo, Zhenyu Wang, Yifan Hu, Francis Bach,
• Abstract summary: We study multi-source unsupervised domain adaptation, where labeled data are drawn from multiple source domains and only unlabeled data from a target domain.<n>We propose a novel Conditional Conditional Optimization (CG-DRO) framework that learns a classifier by minimizing the worst-case cross-entropy loss over the convex combinations of the conditional outcome distributions from the sources.<n>We establish fast statistical convergence rates for the estimator by constructing two surrogate minimax optimization problems that serve as theoretical bridges.
• Score: 9.054486124506521
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In multi-source learning with discrete labels, distributional heterogeneity across domains poses a central challenge to developing predictive models that transfer reliably to unseen domains. We study multi-source unsupervised domain adaptation, where labeled data are drawn from multiple source domains and only unlabeled data from a target domain. To address potential distribution shifts, we propose a novel Conditional Group Distributionally Robust Optimization (CG-DRO) framework that learns a classifier by minimizing the worst-case cross-entropy loss over the convex combinations of the conditional outcome distributions from the sources. To solve the resulting minimax problem, we develop an efficient Mirror Prox algorithm, where we employ a double machine learning procedure to estimate the risk function. This ensures that the errors of the machine learning estimators for the nuisance models enter only at higher-order rates, thereby preserving statistical efficiency under covariate shift. We establish fast statistical convergence rates for the estimator by constructing two surrogate minimax optimization problems that serve as theoretical bridges. A distinguishing challenge for CG-DRO is the emergence of nonstandard asymptotics: the empirical estimator may fail to converge to a standard limiting distribution due to boundary effects and system instability. To address this, we introduce a perturbation-based inference procedure that enables uniformly valid inference, including confidence interval construction and hypothesis testing.

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