Statistical Analysis of Conditional Group Distributionally Robust Optimization with Cross-Entropy Loss
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| Format: | Preprint |
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2025
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| _version_ | 1866908783587885056 |
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| author | Guo, Zijian Wang, Zhenyu Hu, Yifan Bach, Francis |
| author_facet | Guo, Zijian Wang, Zhenyu Hu, Yifan Bach, Francis |
| contents | 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 available from multiple source domains and only unlabeled data are observed from the 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 sources domains. We develop an efficient Mirror Prox algorithm for solving the minimax problem and employ a double machine learning procedure to estimate the risk function, ensuring that errors in nuisance estimation contribute only at higher-order rates. We establish fast statistical convergence rates for the empirical CG-DRO 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 CG-DRO 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_09905 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Statistical Analysis of Conditional Group Distributionally Robust Optimization with Cross-Entropy Loss Guo, Zijian Wang, Zhenyu Hu, Yifan Bach, Francis Methodology Optimization and Control Statistics Theory Machine Learning 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 available from multiple source domains and only unlabeled data are observed from the 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 sources domains. We develop an efficient Mirror Prox algorithm for solving the minimax problem and employ a double machine learning procedure to estimate the risk function, ensuring that errors in nuisance estimation contribute only at higher-order rates. We establish fast statistical convergence rates for the empirical CG-DRO 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 CG-DRO 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. |
| title | Statistical Analysis of Conditional Group Distributionally Robust Optimization with Cross-Entropy Loss |
| topic | Methodology Optimization and Control Statistics Theory Machine Learning |
| url | https://arxiv.org/abs/2507.09905 |