Machine learning algorithms minimizing average risk are susceptible to
distributional shifts. Distributionally Robust Optimization (DRO) addresses
this issue by optimizing the worst-case risk within an uncertainty set.
However, DRO suffers from over-pessimism, leading to low-confidence
predictions, poor parameter estimations as well as poor generalization. In this
work, we conduct a theoretical analysis of a probable root cause of
over-pessimism: excessive focus on noisy samples. To alleviate the impact of
noise, we incorporate data geometry into calibration terms in DRO, resulting in
our novel Geometry-Calibrated DRO (GCDRO) for regression. We establish the
connection between our risk objective and the Helmholtz free energy in
statistical physics, and this free-energy-based risk can extend to standard DRO
methods. Leveraging gradient flow in Wasserstein space, we develop an
approximate minimax optimization algorithm with a bounded error ratio and
elucidate how our approach mitigates noisy sample effects. Comprehensive
experiments confirm GCDRO's superiority over conventional DRO methods.

通过灵活利用数据的几何形态与最小风险优化算法相结合，本研究提出了一种新型的校准方法 —— 几何校准分布鲁棒优化 (Geometry-Calibrated DRO，GCDRO)，该方法减轻了噪声样本的影响，理论分析和广泛实验证明了其相较于常规分布鲁棒优化方法的优越性。