  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.
