Geometry-Calibrated DRO: Combating Over-Pessimism with Free Energy Implications

Distributionally Robust Optimization (DRO) optimizes the worst-case risk within an uncertainty set to resist distribution shifts. However, DRO suffers from over-pessimism, leading to low-confidence predictions, poor parameter estimations as well as poor generalization in practice. In this work, we uncover one 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) \emph{for regression}. We establish that our risk objective aligns with the Helmholtz free energy in statistical physics, which could 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.