An Effective Manifold-based Optimization Method for Distributionally Robust Classification

How to promote the robustness of existing deep learning models is a challenging problem for many practical classification tasks. Recently, Distributionally Robust Optimization (DRO) methods have shown promising potential to tackle this problem. These methods aim to construct reliable models by minimizing the worst-case risk within a local region (called "uncertainty set") around the empirical data distribution. However, conventional DRO methods tend to be overly pessimistic, leading to certain discrepancy between the real data distribution and the uncertainty set, which can degrade the classification performance. To address this issue, we propose a manifold-based DRO method that takes the geometric structure of training data into account for constructing the uncertainty set. Specifically, our method employs a carefully designed "game" that integrates contrastive learning with Jacobian regularization to capture the manifold structure, enabling us to solve DRO problems constrained by the data manifold. By utilizing a novel idea for approximating geodesic distance on manifolds, we also provide the theoretical guarantees for its robustness. Moreover, our proposed method is easy to implement in practice. We conduct a set of experiments on several popular benchmark datasets, where the results demonstrate our advantages in terms of accuracy and robustness.