It has been well documented that high dimensional data approximately resides on low dimensional manifolds.

This paper investigates group distributionally robust optimization (GDRO), with the purpose to learn a model that performs well over $m$ different distributions. First, we formulate GDRO as a stochastic convex-concave saddle-point problem, and demonstrate that stochastic mirror descent (SMD), using $m$ samples in each iteration, achieves an $O(m (\log m)/\epsilon^2)$ sample complexity for finding an $\epsilon$-optimal solution, which matches the $\Omega(m/\epsilon^2)$ lower bound up to a logarithmic factor. Then, we make use of techniques from online learning to reduce the number of samples required in each round from $m$ to $1$, keeping the same sample complexity. Specifically, we cast GDRO as a two-players game where one player simply performs SMD and the other executes an online algorithm for non-oblivious multi-armed bandits. Next, we consider a more practical scenario where the number of samples that can be drawn from each distribution is different, and propose a novel formulation of weighted GDRO, which allows us to derive distribution-dependent convergence rates.

Distributionally Robust Optimization (DRO) serves as a robust alternative to empirical risk minimization (ERM), which optimizes the worst-case distribution in an uncertainty set typically specified by distance metrics including $f$-divergence and the Wasserstein distance. The metrics defined in the ostensible high dimensional space lead to exceedingly large uncertainty sets, resulting in the underperformance of most existing DRO methods. It has been well documented that high dimensional data approximately resides on low dimensional manifolds. In this work, to further constrain the uncertainty set, we incorporate data geometric properties into the design of distance metrics, obtaining our novel Geometric Wasserstein DRO (GDRO). Empowered by Gradient Flow, we derive a generically applicable approximate algorithm for the optimization of GDRO, and provide the bounded error rate of the approximation as well as the convergence rate of our algorithm. We also theoretically characterize the edge cases where certain existing DRO methods are the degeneracy of GDRO. Extensive experiments justify the superiority of our GDRO to existing DRO methods in multiple settings with strong distributional shifts, and confirm that the uncertainty set of GDRO adapts to data geometry.

While being straightforward, this result seems new for GDRO.

We then use these approximate subclass labels as a form of noisy supervision in a distributionally robust optimization objective.

It has been well documented that high dimensional data approximately resides on low dimensional manifolds.