Beyond Minimax Rates in Group Distributionally Robust Optimization via a Novel Notion of Sparsity

The minimax sample complexity of group distributionally robust optimization (GDRO) has been determined up to a $\log(K)$ factor, for $K$ the number of groups. In this work, we venture beyond the minimax perspective via a novel notion of sparsity that we dub $(\lambda, \beta)$-sparsity. In short, this condition means that at any parameter $\theta$, there is a set of at most $\beta$ groups whose risks at $\theta$ all are at least $\lambda$ larger than the risks of the other groups. To find an $\epsilon$-optimal $\theta$, we show via a novel algorithm and analysis that the $\epsilon$-dependent term in the sample complexity can swap a linear dependence on $K$ for a linear dependence on the potentially much smaller $\beta$. This improvement leverages recent progress in sleeping bandits, showing a fundamental connection between the two-player zero-sum game optimization framework for GDRO and per-action regret bounds in sleeping bandits. The aforementioned result assumes having a particular $\lambda$ as input. Perhaps surprisingly, we next show an adaptive algorithm which, up to log factors, gets sample complexity that adapts to the best $(\lambda, \beta)$-sparsity condition that holds. Finally, for a particular input $\lambda$, we also show how to get a dimension-free sample complexity result.