  We develop and analyze algorithms for distributionally robust optimization
(DRO) of convex losses. In particular, we consider group-structured and bounded
$f$-divergence uncertainty sets. Our approach relies on an accelerated method
that queries a ball optimization oracle, i.e., a subroutine that minimizes the
objective within a small ball around the query point. Our main contribution is
efficient implementations of this oracle for DRO objectives. For DRO with $N$
non-smooth loss functions, the resulting algorithms find an $\epsilon$-accurate
solution with $\widetilde{O}\left(N\epsilon^{-2/3} + \epsilon^{-2}\right)$
first-order oracle queries to individual loss functions. Compared to existing
algorithms for this problem, we improve complexity by a factor of up to
$\epsilon^{-4/3}$.
