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 -accurate solution with O N −2/3 + −2 first-order oracle queries to individual loss functions. Compared to existing algorithms for this problem, we improve complexity by a factor of up to −4/3 .

We develop a variant of the Monteiro-Svaiter (MS) acceleration framework that removes the need to solve an expensive implicit equation at every iteration. Consequently, for any p ≥ 2 we improve the complexity of convex optimization with Lipschitz pth derivative by a logarithmic factor, matching a lower bound. We also introduce an MS subproblem solver that requires no knowledge of problem parameters, and implement it as either a second-or first-order method via exact linear system solution or MinRes, respectively. On logistic regression our method outperforms previous second-order acceleration schemes, but under-performs Newton's method; simply iterating our first-order adaptive subproblem solver performs comparably to L-BFGS.∞ regression [8,12], minimizing functions with Hölder continuous higher derivatives [40], and distributionally-robust optimization [13,11].