  We characterize the complexity of minimizing $\max_{i\in[N]} f_i(x)$ for
convex, Lipschitz functions $f_1,\ldots, f_N$. For non-smooth functions,
existing methods require $O(N\epsilon^{-2})$ queries to a first-order oracle to
compute an $\epsilon$-suboptimal point and $\tilde{O}(N\epsilon^{-1})$ queries
if the $f_i$ are $O(1/\epsilon)$-smooth. We develop methods with improved
complexity bounds of $\tilde{O}(N\epsilon^{-2/3} + \epsilon^{-8/3})$ in the
non-smooth case and $\tilde{O}(N\epsilon^{-2/3} + \sqrt{N}\epsilon^{-1})$ in
the $O(1/\epsilon)$-smooth case. Our methods consist of a recently proposed
ball optimization oracle acceleration algorithm (which we refine) and a careful
implementation of said oracle for the softmax function. We also prove an oracle
complexity lower bound scaling as $\Omega(N\epsilon^{-2/3})$, showing that our
dependence on $N$ is optimal up to polylogarithmic factors.
