Acceleration with a Ball Optimization Oracle

Consider an oracle which takes a point $x$ and returns the minimizer of a convex function $f$ in an $\ell2$ ball of radius $r$ around $x$. It is straightforward to show that roughly $r^{{-1}\log\frac{1}{\epsilon}$} calls to the oracle suffice to find an $\epsilon$-approximate minimizer of $f$ in an $\ell2$ unit ball. Perhaps surprisingly, this is not optimal: we design an accelerated algorithm which attains an $\epsilon$-approximate minimizer with roughly $r^{-2/3} \log \frac{1}{\epsilon}$ oracle queries, and give a matching lower bound. Further, we implement ball optimization oracles for functions with locally stable Hessians using a variant of Newton's method. The resulting algorithm applies to a number of problems of practical and theoretical import, improving upon previous results for logistic and $\ell\infty$ regression and achieving guarantees comparable to the state-of-the-art for $\ellp$ regression.