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 l₂ ball of radius r around x. It is straightforward to show that roughly r^-1 log ¹_e calls to the oracle suffice to find an e-approximate minimizer of f in an l₂ unit ball. Perhaps surprisingly, this is not optimal: we design an accelerated algorithm which attains an e-approximate minimizer with roughly r^-2/3 log ¹_e 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 and, in certain cases, stochastic first-order methods. The resulting algorithm applies to a number of problems of practical and theoretical import, improving upon previous results for logistic and l₈ regression and achieving guarantees comparable to the state-of-the-art for l_p regression.