## Abstract

Consider an oracle which takes a point x and returns the minimizer of a convex function f in an l_{2} 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 l_{2} 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 l_{8} regression and achieving guarantees comparable to the state-of-the-art for l_{p} regression.

Original language | English |
---|---|

Journal | Advances in Neural Information Processing Systems |

Volume | 2020-December |

State | Published - 2020 |

Event | 34th Conference on Neural Information Processing Systems, NeurIPS 2020 - Virtual, Online Duration: 6 Dec 2020 → 12 Dec 2020 |