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 1e 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 1e 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 lp regression.
|Advances in Neural Information Processing Systems
|Published - 2020
|34th Conference on Neural Information Processing Systems, NeurIPS 2020 - Virtual, Online
Duration: 6 Dec 2020 → 12 Dec 2020