Consider an oracle which takes a point $x$ and returns the minimizer of a convex function $f$ in an $\ell_2$ 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 $\ell_2$ 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 $\ell_p$ regression.

中文翻译：

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使用 Ball 优化 Oracle 加速

考虑一个预言机，它接受一个点 $x$ 并返回一个在 $x$ 周围半径为 $r$ 的 $\ell_2$ 球中的凸函数 $f$ 的最小值。很容易证明，大约 $r^{-1}\log\frac{1}{\epsilon}$ 调用 oracle 就足以在 $\ell_2 中找到 $f$ 的 $\epsilon$-近似最小值$单位球。也许令人惊讶的是，这不是最优的：我们设计了一个加速算法，该算法通过大约 $r^{-2/3} \log \frac{1}{\epsilon}$ oracle 查询获得 $\epsilon$-近似最小化器，并且给出一个匹配的下限。此外，我们使用牛顿方法的变体为具有局部稳定 Hessian 的函数实现了球优化预言机。由此产生的算法适用于许多具有实践和理论意义的问题，