We develop algorithms for the optimization of convex objectives that have H\"older continuous $q$-th derivatives with respect to a $p$-norm by using a $q$-th order oracle, for $p, q \geq 1$. We can also optimize other structured functions. We do this by developing a non-Euclidean inexact accelerated proximal point method that makes use of an inexact uniformly convex regularizer. We also provide nearly matching lower bounds for any deterministic algorithm that interacts with the function via a local oracle.

翻译：针对具有关于$p$-范数的H\"older连续$q$阶导数的凸目标函数（其中$p, q \geq 1$），我们利用$q$阶预言机开发了相应的优化算法。该方法同样适用于其他结构化函数的优化。我们通过构建一种非欧几里得不精确加速近端点方法实现这一目标，该方法采用了不精确一致凸正则化器。此外，我们为任何通过局部预言机与函数交互的确定性算法提供了近乎匹配的下界。