We design algorithms for minimizing $\max_{i\in[n]} f_i(x)$ over a $d$-dimensional Euclidean or simplex domain. When each $f_i$ is $1$-Lipschitz and $1$-smooth, our method computes an $\epsilon$-approximate solution using $\widetilde{O}(n \epsilon^{-1/3} + \epsilon^{-2})$ gradient and function evaluations, and $\widetilde{O}(n \epsilon^{-4/3})$ additional runtime. For large $n$, our evaluation complexity is optimal up to polylogarithmic factors. In the special case where each $f_i$ is linear -- which corresponds to finding a near-optimal primal strategy in a matrix game -- our method finds an $\epsilon$-approximate solution in runtime $\widetilde{O}(n (d/\epsilon)^{2/3} + nd + d\epsilon^{-2})$. For $n>d$ and $\epsilon=1/\sqrt{n}$ this improves over all existing first-order methods. When additionally $d = \omega(n^{8/11})$ our runtime also improves over all known interior point methods. Our algorithm combines three novel primitives: (1) A dynamic data structure which enables efficient stochastic gradient estimation in small $\ell_2$ or $\ell_1$ balls. (2) A mirror descent algorithm tailored to our data structure implementing an oracle which minimizes the objective over these balls. (3) A simple ball oracle acceleration framework suitable for non-Euclidean geometry.

翻译：我们设计算法以最小化 $\max_{i\in[n]} f_i(x)$，其定义域为 $d$ 维欧氏空间或单纯形域。当每个 $f_i$ 均为 $1$-Lipschitz 且 $1$-光滑时，我们的方法通过 $\widetilde{O}(n \epsilon^{-1/3} + \epsilon^{-2})$ 次梯度与函数评估，以及 $\widetilde{O}(n \epsilon^{-4/3})$ 额外运行时间，即可计算出 $\epsilon$-近似解。对于大规模 $n$，我们的评估复杂度在多项式对数因子内达到最优。在特殊情形下，即每个 $f_i$ 为线性函数——对应矩阵博弈中寻找近似最优原始策略——我们的方法可在 $\widetilde{O}(n (d/\epsilon)^{2/3} + nd + d\epsilon^{-2})$ 运行时间内求得 $\epsilon$-近似解。当 $n>d$ 且 $\epsilon=1/\sqrt{n}$ 时，该结果优于所有现有的一阶方法。此外，若 $d = \omega(n^{8/11})$，我们的运行时间亦优于所有已知的内点方法。本算法融合三项新型原语：(1) 一种动态数据结构，支持在小的 $\ell_2$ 或 $\ell_1$ 球内进行高效随机梯度估计；(2) 一种适配该数据结构的镜像下降算法，实现最小化这些球内目标函数的预言机；(3) 一种适用于非欧几何的简单球预言机加速框架。