A Whole New Ball Game: A Primal Accelerated Method for Matrix Games and Minimizing the Maximum of Smooth Functions

• URL: http://arxiv.org/abs/2311.10886v1
• Date: Fri, 17 Nov 2023 22:07:18 GMT
• Title: A Whole New Ball Game: A Primal Accelerated Method for Matrix Games and Minimizing the Maximum of Smooth Functions
• Authors: Yair Carmon, Arun Jambulapati, Yujia Jin, Aaron Sidford
• Abstract summary: We design algorithms for minimizing $max_iin[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.
• Score: 44.655316553524855
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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