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arXiv · 2311.10886

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

Yair Carmon·Arun Jambulapati·Yujia Jin·Aaron Sidford
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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 $ε$-approximate solution using $\widetilde{O}(n ε^{-1/3} + ε^{-2})$ gradient and function evaluations, and $\widetilde{O}(n ε^{-4/3})$ 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 $ε$-approximate solution in runtime $\widetilde{O}(n (d/ε)^{2/3} + nd + dε^{-2})$. For $n>d$ and $ε=1/\sqrt{n}$ this improves over all existing first-order methods. When additionally $d = ω(n^{8/11})$ 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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Yair Carmon, Arun Jambulapati, Yujia Jin, Aaron Sidford. 2023-11-17. A Whole New Ball Game: A Primal Accelerated Method for Matrix Games and Minimizing the Maximum of Smooth Functions. https://arxiv.org/abs/2311.10886

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