We propose a randomized multiplicative weight update (MWU) algorithm for $\ell_{\infty}$ regression that runs in $\widetilde{O}\left(n^{2+1/22.5} \text{poly}(1/\epsilon)\right)$ time when $\omega = 2+o(1)$, improving upon the previous best $\widetilde{O}\left(n^{2+1/18} \text{poly} \log(1/\epsilon)\right)$ runtime in the low-accuracy regime. Our algorithm combines state-of-the-art inverse maintenance data structures with acceleration. In order to do so, we propose a novel acceleration scheme for MWU that exhibits {\it stabiliy} and {\it robustness}, which are required for the efficient implementations of the inverse maintenance data structures. We also design a faster {\it deterministic} MWU algorithm that runs in $\widetilde{O}\left(n^{2+1/12}\text{poly}(1/\epsilon)\right))$ time when $\omega = 2+o(1)$, improving upon the previous best $\widetilde{O}\left(n^{2+1/6} \text{poly} \log(1/\epsilon)\right)$ runtime in the low-accuracy regime. We achieve this by showing a novel stability result that goes beyond the previous known works based on interior point methods (IPMs). Our work is the first to use acceleration and inverse maintenance together efficiently, finally making the two most important building blocks of modern structured convex optimization compatible.

翻译：我们提出了一种用于$\ell_{\infty}$回归的随机化乘性权重更新（MWU）算法，当$\omega = 2+o(1)$时，其运行时间为$\widetilde{O}\left(n^{2+1/22.5} \text{poly}(1/\epsilon)\right)$，改进了先前在低精度机制下的最佳运行时间$\widetilde{O}\left(n^{2+1/18} \text{poly} \log(1/\epsilon)\right)$。我们的算法将最先进的逆维护数据结构与加速技术相结合。为此，我们提出了一种新颖的MWU加速方案，该方案展现出**稳定性**和**鲁棒性**，这是高效实现逆维护数据结构所必需的。我们还设计了一种更快的**确定性**MWU算法，当$\omega = 2+o(1)$时，其运行时间为$\widetilde{O}\left(n^{2+1/12}\text{poly}(1/\epsilon)\right))$，改进了先前在低精度机制下的最佳运行时间$\widetilde{O}\left(n^{2+1/6} \text{poly} \log(1/\epsilon)\right)$。我们通过展示一个超越先前基于内点法（IPMs）已知工作的新颖稳定性结果来实现这一点。我们的工作是首次将加速与逆维护高效地结合使用，最终使现代结构化凸优化的两个最重要构建模块得以兼容。