低维自适应数据结构加速Frank-Wolfe算法
Accelerating Frank-Wolfe Algorithm using Low-Dimensional and Adaptive Data Structures
摘要：
在本文中，我们研究的问题，加快一类优化算法称为Frank-Wolfe，一个条件梯度法。我们开发并采用了两种新的内积搜索数据结构，改进了[Shrivastava，Song和Xu，NeurIPS 2021]中的现有最快算法。* 第一种数据结构使用低维随机投影将问题降维，然后使用高效的内积数据结构。它具有预处理时间$\tilde O(nd^{ω-1}+dn^{1+o(1)})$和每次迭代成本$\tilde O(d+n^ρ)$小常数$ρ$。* 第二数据结构利用了自适应内积搜索数据结构的最新发展，该数据结构可以输出对所有内积的估计。它具有预处理时间$\tilde O(nd)$和每次迭代成本$\tilde O(d+n)$。第一种算法在所有情况下都改进了现有技术（具有预处理时间$\tilde O(d^2n^{1+o(1)})$和每次迭代成本$\tilde O(dn^ρ)$），而第二种算法提供了甚至更快的预处理时间，并且在迭代次数小时是合适的。
英文摘要：
In this paper, we study the problem of speeding up a type of optimization algorithms called Frank-Wolfe, a conditional gradient method. We develop and employ two novel inner product search data structures, improving the prior fastest algorithm in [Shrivastava, Song and Xu, NeurIPS 2021]. * The first data structure uses low-dimensional random projection to reduce the problem to a lower dimension, then uses efficient inner product data structure. It has preprocessing time $\tilde O(nd^{ω-1}+dn^{1+o(1)})$ and per iteration cost $\tilde O(d+n^ρ)$ for small constant $ρ$. * The second data structure leverages the recent development in adaptive inner product search data structure that can output estimations to all inner products. It has preprocessing time $\tilde O(nd)$ and per iteration cost $\tilde O(d+n)$. The first algorithm improves the state-of-the-art (with preprocessing time $\tilde O(d^2n^{1+o(1)})$ and per iteration cost $\tilde O(dn^ρ)$) in all cases, while the second one provides an even faster preprocessing time and is suitable when the number of iterations is small.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
