Accelerating Frank-Wolfe Algorithm using Low-Dimensional and Adaptive Data Structures

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(ndω-1+dn1+o(1)) and per iteration cost tilde O(d+nρ) 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(d2n1+o(1)) and per iteration cost tilde O(dnρ)) in all cases, while the second one provides an even faster preprocessing time and is suitable when the number of iterations is small.