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

TL;DR

This paper introduces two innovative data structures to accelerate the Frank-Wolfe optimization algorithm, significantly reducing preprocessing and iteration costs, and advancing the state-of-the-art in conditional gradient methods.

Contribution

The paper presents two novel inner product search data structures that improve the efficiency of the Frank-Wolfe algorithm over previous methods.

Findings

First data structure reduces preprocessing and iteration costs.

Second data structure offers faster preprocessing and is ideal for fewer iterations.

Both methods outperform prior algorithms in various scenarios.

Abstract

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 $\tilde O(nd^{\omega-1}+dn^{1+o(1)})$ and per iteration cost $\tilde O(d+n^\rho)$ for small constant $\rho$. * 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 $\tilde O(dn^\rho)$) in all cases, while the second one provides an even faster preprocessing time and is suitable when the number of iterations is small.