# A Multistep Frank-Wolfe Method

arXiv: https://arxiv.org/abs/2210.08110 · HTML: https://mlsanity.com/paper/2210.08110/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 11 references that are in the map.
- First posted on arXiv 14 Oct 2022.

## Builds on (most cited first, 4 of 4)

- [On the Global Linear Convergence of Frank-Wolfe Optimization Variants](https://mlsanity.com/paper/1511.05932/) (2015)
- [Faster Rates for the Frank-Wolfe Method over Strongly-Convex Sets](https://mlsanity.com/paper/1406.1305/) (2014)
- [The Approximate Duality Gap Technique: A Unified Theory of First-Order Methods](https://mlsanity.com/paper/1712.02485/) (2017)
- [New Analysis and Results for the Frank-Wolfe Method](https://mlsanity.com/paper/1307.0873/) (2013)

## Built on by (most cited first, 1 of 1)

- [Empirical evaluation of the Frank-Wolfe methods for constructing white-box adversarial attacks](https://mlsanity.com/paper/2512.10936/) (2025)

## Abstract (arXiv)

The Frank-Wolfe algorithm has regained much interest in its use in structurally constrained machine learning applications. However, one major limitation of the Frank-Wolfe algorithm is the slow local convergence property due to the zig-zagging behavior. We observe the zig-zagging phenomenon in the Frank-Wolfe method as an artifact of discretization, and propose multistep Frank-Wolfe variants where the truncation errors decay as O(Deltap), where p is the method's order. This strategy "stabilizes" the method, and allows tools like line search and momentum to have more benefits. However, our results suggest that the worst case convergence rate of Runge-Kutta-type discretization schemes cannot improve upon that of the vanilla Frank-Wolfe method for a rate depending on k. Still, we believe that this analysis adds to the growing knowledge of flow analysis for optimization methods, and is a cautionary tale on the ultimate usefulness of multistep methods.