Accelerated affine-invariant convergence rates of the Frank-Wolfe algorithm with open-loop step-sizes

Recent papers have shown that the Frank-Wolfe algorithm (FW) with open-loop step-sizes exhibits rates of convergence faster than the iconic O(t-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}(t<^>{-1})$$\end{document} rate. In particular, when the minimizer of a strongly convex function over a polytope lies on the boundary of the polytope, the FW algorithm with open-loop step-sizes eta t=& ell;t+& ell;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta _t = \frac{\ell }{t+\ell }$$\end{document} for & ell;is an element of N >= 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \in \mathbb {N}_{\ge 2}$$\end{document} has accelerated convergence O(t-2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}(t<^>{-2})$$\end{document} in contrast to the rate Omega(t-1-& varepsilon;)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega (t<^>{-1-\epsilon })$$\end{document} attainable with more complex line-search or short-step step-sizes. Given the relevance of this scenario in data science problems, research has grown to explore the settings enabling acceleration in open-loop FW. However, despite FW's well-known affine invariance, existing acceleration results for open-loop FW are affine-dependent. This paper remedies this gap in the literature, by merging two recent research trajectories: affine invariance (Pe & ntilde;a in SIAM J. Optim. 33(4):2654-2674, 2023) and open-loop step-sizes (Wirth et al. in Proceedings of the International Conference on Artificial Intelligence and Statistics, 2023). In particular, we extend all known non-affine-invariant convergence rates for FW with open-loop step-sizes to affine-invariant results.