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Accelerated Affine-Invariant Convergence Rates of the Frank-Wolfe Algorithm with Open-Loop Step-Sizes
Elias Wirth, Javier Pena, Sebastian Pokutta
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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) rate. In particular, when the minimizer of a strongly convex function over a polytope lies in the relative interior of a feasible region face, the FW with open-loop step-sizes ηₜ = ℓ/(t+ℓ) for ℓ∈ℕ_(≥2) has accelerated convergence 𝒪(t⁻²) in contrast to the rate Ω(t^(-1-ϵ)) 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 (Wirth et al., 2023b) and open-loop step-sizes (Pena, 2021). In particular, we extend all known non-affine-invariant convergence rates for FW with open-loop step-sizes to affine-invariant results.
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