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Formal H-duality theory for smooth minimax optimization

Develop a formal H-duality theory for smooth convex-concave minimax optimization that rigorously defines the dual operation and establishes rate correspondences, explaining the observed H-dual relationship between Fast Extragradient (FEG) and Dual-FEG.

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Background

Section 6 shows that Dual-FEG and FEG are H-duals of each other and exhibit identical convergence rates, suggesting a deeper duality structure in smooth minimax optimization.

The authors explicitly state that formalizing this H-duality theory for minimax problems is left for future work.

References

This intriguing H-dual relationship and identical convergence rates of \ref{alg:feg} and \ref{alg:dual-feg} strongly indicate the possible existence of H-duality theory for smooth minimax optimization; we leave its formal treatment to future work.

— Optimal Acceleration for Minimax and Fixed-Point Problems is Not Unique (2404.13228 - Yoon et al., 20 Apr 2024) in Section 6 (Analysis of Dual-FEG for minimax problems)