Unifying fundamental principle of H-duality

Identify a unifying fundamental principle that explains H-duality across algorithmic settings, reconciling the self-duality of residual norms in fixed-point algorithms developed in this work with the previously established duality between function values and gradient norms in smooth convex minimization.

Background

The paper establishes a new H-duality for fixed-point algorithms, showing that OHM and its dual share identical convergence rates with respect to the same residual measure, contrasting with prior convex minimization H-duality where function values are dual to gradient norms.

The authors emphasize that, despite superficial similarities, a single underlying principle connecting these dualities is currently unknown.

References

While the H-duality theory of this work and that of share some superficial similarities, the unifying fundamental principle remains unknown, and finding one is an interesting subject of future research.

— Optimal Acceleration for Minimax and Fixed-Point Problems is Not Unique  (2404.13228 - Yoon et al., 2024) in Section 5 (H-duality for fixed-point algorithms)