New preprint gives a unified theory of H-duality, explaining why reversing stepsizes yields new optimizers
prof_grimmer · x · 2026-09-06
A new arXiv preprint by Kevin Shu and Alex L. Wang provides two complementary explanations of H-duality in first-order methods — the phenomenon where reversing the stepsize order of a fixed-step method yields a paired method with matching worst-case guarantees. The primal explanation: on instance classes with extremal curvature properties (e.g. Huber-type functions), the final iterate is a scalar multiple of the initial point, invariant under time reversal. The dual explanation: when H-duality holds, a simple algebraic transformation converts performance estimation (PEP) certificates, letting researchers automatically convert final-suboptimality guarantees into final-gradient-norm guarantees. First observed by Kim et al., the phenomenon led to new momentum methods like OGM-G; the analysis applies to both gradient methods and contractive fixed-point operators.
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