Tightness conjecture for gradient descent with variable mid-range stepsizes
Let with , and consider gradient-descent iterations from with stepsizes satisfying for . Variable-mid-range-stepsize tightness conjecture. The convergence-rate bound from the preceding theorem is tight:
This claim would establish exact worst-case convergence rates in the nonconstant mid-range stepsize regime. The paper says it is supported by numerical verification of the interpolation conditions, but gives no proof, so it remains open.
References
Primary source
Teodor Rotaru, François Glineur and Panagiotis Patrinos, “Exact worst-case convergence rates of gradient descent: a complete analysis for all constant stepsizes over nonconvex and convex functions”, arXiv:2406.17506 (2026).
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