Tightness conjecture for gradient descent with variable mid-range stepsizes
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.
Sources & referencesView supporting material
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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