Extend adaptive-step asymptotics beyond the proven gain range

Extend the implicit-bias and exact-asymptotic theorem for the adaptive schedule \(\eta_t=\kappa |w_t|\) from \(0<\kappa\le1/4\) to the remaining cone-invariant range \(1/4<\kappa\le1/2\).

Background

For the symmetric two-neuron ReLU classifier, the schedule ηt=κ∣wt∣\eta_t=\kappa|w_t| preserves the active cone for every 0<κ≤1/20<\kappa\le1/2. The paper proves convergence rates, implicit bias, and exact loss asymptotics only for 0<κ≤1/40<\kappa\le1/4. The authors explicitly identify extending these asymptotic results to the larger cone-invariant range as unresolved.

References

Cone invariance holds up to \kappa=1/2 by \Cref{lem:invariance}; extending the asymptotic theorem to 1/4<\kappa\le1/2 remains open.

— A Minimal ReLU Classifier on Non-Linearly Separable Data: Convergence and Implicit Bias at and beyond the Edge of Stability  (2610.23007 - Nguyen et al., 16 Oct 2026) in Section 5, subsection “Adaptive Learning Rate”