arXiv:2608.15202 2026 Dynamics 1 ideas extracted · analyzed Sep 1, 2026
The paper replaces one-step error ratios with comparisons between errors separated by p iterations, targeting convergent sequences whose intermediate iterates oscillate or cycle. This is potentially useful for diagnosing optimizers with momentum, alternating updates, block-coordinate schedules, or multi-step neural operators, where per-step loss can increase even while a cycle-level envelope contracts. The most practical transfer is a phase-aware convergence monitor that estimates p-step contraction online and adjusts learning rate only when cycle-level dynamics deteriorate. The expected impact is moderate because this is primarily a diagnostic and scheduling principle rather than a new optimizer.
Monitor optimizer convergence over a cycle of p updates instead of judging every update independently. Estimate the p-step contraction factor and effective convergence order from parameter or loss errors, then reduce learning rate only when the cycle-level contraction worsens, avoiding false alarms caused by alternating or oscillatory iterates.