A Unified Noise-Curvature View of Loss of Trainability

Loss of trainability refers to a phenomenon in continual learning where parameter updates no longer make progress on the optimization objective, so accuracy stalls or degrades as the learning problem changes over time. In this paper, we analyze loss of trainability through an optimization lens and find that the phenomenon is not reliably predicted by existing individual indicators such as Hessian rank, sharpness level, weight or gradient norms, gradient-to-parameter ratios, and unit-sign entropy. Motivated by our analysis, we introduce two complementary indicators: a batch-size-aware gradient-noise bound and a curvature volatility-controlled bound. We then combine these two indicators into a per-layer adaptive noise threshold on the effective step-size that anticipates trainability behavior. Using this insight, we propose a step-size scheduler that keeps each layer's effective parameter update below this bound, thereby avoiding loss of trainability. We demonstrate that our scheduler can improve the accuracy maintained by previously proposed approaches, such as concatenated ReLU (CReLU), Wasserstein regularizer, and L2 weight decay. Surprisingly, our scheduler produces adaptive step-size trajectories that, without tuning, mirror the manually engineered step-size decay schedules.