Weight-norm Criticality: A Mechanism for Loss Spikes Induced by the Normalization and Weight Decay

Most explanations of training instability focus on learning-rate criticality, typically characterized by the Edge of Stability, beyond which optimization becomes unstable. We argue that, in practical deep neural network training, there is an additional and often overlooked weight-norm criticality. This criticality is induced by the interaction between normalization (which introduces scale-invariant components) and weight decay (which persistently shrinks parameter norms). As the weight decay coefficient increases, the norms of scale-invariant weights are progressively driven toward zero. Meanwhile, the sharpness of the loss landscape increases rapidly, destabilizing the optimization dynamics and resulting in abrupt loss spikes. This perspective provides a rationale for why weight penalties can improve generalization yet cannot be made arbitrarily strong: excessive decay drives scale-invariant weight norms past a critical boundary and destabilizes training. Our work provides a new mechanistic understanding of loss spikes through the lens of weight-norm criticality. Moreover, weight-norm criticality yields testable predictions that we validate empirically in networks with scale-invariant components, providing empirical support for the proposed mechanism.