Intermittent ``spikes'' in loss, during the pre-training of Large Language Models (LLMs), remains a mysterious phenomenon both theoretically and empirically. In this paper, we show that simple quadratic models with heavy-tailed, power-law distributed inputs can replicate these spikes. We introduce a theoretical framework and simulation algorithm to model per-layer training dynamics of Deep Neural Networks (DNNs), using such quadratic models. In a case study on an Adam hyperparameter configuration (where $β_2$ is set close to $1$) known to induce spikes in Transformer LMs, our simulation successfully predicts bumps indicating spiky behavior from a given checkpoint. Furthermore, we derive an appropriate $β_2$ schedule from observed metrics, which empirically improves performance.

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