The cost-accuracy trade-off has been a longstanding and critical issue for deep neural networks.

Then dynamics ofwt is like a physical process - a satellite's motion around the earth (see illustration in Fig.1): according to

Decoupled weight decay, solely responsible for the performance advantage of AdamW over Adam, has long been set to proportional to learning rate γ without questioning. To the contrary, we find that eliminating the contribution of the perpendicular component of the update to the weight norm leads to little change to the training dynamics. For adaptive gradient methods such as SGD with momentum (Sutskever et al., 2013) and Adam (Kingma & Ba, 2015), weight decay is no longer equivalent to L Nevertheless, Defazio (2025) presents experiments on Llama 3 architecture (Grattafiori et al., 2024) in which most layers are not immediately followed by normalization. It states that "we consider every linear layer as normalized, excluding the output layer of the network" for the purpose of applying such corrected weight decay, and AdamC results in more stable weight and gradient norms than the AdamW baseline regardless. Consider the "Renormalized" AdamW optimizer above (Algorithm 1) which eliminates the contribution of u We train a variant of ViT -S/16 based on the setup described in Beyer et al. (2022) on the ImageNet-1k dataset (Russakovsky et al., 2015) for 90 epochs and instead observe almost no differences in relevant metrics (Figure 1).

In this paper, we investigate the phenomenon of grokking, where models exhibit delayed generalization following overfitting on training data. We focus on data-scarce regimes where the number of training samples falls below the critical threshold, making grokking unobservable, and on practical scenarios involving distribution shift. We first show that Knowledge Distillation (KD) from a model that has already grokked on a distribution (p1) can induce and accelerate grokking on a different distribution (p2), even when the available data lies below the critical threshold. This highlights the value of KD for deployed models that must adapt to new distributions under limited data. We then study training on the joint distribution (p1, p2) and demonstrate that while standard supervised training fails when either distribution has insufficient data, distilling from models grokked on the individual distributions enables generalization. Finally, we examine a continual pretraining setup, where a grokked model transitions from p1 to p2, and find that KD both accelerates generalization and mitigates catastrophic forgetting, achieving strong performance even with only 10% of the data. Together, our results provide new insights into the mechanics of grokking under knowledge transfer and underscore the central role of KD in enabling generalization in low-data and evolving distribution settings.