Neural Weight Norm = Kolmogorov Complexity

Training large-scale deep neural networks (DNNs) is resource-intensive, making model compression a practical necessity. The widely accepted ''learning as compression'' hypothesis posits that training induces structure in network weights, which enables compression. Measuring this structure through Kolmogorov-Chaitin-Solomonoff (KCS) complexity is appealing, but existing estimators based on the Coding Theorem Method (CTM) and the Block Decomposition Method (BDM) are limited to small binary objects and do not scale to modern DNNs. We introduce the Quantized Block Decomposition method (QuBD), which extends algorithmic complexity estimation to any $k$-ary object. QuBD first quantizes the network weights to a finite alphabet, then estimates the KCS complexity by aggregating per bit-plane CTM estimates. We show theoretically that QuBD yields a strictly tighter estimation gap with respect to true KCS complexity than binarization-based methods. Using QuBD, we study how the algorithmic complexity of neural network weights evolves during training, showing that it decreases as models learn, scales with data budget, increases during overfitting, follows the delayed generalization observed during grokking, and correlates with generalization performance. We further show that algorithmic information resides predominantly in the most significant bit-planes, which can serve as a practical diagnostic for determining appropriate post-training quantization levels. This work offers novel insights into learning mechanisms in DNNs by providing the first scalable, tractable estimates of KCS complexity for large, non-binary objects such as DNN weights.

The discovery of scaling laws has motivated training neural networks on ever increasing quantities of data. This is typically done with a constant decoupled weight decay which causes the network weights to shrink steadily over the course of training. Taking inspiration from the Robbins--Monro conditions, we propose to scale weight decay by the fraction of the peak learning rate $η/η_{\max}$. We prove that this scaled weight decay preserves the asymptotic stationarity guarantees of the corresponding unregularized methods for both stochastic gradient descent and the non-Euclidean spectral optimizer Muon, thereby avoiding the additional asymptotic bias introduced by constant decoupled weight decay. This retains the stability benefits of weight decay without changing the asymptotic optimization target. Using a steady-state analysis, we explain why under standard weight decay the weight norm shrinks steadily as training proceeds, whereas under scaled weight decay it settles to a roughly constant value. When applied to the training of mixture-of-experts models, Muon with scaled weight decay (Muon-SW) consistently outpaces Muon with identical hyperparameters, reaching the same validation loss $\mathbf{30\%}$ faster at our largest scale across models from $72 - 930$ million parameters trained at $\sim 600$ tokens per active parameter. If this trend continues to hold, the method promises to substantially accelerate the pre-training of frontier models while requiring only a few lines of code to implement.

Most explanations of training instability focus on \emph{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 \emph{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 \emph{weight-norm criticality}. Moreover, \emph{weight-norm criticality} yields testable predictions that we validate empirically in networks with scale-invariant components, providing empirical support for the proposed mechanism.