Qingyue Zhao

Zixiang Chen, Greg Yang, Qingyue Zhao et al.

Despite deep neural networks' powerful representation learning capabilities, theoretical understanding of how networks can simultaneously achieve meaningful feature learning and global convergence remains elusive. Existing approaches like the neural tangent kernel (NTK) are limited because features stay close to their initialization in this parametrization, leaving open questions about feature properties during substantial evolution. In this paper, we investigate the training dynamics of infinitely wide, $L$-layer neural networks using the tensor program (TP) framework. Specifically, we show that, when trained with stochastic gradient descent (SGD) under the Maximal Update parametrization ($μ$P) and mild conditions on the activation function, SGD enables these networks to learn linearly independent features that substantially deviate from their initial values. This rich feature space captures relevant data information and ensures that any convergent point of the training process is a global minimum. Our analysis leverages both the interactions among features across layers and the properties of Gaussian random variables, providing new insights into deep representation learning. We further validate our theoretical findings through experiments on real-world datasets.

Zhiyuan Fan, Yifeng Liu, Qingyue Zhao et al.

Empirical scaling laws prescribe how to allocate parameters, data, and compute, while maximal-update parameterization ($μ$P) enables learning-rate transfer across widths by equalizing early-time update magnitudes. However, in modern scale-invariant architectures, training quickly enters an optimizer-governed steady state where normalization layers create backward scale sensitivity and the effective learning rate becomes width dependent, degrading $μ$P transfer. We address this by introducing a weight-decay scaling rule for AdamW that preserves sublayer gain across widths. Empirically, the singular-value spectrum of each matrix parameter scales in norm as $\sqrt{η/λ}$ with an approximately invariant shape; under width scaling $d$, we observe that the top singular value scales approximately as $\sqrt{η/λ}\cdot d^{0.75}$. Combining this observation with the $μ$P learning-rate rule $η_2\propto d^{-1}$ for matrix-like parameters implies an empirical weight-decay scaling rule $λ_2\propto \sqrt{d}$ that approximately keeps sublayer gains width invariant. Together with vector-like parameters trained at $η_1=Θ_d(1)$ and $λ_1=0$, this yields \emph{zero-shot} transfer of both learning rate and weight decay from proxy to target widths, removing per-width sweeps. We validate the rule on LLaMA-style Transformers and in a minimal synthetic setting, and we provide a simple diagnostic, matching top singular values, to check sublayer-gain invariance. Our results extend $μ$P beyond the near-init regime by explicitly controlling steady-state scales set by the optimizer, offering a practical recipe for width-robust hyperparameter transfer under AdamW.