Arithmetic-Mean $μ$P for Modern Architectures: A Unified Learning-Rate Scale for CNNs and ResNets

• URL: http://arxiv.org/abs/2510.04327v1
• Date: Sun, 05 Oct 2025 19:22:50 GMT
• Title: Arithmetic-Mean $μ$P for Modern Architectures: A Unified Learning-Rate Scale for CNNs and ResNets
• Authors: Haosong Zhang, Shenxi Wu, Yichi Zhang, Wei Lin,
• Abstract summary: Arithmetic-Mean $mu$P constrains not each individual layer but the network-wide average one-step pre-activation second moment to a constant scale.<n>We prove that, for one- and two-dimensional convolutional networks, the maximal-update learning rate satisfies $etastar(L)propto L-3/2$; with zero padding, boundary effects are constant-level as $Ngg k$.
• Score: 9.94514344279733
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Choosing an appropriate learning rate remains a key challenge in scaling depth of modern deep networks. The classical maximal update parameterization ($\mu$P) enforces a fixed per-layer update magnitude, which is well suited to homogeneous multilayer perceptrons (MLPs) but becomes ill-posed in heterogeneous architectures where residual accumulation and convolutions introduce imbalance across layers. We introduce Arithmetic-Mean $\mu$P (AM-$\mu$P), which constrains not each individual layer but the network-wide average one-step pre-activation second moment to a constant scale. Combined with a residual-aware He fan-in initialization - scaling residual-branch weights by the number of blocks ($\mathrm{Var}[W]=c/(K\cdot \mathrm{fan\text{-}in})$) - AM-$\mu$P yields width-robust depth laws that transfer consistently across depths. We prove that, for one- and two-dimensional convolutional networks, the maximal-update learning rate satisfies $\eta^\star(L)\propto L^{-3/2}$; with zero padding, boundary effects are constant-level as $N\gg k$. For standard residual networks with general conv+MLP blocks, we establish $\eta^\star(L)=\Theta(L^{-3/2})$, with $L$ the minimal depth. Empirical results across a range of depths confirm the $-3/2$ scaling law and enable zero-shot learning-rate transfer, providing a unified and practical LR principle for convolutional and deep residual networks without additional tuning overhead.

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