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| Hauptverfasser: | , , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2510.04327 |
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| _version_ | 1866911291415724032 |
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| author | Zhang, Haosong Wu, Shenxi Zhang, Yichi Chen, Xi Lin, Wei |
| author_facet | Zhang, Haosong Wu, Shenxi Zhang, Yichi Chen, Xi Lin, Wei |
| contents | Choosing an appropriate learning rate remains a key challenge in scaling depth of modern deep networks. The classical maximal update parameterization ($μ$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 $μ$P (AM-$μ$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-$μ$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 $η^\star(L)\propto L^{-3/2}$; with zero padding, boundary effects are constant-level as $N\gg k$. For standard residual networks with general conv+MLP blocks, we establish $η^\star(L)=Θ(L^{-3/2})$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_04327 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Arithmetic-Mean $μ$P for Modern Architectures: A Unified Learning-Rate Scale for CNNs and ResNets Zhang, Haosong Wu, Shenxi Zhang, Yichi Chen, Xi Lin, Wei Machine Learning Choosing an appropriate learning rate remains a key challenge in scaling depth of modern deep networks. The classical maximal update parameterization ($μ$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 $μ$P (AM-$μ$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-$μ$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 $η^\star(L)\propto L^{-3/2}$; with zero padding, boundary effects are constant-level as $N\gg k$. For standard residual networks with general conv+MLP blocks, we establish $η^\star(L)=Θ(L^{-3/2})$, 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. |
| title | Arithmetic-Mean $μ$P for Modern Architectures: A Unified Learning-Rate Scale for CNNs and ResNets |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2510.04327 |