On the Depth Transfer of Neural Optimizers via L-Smoothness

Scaling deep learning models efficiently requires predictable performance transfer across compute regimes. Establishing reliable hyperparameter transfer allows optimal training dynamics and structural choices to be discovered on low-cost proxy models, avoiding catastrophic failures during massive-scale training. Existing depth-transfer analyses are architecture-specific and heuristic: hyperparameter transfer is only justified through toy blocks, and both rely on infinite-depth dynamics, a specific initialization, and feature-learning criteria. Building on $L$-smoothness framework of Xu et al. (2026), we understand hyperparameter transfer as how do the Lipschitz and $L$-smoothness constants of the loss scale with the depth $K$? We show that inverse-depth residual branch scaling $α=Θ(1/K)$ is sufficient to make both constants depth-independent at fixed width, uniformly over the bounded parameter domain and for arbitrary $p\to q$ geometries. Because the bounds depend on each residual branch only through its Lipschitz and smoothness constants, the argument applies to general residual networks (including branches of block depth $\geq 2$ such as transformer and ResNet blocks) without assumptions on initialization or correlations between layer weights. Since the smoothness constant controls the curvature penalty in the descent inequality, our bounds motivate a learning-rate scale without an additional depth-dependent curvature correction for updates bounded in the prescribed norm. This provides a geometric explanation of inverse-depth residual scaling and its application to the operator-norm geometries associated with AdamW, Muon, and MOGA. Learning-rate sweeps on GPT-2 (depths 2--128) and CIFAR-100 ResNets (8--32 residual blocks) confirm that Muon and MOGA exhibit stable learning-rate transfer under these prescriptions, with a common optimal rate across a 64-fold increase in depth.