By classifying infinite-width neural networks and identifying the *optimal* limit, Tensor Programs IV and V demonstrated a universal way, called $\mu$P, for *widthwise hyperparameter transfer*, i.e., predicting optimal hyperparameters of wide neural networks from narrow ones. Here we investigate the analogous classification for *depthwise parametrizations* of deep residual networks (resnets). We classify depthwise parametrizations of block multiplier and learning rate by their infinite-width-then-depth limits. In resnets where each block has only one layer, we identify a unique optimal parametrization, called Depth-$\mu$P that extends $\mu$P and show empirically it admits depthwise hyperparameter transfer. We identify *feature diversity* as a crucial factor in deep networks, and Depth-$\mu$P can be characterized as maximizing both feature learning and feature diversity. Exploiting this, we find that absolute value, among all homogeneous nonlinearities, maximizes feature diversity and indeed empirically leads to significant better performance. However, if each block is deeper (such as modern transformers), then we find fundamental limitations in all possible infinite-depth limits of such parametrizations, which we illustrate both theoretically and empirically on simple networks as well as Megatron transformer trained on Common Crawl.

翻译：通过分类无限宽神经网络并识别*最优*极限，张量程序IV和V展示了一种通用方法（称为μP），用于*宽度方向超参数迁移*，即从窄神经网络预测宽神经网络的最优超参数。本文研究了深度残差网络（resnets）中*深度方向参数化*的类似分类。我们根据其无限宽度-深度极限对块乘子和学习率的深度方向参数化进行分类。在每块仅含一层的残差网络中，我们识别出一种独特的最优参数化——Depth-μP（它扩展了μP），并通过实验证明其支持深度方向超参数迁移。我们提出*特征多样性*是深度网络的关键因素，而Depth-μP可被表征为同时最大化特征学习和特征多样性。利用这一特性，我们发现绝对值（在所有齐次非线性函数中）能最大化特征多样性，并在实验中显著提升性能。然而，若每个块更深（如现代Transformer），此类参数化的所有可能无限深度极限均存在根本性限制——我们通过简单网络和在Common Crawl上训练的Megatron Transformer从理论与实验两方面阐述了这一点。