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 significantly 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：通过对无穷宽度神经网络进行分类并识别出*最优*极限，张量程序 IV 和 V 展示了一种称为 μP 的统一方法，用于实现*宽度方向超参数迁移*，即从窄网络预测宽网络的最优超参数。本文我们针对深度残差网络（resnet）的*深度方向参数化*进行类似的分类研究。我们根据其无穷宽度-然后-深度极限，对块乘子和学习率的深度方向参数化进行分类。在每个块仅包含一层的残差网络中，我们识别出一种唯一的最优参数化，称为 Depth-μP，它扩展了 μP，并实验证明其具有深度方向超参数迁移能力。我们识别出*特征多样性*是深度网络中的关键因素，而 Depth-μP 可被表征为同时最大化特征学习和特征多样性。利用这一点，我们发现，在所有齐次非线性函数中，绝对值函数能最大化特征多样性，且实验上确实能带来显著更优的性能。然而，如果每个块更深（例如现代 Transformer），我们发现此类参数化所有可能的深度无穷极限均存在根本性限制，我们通过简单网络以及基于 Common Crawl 训练的 Megatron Transformer 从理论和实验两方面阐明了这一点。