证明深度残差网络在大尺度极限下训练动态收敛，给出可验证误差界。
ResNets of All Shapes and Sizes: Convergence of Training Dynamics in the Large-scale Limit
- 利用深度两层残差块结构，结合壳映射与混沌传播方法分析极限行为。
- 误差率可达 O(1/L + sqrt(D/(L M)) + 1/sqrt(D))，参数预算下最优收敛速率为 P^(-1/6)。
- 适用于主流架构如 Transformer，首次实现 DMFT 型极限的严格定量分析。
我们建立了残差神经网络（ResNets）在无限深度 L、隐层宽度 M 与嵌入维度 D 极限下的训练动态收敛性。具体而言，考虑采用双层感知机残差块且处于最大局部特征更新（MLU）模式的 ResNets，证明在有限训练步数后，其与大尺度极限之间的误差为 O(1/L + sqrt(D/(L M)) + 1/sqrt(D))，该误差率在嵌入空间中具有经验紧性。当总参数量 P = Theta(L M D) 时，对应最优缩放下收敛速率达 O(P^(-1/6))。分析关键依赖于残差块的深度两层结构，并可形式化应用于包括键值维数有界的 Transformer 在内的多种先进架构。从技术角度看，本工作完成 [Chi25] 中开启的研究：此前已证固定嵌入维数下收敛至均值微分方程（Mean ODE）速率为 O(1/L + sqrt(D)/sqrt(L M))；本文研究该均值微分方程在大维数极限下的行为，建立其收敛速率 O(1/sqrt(D))，通过三角不等式得到最终界。为处理极限动力学中复杂的概率结构并实现首个针对 DMFT 型极限的严格定量结果，我们结合了腔方法与函数层面的混沌传播论证，基于称为‘骨架映射’的表达式——权重更新被表示为过去若干项满足中心极限定理型和的函数。
原文摘要 · Abstract (English)
We establish convergence of the training dynamics of residual neural networks (ResNets) to their joint infinite depth L, hidden width M, and embedding dimension D limit. Specifically, we consider ResNets with two-layer perceptron blocks in the maximal local feature update (MLU) regime and prove that, after a bounded number of training steps, the error between the ResNet and its large-scale limit is O(1/L + sqrt(D/(L M)) + 1/sqrt(D)). This error rate is empirically tight when measured in embedding space. For a budget of P = Theta(L M D) parameters, this yields a convergence rate O(P^(-1/6)) for the scalings of (L, M, D) that minimize the bound. Our analysis exploits in an essential way the depth-two structure of residual blocks and applies formally to a broad class of state-of-the-art architectures, including Transformers with bounded key-query dimension. From a technical viewpoint, this work completes the program initiated in the companion paper [Chi25] where it is proved that for a fixed embedding dimension D, the training dynamics converges to a Mean ODE dynamics at rate O(1/L + sqrt(D)/sqrt(L M)). Here, we study the large-D limit of this Mean ODE model and establish convergence at rate O(1/sqrt(D)), yielding the above bound by a triangle inequality. To handle the rich probabilistic structure of the limit dynamics and obtain one of the first rigorous quantitative convergence for a DMFT-type limit, we combine the cavity method with propagation of chaos arguments at a functional level on so-called skeleton maps, which express the weight updates as functions of CLT-type sums from the past.
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