# The Future is Log-Gaussian: ResNets and Their Infinite-Depth-and-Width Limit at Initialization

arXiv: https://arxiv.org/abs/2106.04013 · HTML: https://mlsanity.com/paper/2106.04013/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- Builds on 40 of its 57 references that are in the map.
- First posted on arXiv 7 Jun 2021.

## Builds on (most cited first, 30 of 40)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [A Convergence Theory for Deep Learning via Over-Parameterization](https://mlsanity.com/paper/1811.03962/) (2018)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [Gradient Descent Finds Global Minima of Deep Neural Networks](https://mlsanity.com/paper/1811.03804/) (2018)
- [On Exact Computation with an Infinitely Wide Neural Net](https://mlsanity.com/paper/1904.11955/) (2019)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Learning and Generalization in Overparameterized Neural Networks, Going Beyond Two Layers](https://mlsanity.com/paper/1811.04918/) (2018)
- [Gaussian Process Behaviour in Wide Deep Neural Networks](https://mlsanity.com/paper/1804.11271/) (2018)
- [Fixup Initialization: Residual Learning Without Normalization](https://mlsanity.com/paper/1901.09321/) (2019)
- [Feature Learning in Infinite-Width Neural Networks](https://mlsanity.com/paper/2011.14522/) (2020)
- [Mean-field theory of two-layers neural networks: dimension-free bounds and kernel limit](https://mlsanity.com/paper/1902.06015/) (2019)
- [Bayesian Deep Convolutional Networks with Many Channels are Gaussian Processes](https://mlsanity.com/paper/1810.05148/) (2018)
- [The Shattered Gradients Problem: If resnets are the answer, then what is the question?](https://mlsanity.com/paper/1702.08591/) (2017)
- [Tensor Programs I: Wide Feedforward or Recurrent Neural Networks of Any Architecture are Gaussian Processes](https://mlsanity.com/paper/1910.12478/) (2019)
- [Scaling Limits of Wide Neural Networks with Weight Sharing: Gaussian Process Behavior, Gradient Independence, and Neural Tangent Kernel Derivation](https://mlsanity.com/paper/1902.04760/) (2019)
- [Tensor Programs II: Neural Tangent Kernel for Any Architecture](https://mlsanity.com/paper/2006.14548/) (2020)
- [Stochastic Gradient Descent Optimizes Over-parameterized Deep ReLU Networks](https://mlsanity.com/paper/1811.08888/) (2018)
- [Finite Depth and Width Corrections to the Neural Tangent Kernel](https://mlsanity.com/paper/1909.05989/) (2019)
- [How to Start Training: The Effect of Initialization and Architecture](https://mlsanity.com/paper/1803.01719/) (2018)
- [Trainability and Accuracy of Neural Networks: An Interacting Particle System Approach](https://mlsanity.com/paper/1805.00915/) (2018)
- [Which Neural Net Architectures Give Rise To Exploding and Vanishing Gradients?](https://mlsanity.com/paper/1801.03744/) (2018)

## Built on by (most cited first, 30 of 33)

- [Depthwise Hyperparameter Transfer in Residual Networks: Dynamics and Scaling Limit](https://mlsanity.com/paper/2309.16620/) (2023)
- [The Shaped Transformer: Attention Models in the Infinite Depth-and-Width Limit](https://mlsanity.com/paper/2306.17759/) (2023)
- [Dynamics of Finite Width Kernel and Prediction Fluctuations in Mean Field Neural Networks](https://mlsanity.com/paper/2304.03408/) (2023)
- [The Neural Covariance SDE: Shaped Infinite Depth-and-Width Networks at Initialization](https://mlsanity.com/paper/2206.02768/) (2022)
- [Dynamic Sparse Training with Structured Sparsity](https://mlsanity.com/paper/2305.02299/) (2023)
- [Neural Tangent Kernel Beyond the Infinite-Width Limit: Effects of Depth and Initialization](https://mlsanity.com/paper/2202.00553/) (2022)
- [Precise characterization of the prior predictive distribution of deep ReLU networks](https://mlsanity.com/paper/2106.06615/) (2021)
- [Mind the spikes: Benign overfitting of kernels and neural networks in fixed dimension](https://mlsanity.com/paper/2305.14077/) (2023)
- [Super Consistency of Neural Network Landscapes and Learning Rate Transfer](https://mlsanity.com/paper/2402.17457/) (2024)
- [On the infinite-depth limit of finite-width neural networks](https://mlsanity.com/paper/2210.00688/) (2022)
- [Width and Depth Limits Commute in Residual Networks](https://mlsanity.com/paper/2302.00453/) (2023)
- [Meta-Principled Family of Hyperparameter Scaling Strategies](https://mlsanity.com/paper/2210.04909/) (2022)
- [Wide Neural Networks as Gaussian Processes: Lessons from Deep Equilibrium Models](https://mlsanity.com/paper/2310.10767/) (2023)
- [On the impact of activation and normalization in obtaining isometric embeddings at initialization](https://mlsanity.com/paper/2305.18399/) (2023)
- [The Feature Speed Formula: a flexible approach to scale hyper-parameters of deep neural networks](https://mlsanity.com/paper/2311.18718/) (2023)
- [Deep Stable neural networks: large-width asymptotics and convergence rates](https://mlsanity.com/paper/2108.02316/) (2021)
- [On the Neural Tangent Kernel of Equilibrium Models](https://mlsanity.com/paper/2310.14062/) (2023)
- [Adversarially Robust Out-of-Distribution Detection Using Lyapunov-Stabilized Embeddings](https://mlsanity.com/paper/2410.10744/) (2024)
- [Theoretical Limitations of Ensembles in the Age of Overparameterization](https://mlsanity.com/paper/2410.16201/) (2024)
- [Bounding generalization error with input compression: An empirical study with infinite-width networks](https://mlsanity.com/paper/2207.09408/) (2022)
- [Differential Equation Scaling Limits of Shaped and Unshaped Neural Networks](https://mlsanity.com/paper/2310.12079/) (2023)
- [Generalization of Scaled Deep ResNets in the Mean-Field Regime](https://mlsanity.com/paper/2403.09889/) (2024)
- [A Proof of Learning Rate Transfer under μP](https://mlsanity.com/paper/2511.01734/) (2025)
- [On Bridging the Gap between Mean Field and Finite Width in Deep Random Neural Networks with Batch Normalization](https://mlsanity.com/paper/2205.13076/) (2022)
- [Revisiting Glorot Initialization for Long-Range Linear Recurrences](https://mlsanity.com/paper/2505.19827/) (2025)
- [Random feature approximation for general spectral methods](https://mlsanity.com/paper/2506.16283/) (2025)
- [Commutative Width and Depth Scaling in Deep Neural Networks](https://mlsanity.com/paper/2310.01683/) (2023)
- [Random Features for Operator-Valued Kernels: Bridging Kernel Methods and Neural Operators](https://mlsanity.com/paper/2603.00971/) (2026)
- [Dynamical Isometry for Residual Networks](https://mlsanity.com/paper/2210.02411/) (2022)
- [Ortho-ODE: Enhancing Robustness and of Neural ODEs against Adversarial Attacks](https://mlsanity.com/paper/2305.09179/) (2023)

## Abstract (arXiv)

Theoretical results show that neural networks can be approximated by Gaussian processes in the infinite-width limit. However, for fully connected networks, it has been previously shown that for any fixed network width, n, the Gaussian approximation gets worse as the network depth, d, increases. Given that modern networks are deep, this raises the question of how well modern architectures, like ResNets, are captured by the infinite-width limit. To provide a better approximation, we study ReLU ResNets in the infinite-depth-and-width limit, where both depth and width tend to infinity as their ratio, d/n, remains constant. In contrast to the Gaussian infinite-width limit, we show theoretically that the network exhibits log-Gaussian behaviour at initialization in the infinite-depth-and-width limit, with parameters depending on the ratio d/n. Using Monte Carlo simulations, we demonstrate that even basic properties of standard ResNet architectures are poorly captured by the Gaussian limit, but remarkably well captured by our log-Gaussian limit. Moreover, our analysis reveals that ReLU ResNets at initialization are hypoactivated: fewer than half of the ReLUs are activated. Additionally, we calculate the interlayer correlations, which have the effect of exponentially increasing the variance of the network output. Based on our analysis, we introduce Balanced ResNets, a simple architecture modification, which eliminates hypoactivation and interlayer correlations and is more amenable to theoretical analysis.