# Vanishing Curvature and the Power of Adaptive Methods in Randomly Initialized Deep Networks

arXiv: https://arxiv.org/abs/2106.03763 · HTML: https://mlsanity.com/paper/2106.03763/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 47 of its 83 references that are in the map.
- First posted on arXiv 7 Jun 2021.

## Builds on (most cited first, 30 of 47)

- [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)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [EfficientNet: Rethinking Model Scaling for Convolutional Neural Networks](https://mlsanity.com/paper/1905.11946/) (2019)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Analyzing and Improving the Image Quality of StyleGAN](https://mlsanity.com/paper/1912.04958/) (2019)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Large Scale GAN Training for High Fidelity Natural Image Synthesis](https://mlsanity.com/paper/1809.11096/) (2018)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [Semantic Image Synthesis with Spatially-Adaptive Normalization](https://mlsanity.com/paper/1903.07291/) (2019)
- [Training Generative Adversarial Networks with Limited Data](https://mlsanity.com/paper/2006.06676/) (2020)
- [MnasNet: Platform-Aware Neural Architecture Search for Mobile](https://mlsanity.com/paper/1807.11626/) (2018)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [A Convergence Theory for Deep Learning via Over-Parameterization](https://mlsanity.com/paper/1811.03962/) (2018)
- [On the Variance of the Adaptive Learning Rate and Beyond](https://mlsanity.com/paper/1908.03265/) (2019)
- [Exact solutions to the nonlinear dynamics of learning in deep linear neural networks](https://mlsanity.com/paper/1312.6120/) (2013)
- [Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming](https://mlsanity.com/paper/1309.5549/) (2013)
- [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)
- [New insights and perspectives on the natural gradient method](https://mlsanity.com/paper/1412.1193/) (2014)
- [Why gradient clipping accelerates training: A theoretical justification for adaptivity](https://mlsanity.com/paper/1905.11881/) (2019)
- [Lower Bounds for Non-Convex Stochastic Optimization](https://mlsanity.com/paper/1912.02365/) (2019)
- [Momentum-Based Variance Reduction in Non-Convex SGD](https://mlsanity.com/paper/1905.10018/) (2019)
- [PyHessian: Neural Networks Through the Lens of the Hessian](https://mlsanity.com/paper/1912.07145/) (2019)
- [Benefits of depth in neural networks](https://mlsanity.com/paper/1602.04485/) (2016)
- [The Marginal Value of Adaptive Gradient Methods in Machine Learning](https://mlsanity.com/paper/1705.08292/) (2017)
- [Limitations of the Empirical Fisher Approximation for Natural Gradient Descent](https://mlsanity.com/paper/1905.12558/) (2019)

## Built on by (most cited first, 3 of 3)

- [Signal Propagation in Transformers: Theoretical Perspectives and the Role of Rank Collapse](https://mlsanity.com/paper/2206.03126/) (2022)
- [Super Consistency of Neural Network Landscapes and Learning Rate Transfer](https://mlsanity.com/paper/2402.17457/) (2024)
- [Dissecting adaptive methods in GANs](https://mlsanity.com/paper/2210.04319/) (2022)

## Abstract (arXiv)

This paper revisits the so-called vanishing gradient phenomenon, which commonly occurs in deep randomly initialized neural networks. Leveraging an in-depth analysis of neural chains, we first show that vanishing gradients cannot be circumvented when the network width scales with less than O(depth), even when initialized with the popular Xavier and He initializations. Second, we extend the analysis to second-order derivatives and show that random i.i.d. initialization also gives rise to Hessian matrices with eigenspectra that vanish as networks grow in depth. Whenever this happens, optimizers are initialized in a very flat, saddle point-like plateau, which is particularly hard to escape with stochastic gradient descent (SGD) as its escaping time is inversely related to curvature. We believe that this observation is crucial for fully understanding (a) historical difficulties of training deep nets with vanilla SGD, (b) the success of adaptive gradient methods (which naturally adapt to curvature and thus quickly escape flat plateaus) and (c) the effectiveness of modern architectural components like residual connections and normalization layers.