Infinite width limits of neural networks

Large-width limits of neural nets.

Neural Network Gaussian Process

For now: See Neural network Gaussian process on Wikipedia.

The field that sprang from the insight (Neal 1996a) that in the infinite limit, random neural nets with Gaussian weights and appropriate scaling asymptotically approach Gaussian processes, and there are useful conclusions we can draw from that.

More generally we might consider correlated and/or non-Gaussian weights, and deep networks. Unless otherwise stated though, I am thinking about i.i.d. Gaussian weights, and a single hidden layer.

In this single-hidden-layer case we get tractable covariance structure. See NN kernels.

For some reason this multi-layer wide NNs for me

Neural Network Tangent Kernel

See Neural tangent kernel on Wikipedia. Ferenc Huszár provides some Intuition on the Neural Tangent Kernel, i.e. the paper (Lee et al. 2019).

It turns out the neural tangent kernel becomes particularly useful when studying learning dynamics in infinitely wide feed-forward neural networks. Why? Because in this limit, two things happen:

  1. First: if we initialize \(θ_0\) randomly from appropriately chosen distributions, the initial NTK of the network \(k_{θ_0}\) approaches a deterministic kernel as the width increases. This means, that at initialization, \(k_{θ_0}\) doesn’t really depend on \(k_{θ_0}\) but is a fixed kernel independent of the specific initialization.-
  2. Second: in the infinite limit the kernel \(k_{θ_t}\) stays constant over time as we optimise \(\theta_t\). This removes the parameter dependence during training.

These two facts put together imply that gradient descent in the infinitely wide and infinitesimally small learning rate limit can be understood as a pretty simple algorithm called kernel gradient descent with a fixed kernel function that depends only on the architecture (number of layers, activations, etc).

These results, taken together with an older known result (Neal 1996b), allows us to characterise the probability distribution of minima that gradient descent converges to in this infinite limit as a Gaussian process.

Implicit regularization

Here’s one interesting perspective on wide nets (Zhang et al. 2017) which looks rather like the NTK model, but is it? To read.

  • The effective capacity of neural networks is large enough for a brute-force memorization of the entire data set.

  • Even optimization on random labels remains easy. In fact, training time increases only by a small constant factor compared with training on the true labels.

  • Randomizing labels is solely a data transformation, leaving all other properties of the learning problem unchanged.

[…] Explicit regularization may improve generalization performance, but is neither necessary nor by itself sufficient for controlling generalization error. […] Appealing to linear models, we analyze how SGD acts as an implicit regularizer.


Dropout is sometimes presumed to simulate from a certain kind of Gaussian process out of a neural net. See Dropout.

As stochastic DEs

We can find an SDE for a given NN-style kernel if we can find Green’s functions \(\sigma^2_\varepsilon \langle G_\cdot(\mathbf{x}_p), G_\cdot(\mathbf{x}_q)\rangle = \mathbb{E} \big[ \psi\big(Z_p\big) \psi\big(Z_q \big) \big].\) Russell Tsuchida observes: if you set \(G_\mathbf{s}(\mathbf{x}_p) = \psi(\mathbf{s}^\top \mathbf{x}_p) \sqrt{\phi(\mathbf{s})}\), where \(\phi\) is the pdf of an independent standard multivariate normal vector is a solution.


Adlam, Ben, Jaehoon Lee, Lechao Xiao, Jeffrey Pennington, and Jasper Snoek. 2020. “Exploring the Uncertainty Properties of Neural Networks’ Implicit Priors in the Infinite-Width Limit.” arXiv:2010.07355 [cs, Stat], October.
Arora, Sanjeev, Simon S Du, Wei Hu, Zhiyuan Li, Russ R Salakhutdinov, and Ruosong Wang. 2019. “On Exact Computation with an Infinitely Wide Neural Net.” In Advances in Neural Information Processing Systems, 10.
Bai, Yu, and Jason D. Lee. 2020. “Beyond Linearization: On Quadratic and Higher-Order Approximation of Wide Neural Networks.” arXiv:1910.01619 [cs, Math, Stat], February.
Belkin, Mikhail, Siyuan Ma, and Soumik Mandal. 2018. “To Understand Deep Learning We Need to Understand Kernel Learning.” In International Conference on Machine Learning, 541–49.
Chen, Lin, and Sheng Xu. 2020. “Deep Neural Tangent Kernel and Laplace Kernel Have the Same RKHS.” arXiv:2009.10683 [cs, Math, Stat], October.
Chen, Minshuo, Yu Bai, Jason D. Lee, Tuo Zhao, Huan Wang, Caiming Xiong, and Richard Socher. 2021. “Towards Understanding Hierarchical Learning: Benefits of Neural Representations.” arXiv:2006.13436 [cs, Stat], March.
Cho, Youngmin, and Lawrence K. Saul. 2009. “Kernel Methods for Deep Learning.” In Proceedings of the 22nd International Conference on Neural Information Processing Systems, 22:342–50. NIPS’09. Red Hook, NY, USA: Curran Associates Inc.
Domingos, Pedro. 2020. “Every Model Learned by Gradient Descent Is Approximately a Kernel Machine.” arXiv:2012.00152 [cs, Stat], November.
Fan, Zhou, and Zhichao Wang. 2020. “Spectra of the Conjugate Kernel and Neural Tangent Kernel for Linear-Width Neural Networks.” In Advances in Neural Information Processing Systems, 33:12.
Fort, Stanislav, Gintare Karolina Dziugaite, Mansheej Paul, Sepideh Kharaghani, Daniel M. Roy, and Surya Ganguli. 2020. “Deep Learning Versus Kernel Learning: An Empirical Study of Loss Landscape Geometry and the Time Evolution of the Neural Tangent Kernel.” In Advances in Neural Information Processing Systems. Vol. 33.
Gal, Yarin, and Zoubin Ghahramani. 2015. “Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning.” In Proceedings of the 33rd International Conference on Machine Learning (ICML-16).
———. 2016. “A Theoretically Grounded Application of Dropout in Recurrent Neural Networks.” In arXiv:1512.05287 [stat].
Geifman, Amnon, Abhay Yadav, Yoni Kasten, Meirav Galun, David Jacobs, and Ronen Basri. 2020. “On the Similarity Between the Laplace and Neural Tangent Kernels.” In arXiv:2007.01580 [cs, Stat].
Ghahramani, Zoubin. 2013. “Bayesian Non-Parametrics and the Probabilistic Approach to Modelling.” Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 371 (1984): 20110553.
Girosi, Federico, Michael Jones, and Tomaso Poggio. 1995. “Regularization Theory and Neural Networks Architectures.” Neural Computation 7 (2): 219–69.
Giryes, R., G. Sapiro, and A. M. Bronstein. 2016. “Deep Neural Networks with Random Gaussian Weights: A Universal Classification Strategy?” IEEE Transactions on Signal Processing 64 (13): 3444–57.
He, Bobby, Balaji Lakshminarayanan, and Yee Whye Teh. 2020. “Bayesian Deep Ensembles via the Neural Tangent Kernel.” In Advances in Neural Information Processing Systems. Vol. 33.
Jacot, Arthur, Franck Gabriel, and Clement Hongler. 2018. “Neural Tangent Kernel: Convergence and Generalization in Neural Networks.” In Advances in Neural Information Processing Systems, 31:8571–80. NIPS’18. Red Hook, NY, USA: Curran Associates Inc.
Karakida, Ryo, and Kazuki Osawa. 2020. “Understanding Approximate Fisher Information for Fast Convergence of Natural Gradient Descent in Wide Neural Networks.” Advances in Neural Information Processing Systems 33.
Kristiadi, Agustinus, Matthias Hein, and Philipp Hennig. 2021. “An Infinite-Feature Extension for Bayesian ReLU Nets That Fixes Their Asymptotic Overconfidence.” arXiv:2010.02709 [cs, Stat], May.
Lee, Jaehoon, Yasaman Bahri, Roman Novak, Samuel S. Schoenholz, Jeffrey Pennington, and Jascha Sohl-Dickstein. 2018. “Deep Neural Networks as Gaussian Processes.” In ICLR.
Lee, Jaehoon, Lechao Xiao, Samuel S. Schoenholz, Yasaman Bahri, Roman Novak, Jascha Sohl-Dickstein, and Jeffrey Pennington. 2019. “Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent.” In Advances in Neural Information Processing Systems, 8570–81.
Matthews, Alexander Graeme de Garis, Mark Rowland, Jiri Hron, Richard E. Turner, and Zoubin Ghahramani. 2018. “Gaussian Process Behaviour in Wide Deep Neural Networks.” arXiv:1804.11271 [cs, Stat], August.
Meronen, Lassi, Christabella Irwanto, and Arno Solin. 2020. “Stationary Activations for Uncertainty Calibration in Deep Learning.” In Advances in Neural Information Processing Systems. Vol. 33.
Neal, Radford M. 1996a. “Bayesian Learning for Neural Networks.” Secaucus, NJ, USA: Springer-Verlag New York, Inc.
———. 1996b. “Priors for Infinite Networks.” In Bayesian Learning for Neural Networks, edited by Radford M. Neal, 29–53. Lecture Notes in Statistics. New York, NY: Springer.
Novak, Roman, Lechao Xiao, Jiri Hron, Jaehoon Lee, Alexander A. Alemi, Jascha Sohl-Dickstein, and Samuel S. Schoenholz. 2019. “Neural Tangents: Fast and Easy Infinite Neural Networks in Python.” arXiv:1912.02803 [cs, Stat], December.
Novak, Roman, Lechao Xiao, Jaehoon Lee, Yasaman Bahri, Greg Yang, Jiri Hron, Daniel A. Abolafia, Jeffrey Pennington, and Jascha Sohl-Dickstein. 2020. “Bayesian Deep Convolutional Networks with Many Channels Are Gaussian Processes.” In The International Conference on Learning Representations.
Pearce, Tim, Russell Tsuchida, Mohamed Zaki, Alexandra Brintrup, and Andy Neely. 2019. “Expressive Priors in Bayesian Neural Networks: Kernel Combinations and Periodic Functions.” In Uncertainty in Artificial Intelligence, 11.
Williams, Christopher K. I. 1996. “Computing with Infinite Networks.” In Proceedings of the 9th International Conference on Neural Information Processing Systems, 295–301. NIPS’96. Cambridge, MA, USA: MIT Press.
Yang, Greg. 2019. “Tensor Programs I: Wide Feedforward or Recurrent Neural Networks of Any Architecture Are Gaussian Processes.” arXiv:1910.12478 [cond-Mat, Physics:math-Ph], December.
Yang, Greg, and Edward J. Hu. 2020. “Feature Learning in Infinite-Width Neural Networks.” arXiv:2011.14522 [cond-Mat], November.
Zhang, Chiyuan, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. 2017. “Understanding Deep Learning Requires Rethinking Generalization.” In Proceedings of ICLR.

No comments yet. Why not leave one?

GitHub-flavored Markdown & a sane subset of HTML is supported.