Infinite width limits of neural networks



Large-width limits of neural nets. An interesting way of considering overparameterization.

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 HuszΓ‘r 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 \(ΞΈ_0\) randomly from appropriately chosen distributions, the initial NTK of the network \(k_{ΞΈ_0}\) approaches a deterministic kernel as the width increases. This means, that at initialization, \(k_{ΞΈ_0}\) doesn’t really depend on \(k_{ΞΈ_0}\) but is a fixed kernel independent of the specific initialization.-
  2. Second: in the infinite limit the kernel \(k_{ΞΈ_t}\) 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

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.

References

Adlam, Ben, Jaehoon Lee, Lechao Xiao, Jeffrey Pennington, and Jasper Snoek. 2020. β€œExploring 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. β€œOn Exact Computation with an Infinitely Wide Neural Net.” In Advances in Neural Information Processing Systems, 10.
Bai, Yu, and Jason D. Lee. 2020. β€œBeyond 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. β€œTo Understand Deep Learning We Need to Understand Kernel Learning.” In International Conference on Machine Learning, 541–49.
Chen, Lin, and Sheng Xu. 2020. β€œDeep 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. β€œTowards Understanding Hierarchical Learning: Benefits of Neural Representations.” arXiv:2006.13436 [Cs, Stat], March.
Cho, Youngmin, and Lawrence K. Saul. 2009. β€œKernel 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. β€œEvery Model Learned by Gradient Descent Is Approximately a Kernel Machine.” arXiv:2012.00152 [Cs, Stat], November.
Fan, Zhou, and Zhichao Wang. 2020. β€œSpectra of the Conjugate Kernel and Neural Tangent Kernel for Linear-Width Neural Networks.” In Advances in Neural Information Processing Systems, 33:12.
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