Neural tangent kernel

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In the study of artificial neural networks (ANNs), the neural tangent kernel (NTK) is a kernel that describes the evolution of deep artificial neural networks during their training by gradient descent. It allows ANNs to be studied using theoretical tools from kernel methods.

For most common neural network architectures, in the limit of large layer width the NTK becomes constant. This enables simple closed form statements to be made about neural network predictions, training dynamics, generalization, and loss surfaces. For example, it guarantees that wide enough ANNs converge to a global minimum when trained to minimize an empirical loss. The NTK of large width networks is also related to several other large width limits of neural networks.

The NTK was introduced in 2018 by Arthur Jacot, Franck Gabriel and Clément Hongler.[1] It was implicit in contemporaneous work on overparameterization.[2][3][4]

Definition[edit]

Scalar output case[edit]

An ANN with scalar output consists of a family of functions parametrized by a vector of parameters .

The NTK is a kernel defined by

In the language of kernel methods, the NTK is the kernel associated with the feature map .

Vector output case[edit]

An ANN with vector output of size consists in a family of functions parametrized by a vector of parameters .

In this case, the NTK is a matrix-valued kernel, with values in the space of matrices, defined by

Derivation[edit]

When optimizing the parameters of an ANN to minimize an empirical loss through gradient descent, the NTK governs the dynamics of the ANN output function throughout the training.

Scalar output case[edit]

For a dataset with scalar labels and a loss function , the associated empirical loss, defined on functions , is given by

When the ANN is trained to fit the dataset (i.e. minimize ) via continuous-time gradient descent, the parameters evolve through the ordinary differential equation:

During training the ANN output function follows an evolution differential equation given in terms of the NTK:

This equation shows how the NTK drives the dynamics of in the space of functions during training.

Vector output case[edit]

For a dataset with vector labels and a loss function , the corresponding empirical loss on functions is defined by

The training of through continuous-time gradient descent yields the following evolution in function space driven by the NTK:

Interpretation[edit]

The NTK represents the influence of the loss gradient with respect to example on the evolution of ANN output through a gradient descent step: in the scalar case, this reads

In particular, each data point influences the evolution of the output for each throughout the training, in a way that is captured by the NTK .

Large-width limit[edit]

Recent theoretical and empirical work in deep learning has shown the performance of ANNs to strictly improve as their layer widths grow larger.[5][6] For various ANN architectures, the NTK yields precise insight into the training in this large-width regime.[1][7][8][9][10][11]

Wide fully-connected ANNs have a deterministic NTK, which remains constant throughout training[edit]

Consider an ANN with fully-connected layers of widths , so that , where is the composition of an affine transformation with the pointwise application of a nonlinearity , where parametrizes the maps . The parameters are initialized randomly, in an independent, identically distributed way.

As the widths grow, the NTK's scale is affected by the exact parametrization of the 's and by the parameter initialization. This motivates the so-called NTK parametrization . This parametrization ensures that if the parameters are initialized as standard normal variables, the NTK has a finite nontrivial limit. In the large-width limit, the NTK converges to a deterministic (non-random) limit , which stays constant in time.

The NTK is explicitly given by , where is determined by the set of recursive equations:

where denotes the kernel defined in terms of the Gaussian expectation:

In this formula the kernels are the ANN's so-called activation kernels.[12][13][14]

Wide fully connected networks are linear in their parameters throughout training[edit]

The NTK describes the evolution of neural networks under gradient descent in function space. Dual to this perspective is an understanding of how neural networks evolve in parameter space, since the NTK is defined in terms of the gradient of the ANN's outputs with respect to its parameters. In the infinite width limit, the connection between these two perspectives becomes especially interesting. The NTK remaining constant throughout training at large widths co-occurs with the ANN being well described throughout training by its first order Taylor expansion around its parameters at initialization:[9]

Other architectures[edit]

The NTK can be studied for various ANN architectures,[10] in particular convolutional neural networks (CNNs),[15] recurrent neural networks (RNNs) and transformers.[16] In such settings, the large-width limit corresponds to letting the number of parameters grow, while keeping the number of layers fixed: for CNNs, this involves letting the number of channels grow.

Applications[edit]

Convergence to a global minimum[edit]

For a convex loss functional with a global minimum, if the NTK remains positive-definite during training, the loss of the ANN converges to that minimum as . This positive-definiteness property has been shown in a number of cases, yielding the first proofs that large-width ANNs converge to global minima during training.[1][7][17]

Kernel methods[edit]

The NTK gives a rigorous connection between the inference performed by infinite-width ANNs and that performed by kernel methods: when the loss function is the least-squares loss, the inference performed by an ANN is in expectation equal to the kernel ridge regression (with zero ridge) with respect to the NTK . This suggests that the performance of large ANNs in the NTK parametrization can be replicated by kernel methods for suitably chosen kernels.[1][10]

Software libraries[edit]

Neural Tangents is a free and open-source Python library used for computing and doing inference with the infinite width NTK and neural network Gaussian process (NNGP) corresponding to various common ANN architectures.[18]

References[edit]

  1. ^ a b c d Jacot, Arthur; Gabriel, Franck; Hongler, Clement (2018), Bengio, S.; Wallach, H.; Larochelle, H.; Grauman, K. (eds.), "Neural Tangent Kernel: Convergence and Generalization in Neural Networks" (PDF), Advances in Neural Information Processing Systems 31, Curran Associates, Inc., pp. 8571–8580, arXiv:1806.07572, Bibcode:2018arXiv180607572J, retrieved 2019-11-27
  2. ^ Li, Yuanzhi; Liang, Yingyu (2018). "Learning overparameterized neural networks via stochastic gradient descent on structured data". Advances in Neural Information Processing Systems. arXiv:1808.01204.
  3. ^ Allen-Zhu, Zeyuan; Li, Yuanzhi; Song, Zhao (2018). "A convergence theory for deep learning via overparameterization". International Conference on Machine Learning. arXiv:1811.03962.
  4. ^ Du, Simon S; Zhai, Xiyu; Poczos, Barnabas; Aarti, Singh (2019). "Gradient descent provably optimizes over-parameterized neural networks". International Conference on Learning Representations. arXiv:1810.02054.
  5. ^ Novak, Roman; Bahri, Yasaman; Abolafia, Daniel A.; Pennington, Jeffrey; Sohl-Dickstein, Jascha (2018-02-15). "Sensitivity and Generalization in Neural Networks: an Empirical Study". arXiv:1802.08760. Bibcode:2018arXiv180208760N. {{cite journal}}: Cite journal requires |journal= (help)
  6. ^ Canziani, Alfredo; Paszke, Adam; Culurciello, Eugenio (2016-11-04). "An Analysis of Deep Neural Network Models for Practical Applications". arXiv:1605.07678. Bibcode:2016arXiv160507678C. {{cite journal}}: Cite journal requires |journal= (help)
  7. ^ a b Allen-Zhu, Zeyuan; Li, Yuanzhi; Song, Zhao (2018-11-09). "A Convergence Theory for Deep Learning via Over-Parameterization". International Conference on Machine Learning: 242–252. arXiv:1811.03962.
  8. ^ Du, Simon; Lee, Jason; Li, Haochuan; Wang, Liwei; Zhai, Xiyu (2019-05-24). "Gradient Descent Finds Global Minima of Deep Neural Networks". International Conference on Machine Learning: 1675–1685. arXiv:1811.03804.
  9. ^ a b Lee, Jaehoon; Xiao, Lechao; Schoenholz, Samuel S.; Bahri, Yasaman; Novak, Roman; Sohl-Dickstein, Jascha; Pennington, Jeffrey (2020). "Wide neural networks of any depth evolve as linear models under gradient descent". Journal of Statistical Mechanics: Theory and Experiment. 2020 (12): 124002. arXiv:1902.06720. Bibcode:2020JSMTE2020l4002L. doi:10.1088/1742-5468/abc62b. S2CID 62841516.
  10. ^ a b c Arora, Sanjeev; Du, Simon S; Hu, Wei; Li, Zhiyuan; Salakhutdinov, Russ R; Wang, Ruosong (2019), "On Exact Computation with an Infinitely Wide Neural Net", NeurIPS: 8139–8148, arXiv:1904.11955
  11. ^ Huang, Jiaoyang; Yau, Horng-Tzer (2019-09-17). "Dynamics of Deep Neural Networks and Neural Tangent Hierarchy". arXiv:1909.08156 [cs.LG].
  12. ^ Cho, Youngmin; Saul, Lawrence K. (2009), Bengio, Y.; Schuurmans, D.; Lafferty, J. D.; Williams, C. K. I. (eds.), "Kernel Methods for Deep Learning" (PDF), Advances in Neural Information Processing Systems 22, Curran Associates, Inc., pp. 342–350, retrieved 2019-11-27
  13. ^ Daniely, Amit; Frostig, Roy; Singer, Yoram (2016), Lee, D. D.; Sugiyama, M.; Luxburg, U. V.; Guyon, I. (eds.), "Toward Deeper Understanding of Neural Networks: The Power of Initialization and a Dual View on Expressivity" (PDF), Advances in Neural Information Processing Systems 29, Curran Associates, Inc., pp. 2253–2261, arXiv:1602.05897, Bibcode:2016arXiv160205897D, retrieved 2019-11-27
  14. ^ Lee, Jaehoon; Bahri, Yasaman; Novak, Roman; Schoenholz, Samuel S.; Pennington, Jeffrey; Sohl-Dickstein, Jascha (2018-02-15). "Deep Neural Networks as Gaussian Processes". {{cite journal}}: Cite journal requires |journal= (help)
  15. ^ Yang, Greg (2019-02-13). "Scaling Limits of Wide Neural Networks with Weight Sharing: Gaussian Process Behavior, Gradient Independence, and Neural Tangent Kernel Derivation". arXiv:1902.04760 [cs.NE].
  16. ^ Hron, Jiri; Bahri, Yasaman; Sohl-Dickstein, Jascha; Novak, Roman (2020-06-18). "Infinite attention: NNGP and NTK for deep attention networks". International Conference on Machine Learning. 2020. arXiv:2006.10540. Bibcode:2020arXiv200610540H.
  17. ^ Allen-Zhu, Zeyuan; Li, Yuanzhi; Song, Zhao (2018-10-29). "On the convergence rate of training recurrent neural networks". NeurIPS. arXiv:1810.12065.
  18. ^ Novak, Roman; Xiao, Lechao; Hron, Jiri; Lee, Jaehoon; Alemi, Alexander A.; Sohl-Dickstein, Jascha; Schoenholz, Samuel S. (2019-12-05), "Neural Tangents: Fast and Easy Infinite Neural Networks in Python", International Conference on Learning Representations (ICLR), vol. 2020, arXiv:1912.02803, Bibcode:2019arXiv191202803N

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