NeurIPS 2020

On the Similarity between the Laplace and Neural Tangent Kernels

Review 1

Summary and Contributions: The paper studies properties of neural tangent kernels and relates them to those of the Laplace kernel when restricted to the sphere. In particular, it is shown that the two kernels lead to the same functional space, by having essentially the same eigenvalue decays for the uniform measure on the sphere. The paper provides additional results beyond the sphere, as well as various experiments that highlight the comparable performance of NTK and Laplace kernels, and in many cases improvements in performance by using the gamma-exponential kernel.

Strengths: I find the work quite interesting and significant as it provides a better understanding of the NTK regime for over-parameterized neural networks, and relates the NTK to the more traditional Laplace kernels, which had been empirically observed to behave similarly to deep networks in previous work by Belkin et al. The results on the decay of the Laplace kernel restricted to the sphere are also novel to my knowledge, and of interest by themselves. The experiments are also interesting in that they show the limitations of the NTK compared to usual kernels, in particular that modifications of the standard Laplace kernel tend to outperform the NTK when tuning hyperparameters. The claims are both theoretically and empirically justified, and the paper is clear. I am thus in favor of acceptance.

Weaknesses: I do not see major weaknesses in the paper. Perhaps a minor weakness is that the obtained accuracies on Cifar10 seem quite low, even compared to other works on convolutional kernels. There could also be more discussion on the limitations of the kernel approach/NTK for explaining the success of neural networks, given that the paper shows that the NTK behaves essentially the same as a simple Laplace kernel (this is briefly mentioned in the discussion section, but it seems important to emphasize this from the introduction).

Correctness: yes

Clarity: The paper is well written and pleasant to read.

Relation to Prior Work: yes

Reproducibility: Yes

Additional Feedback: - L118-124: perhaps it should be mentioned here that the work considers the NTK for the ReLU activation? - L201: the meaning of this bound and the following lower bounds could be further clarified. The dependence on the number of examples is also unclear - perhaps standard excess risk bounds for ridge regression would be more appropriate here? - Thm 5: Can something also be said about how the eigenvalues are affected compared to the spherical case? **** update after rebuttal **** Thanks to the authors for their response. One minor detail to add is that it would be useful to include a few more experimental details in the main paper, as opposed to in the appendix (e.g. hyperparameter selection in the convolutional case).

Review 2

Summary and Contributions: The paper theoretically proves that NTK corresponding to a two-layer FC neural network with bias has the same RKHS of Laplace Kernel for normalized data on hyperspheres. The paper also proves the RKHS of Laplace Kernel is a subset of RKHS of NTK corresponding to multi-layer FC NN, and conjecture these two are also the same by experimental evidence. The experiments show that the Laplace kernel performs similarly to NTK in three settings.

Strengths: - The theory of the paper is sound to me. The result is novel and interesting. - The experiments are mostly complete and follow standard cross-validation.

Weaknesses: - The theoretical contribution of this paper is questionable. The paper mentioned similar results in the paper but doesn't give a comparison of the proof techniques, especially for Therorem 1. - For the experiment on convolutional kernels, the author doesn't mention why they need additional parameters a, b, c. It makes me feel like the authors are just try to manually approximate NTK better by adding these parameters. The authors also didn't mention how they choose the hyperparameter beta.

Correctness: I didn't find flaws in the proof, and the experiments are mostly sound.

Clarity: The paper is mostly well-written. Minor comments: - line 108, should use \langle and \rangle for inner product. - A better definition of RKHS is required. - line 141, probably need to define or explain spherical harmonics.

Relation to Prior Work: The paper have listed previous works. I suggest a more detailed comparison to those works.

Reproducibility: Yes

Additional Feedback: Can you clarify why you need a, b, c in the experiment on convolutional kernels? And how is beta chosen? ----- post author response ---- The authors addressed most of my concerns, but I believe choosing a, b, c to explicitly fit NTK is not a good approach for C-exp. You should modify the way you choose a, b, c and re-run the experiments in Sec 4.3.

Review 3

Summary and Contributions: This paper derives new results for the eigenvalues/eigenfunctions of the Neural Tangent Kernel for fully-connected networks (deeper networks, with bias), as well as the Laplace kernel, and demonstrates the similarity between the two kernels both theoretically and empirically. More specifically: --Theoretical proof that NTK eigenvalues, when data lies uniformly on the hypersphere in d-dimensions, decay like O(k^-d) for networks with one-hidden layer and no faster than O(k^-d) for deeper networks [Theorem 1]. Laplace kernels in the same setting decay also as O(k^-d) [Theorem 2]. Hence there is an equivalence between the RKHS of the two kernels, which is also larger than that of the Gaussian kernel [Theorem 3]. --Results for the eigenfunctions of the fully-connected NTK in R^d. --Empirical comparison of the performance of NTK (fully-connected network), Laplace, Gaussian, and \gamma-exponential kernels (as well as homogeneous versions where applicable) on the UCI dataset as well as larger datasets. Indeed, the NTK kernels perform similarly to Laplace. --Empirical comparison, on CIFAR-10, between convolutional NTK (CNTK) and "convolutional" versions of the {Laplace, Gaussian, \gamma-exponential} kernels obtained by recursive application of these kernels to image patches. Interestingly, these latter kernels can perform similarly to CNTK.

Strengths: Soundness of the claims: The results are clearly stated, seem to be theoretically well-supported, and are also confirmed empirically (numerical comparison of the spectra, angular dependence in Fig. 1, as well as indirectly via the performance comparisons in Section 4). Significance, relevance: I think the results in the paper will be of interest to the community studying overparameterized networks as well those working on kernel methods. Naively, one might think that the new class of Neural Tangent Kernels, by virtue of being connected to the compositional/recursive nature of fully-connected networks, would be a richer kernel class than previously known. This paper refutes this for the particular class of NTKs connected to fully-connected architectures. It also performs an early investigation touching on how to construct a powerful class of convolutional-like kernels from existing primitives (\gamma-exponential kernels) by working with image patches. I think the paper also does a nice job balancing theory and empirical work, and I appreciate that the authors did performance comparisons in Section 4 for the variety of kernels they discuss.

Weaknesses: Some treatment (which could be purely empirical) of non-uniform data on the sphere or in R^d would enhance the paper a bit, as it is closer to a realistic setting. (As far as I could gather, the distributions considered are all uniform.) For instance, how do the NTK and Laplace kernel eigenvalues/eigenfunctions compare empirically for the CIFAR-10 dataset? (The performance comparisons are suggestive but only indirect.) Additionally, the main text seems to not discuss which activation functions are being treated, and the supplementary seems to treat the case of Relu. If that is the case, this should be mentioned early on in the main text and abstract; also, some discussion (empirical or theoretical) of any changes in the main conclusions for alternative common activations would be beneficial. Additionally, the discussion of related works could be improved (see "Relation to prior work").

Correctness: The claims and methods appear to be correct, though I have not checked in detail the theory proofs in the supplementary.

Clarity: The paper is clearly written.

Relation to Prior Work: Ref. 4 and 5 in the bibliography are repeated. Some of the related works are not accurately cited and should be corrected. Ref. 41, 35 treat Bayesian inference and do not discuss last layer training under gradient descent. The correspondence with GPs in Ref. 35 was done for single hidden layer fully-connected networks only. The papers and (both in ICLR 2018) derived the GP correspondence for deep fully-connected networks. Ref. 17 derived the arccosine kernel but did not discuss GP correspondences. I'm also not certain that Ref. 1 in its published form (note the arxiv analog spans several different versions) discussed the NTK.

Reproducibility: Yes

Additional Feedback:

Review 4

Summary and Contributions: new connections shown between neural tangent kernel and Laplace kernel and improved results with gamma exponential kernel

Strengths: - new connections shown at the theoretical level and through empirical evaluation - excellent relevance to NeurIPS

Weaknesses: - see below for suggestions for improvement

Correctness: - see below - kernel regression (2) does not correspond with the empirical section

Clarity: overall well written

Relation to Prior Work: can be improved

Reproducibility: Yes

Additional Feedback: Studying the connection between neural networks and kernel methods is important and interesting in general. The authors make a nice contribution in this direction. - in the abstract it is stated that the same eigenfunctions and eigenvalues lead to the same function classes. Later in the paper this is somewhat explained, but it would be good to give additional references on this from learning theory. It is only based on [26] which is more related to approximation theory. - section 2: other early work on the connection between SVM and neural networks is: Cortes C,, Vapnik V., Support vector networks, Machine Learning 1995 (with the use of tanh kernel) Suykens J., Vandewalle J., Training multilayer perceptron classifiers based on a modified support vector method'', IEEE Transactions on Neural Networks, 1999 (defining the hidden layer as feature map in the primal) - eq (2) is rather an interpolation problem. This is not the standard kernel regression approach which allows errors with use of a loss function. It is also not corresponding to the kernel ridge regression used in the experiments. - Lemma 1: please explain what a zonal kernel is and give reference(s). - l.188: convexity versus non-convexity is overlooked here. Please comment on this. - broader impact: this remains to be completed. Possibly the authors may comment here on interpretability of models - A l.17 why is the notation \dot{Sigma} used? Is the dot related to a derivative? - l.49 slightly superior -> slightly better Thanks to authors for the replies, I have taken it into consideration.