NeurIPS 2020

Neural Networks Fail to Learn Periodic Functions and How to Fix It

Review 1

Summary and Contributions: # UPDATE I have read all the rebuttal and other reviews and after the discussion I would like to still recommend acceptance of this paper (7). # REVIEW This paper studies the problem of extrapolation in Neural Networks. In general, extrapolation problem is ill defined, as we ask a model to predict results outside its training regime. However, there are inductive biases that govern many aspects of real life, such as periodicity, that authors focus on in this paper. The main contribution is to propose a simple activation function, that on one hand provided periodicity bias, and on the other avoids issues of training with it that has been encountered in numerous works in the past using sin or cosine activations.

Strengths: - authors identify and propose unified solution for introducing periodicity bias and avoiding typical optimisation issues of this challenge - proposed solution is extremely simple and ready to use out of the box - authors provide simple theoretical foundations of their approach, in an easy to digest way - proposed method is evaluated on a wide range of tasks from different disciplines, including those not having a periodic structure

Weaknesses: - empirical validation could use more examples of clearly non-periodic problems apart from basic computer vision task, but this should not be seen as an argument against accepting this paper, but rather an a valuable addon that would provide even better picture of the proposed solution and test it wide applicability - presentation of proofs in the appendix is of very low quality. It is full of typos, non-gramatical statements, logical issues (authors talk about representing sine and then write a lemma about cosine). These do not lead to invalid results, but look extremely rushed and of low quality.

Correctness: - please consider renaming "Universal Extrapolation Theorem" to "Universal Periodic Extrapolation Theorem" for correctness

Clarity: Overall paper is well written, with small presentation issues (e.g. why is Function capitalised? There are some extra white spaces floating around)

Relation to Prior Work: Paper cites related prior work.

Reproducibility: Yes

Additional Feedback:

Review 2

Summary and Contributions: After rebuttal, I somewhat disagree with the author's disagreement with regards to RNNs. Although I agree that 2D or higher periodic functions are certainly not handled with RNNs, I point out the ubiquity of 1D (i.e. time) periodic functions. So much so that indeed the authors have validated against solely 1D periodic functions. I am hoping the authors will consider 1D periodic as a special case and address comparisons against RNNs more thoroughly. For example, another reviewer suggested the computational efficiency and vanishing gradient problem likely to occur for periodic functions with a large wavelength. The paper proposes a novel activation function to model data with [possible] periodic elements. The idea is justified with some theoretical proofs demonstrating the inability of traditional activation functions to capture periodic data. Validation is performed on several datasets containing periodic elements showing empirically that indeed the activation function is able to learn periodic functions better than traditional activation functions.

Strengths: This paper identifies a significant shortcoming in traditional feedforward networks, in particular their inability to generalize beyond a compact domain. That being said, this is by design as the data is assumed to be i.i.d. with the training dataset sampled from the same distribution as the test dataset. Alternatively this paper considers i.i.d. sampled data in fourier space (restricted to a compact region in R^d), as opposed to R^d space. The intention of this is to allow generalization beyond the compact set within which the training data lies. A straightforward solution is provided to capture this shortcoming with the aid of a particular activation function of a periodic nature. The universal approximation theorem is extended to this activation function. A decent amount of validation is performed with respect to this activation function.

Weaknesses: A significant shortcoming in the approach is the lack of proper and thorough validation with recurrent neural networks. The stated problem (i.i.d. in fourier space, restricted to a compact region in R^d) can be tackled with the auto-regressive approach provided by RNNs. However this is not mentioned, or compared against in the paper. I note that RNNs are somewhat compared against in the appendix, but this is with respect to RNN+Snake vs RNN. Above all this paper needs a comparison showing the ability of feedforward + Snake networks to outperform vanilla RNNs. Beyond this, RNNs should also make up at least some of the related work such that the two competing approaches (autoregressive vs feedforward + Snake) are well compared and contrasted. Although I imagine that this approach *does* solve some of the shortcomings of RNNs (i.e. vanishing gradient across long sequences), a thorough validation and comparison is needed to verify this. Section 3 text should be significantly improved. I found the rather vague and imprecise argument with respect to alternatives unconvincing. It's difficult to evaluate the merits of a mathematical argument which is not formally stated. I also remain unconvinced by the presence of second order terms in the taylor expansion around 0 to be sufficient to explain improvement. This should be formally or empirically explained in more detail. (e.g. by validating against the specific taylor expansion with/without the second order term).

Correctness: I agree with the authors in the shortcoming of traditional feedforward networks to capture periodic data. I somewhat question the inclusion of the 'universality' guarantees provided by this new activation function. I think Cybenko's result can be easily extended to almost any activation function. Perhaps this belongs solely in the appendix with further validation in the main text. I found the financial data prediction validation somewhat less than convincing. With regards to the current health situation, it is not yet clear whether current conditions are 'too remarkable' to be admitted as a suitable test set. Perhaps validation with previous recessions would have been a better idea.

Clarity: The paper is well written enough for my taste.

Relation to Prior Work: This paper is framed 'ok', with comparison provided to the most directly related work. However, I would have appreciated a more broader comparison with respect to both Fourier networks, and RNNs.

Reproducibility: Yes

Additional Feedback: I think this paper is promising given additional work. In particular it must be significantly motivated and empirically validated why an auto-regressive approach (i.e. RNNs) do not work well in this case. However, without this specific validation, I cannot see this paper as ready for publication.

Review 3

Summary and Contributions: The researchers develop their Snake activation which they assert is a better activation function for learning periodic functions (such as sine waves).

Strengths: The problem of feed-forward network extrapolation on periodic functions is a well-known and important problem, which this research seeks to address. The authors provide good heuristics for initialization, which I believe is quite useful for researchers hoping to utilize this methodology on different classes of problems. The authors evaluate, I believe, on a varied set of problems, lending credence to the generalizability of this work.

Weaknesses: The methodology, although well explained, leaves too much uncertainty on the table as far as correctness. Specifically the lack of any discussion for the stopping criteria. It is not clear, at least from my read, whether the researchers cherry-picked their stopping criteria to give their algorithm the best results. I doubt this is what happened, sincerely. However, for NeurIPS quality papers, there should be no doubt about the correctness of the methodology for such basic tasks as stopping criteria.

Correctness: To expound a bit more on the weaknesses above-- For the first problem (3.1), using a set number of epochs to compare the different activation functions is a sub-optimal approach, as we do not know if any of the networks suffered from over-fitting. The authors should use validation loss and early stopping in this particular problem. I don't think the problem with extrapolation is due to over-fitting, however, as I have performed these experiments, myself, long ago, and this is a commonly known problem. Perhaps the authors believed this to be the case and took a shortcut here. For the other examples, it should be clear that the network is stopping based on validation loss, and not the authors just picking a stopping point that works for their model. They should also ensure that the parameters that led to that stopping point are used throughout and not cherry-picked per problem to yield the best results. Though I felt I was careful, it is possible that I somehow overlooked the authors use of correct validation set usage. If I have, please forgive me, and I would be amenable to changing my score.

Clarity: The paper is clearly written. There is a typo for the word "daily" on line 284.

Relation to Prior Work: The authors motivate their problem well. It would have been nice to see a comparison against "Phase-functioned neural networks for character control" Also, it would be nice to see RNNs in general compared to the efficacy of Snake in a standard FFN. Even if RNNs worked better, it still is, in my opinion, wonderful to see a unique solution to periodic problems using only FFNs.

Reproducibility: Yes

Additional Feedback: The list many parameters to their networks, including optimizer, epochs, etc. I think reproduction wouldn't be too difficult, sans the weaknesses due to validation set.

Review 4

Summary and Contributions: In this paper, a novel activation function - Snake – is proposed, aiming to account for extrapolation in periodic signals.

Strengths: (+) introducing a periodic and monotonic activation function (+) the research around new and better activation functions is very relevant

Weaknesses: (-) Some claims are way too strong. *“we study and prove the incapability of standard activation functions to extrapolate” is debatable. The proof is mainly related with ReLU and tanh. * After Corollary 1, row 164: “the proposed activation function is a more general method than the ones previously studied”. In what sense a periodic function is more general? (-) The main problem is the evaluation of the method. The paper does not show that the proposed method provides clear benefits against existing methods. Experiments do not show that Snake activation is better than the existing activation functions - ReLU and their variants (e.g. Leaky ReLU) on any benchmark database. From Figure 5, I can conclude that Swish and Leaky ReLU are comparable with the proposed method (or even better) for CIFAR-10. Thus, why these methods are not also used in the last two experiments (6.2 and 6.3) where Snake activation obtains the best results? The same omission is hold for MNIST (figure 3). The above reflects my understanding, and I may have missed something. But if I am correct, the experiments in this paper fail to demonstrate the usefulness of the method.

Correctness: The method is clear and the proof is mainly related whit ReLU and tanh. Overall, the claims are too strong and are not mirrored by the results. The empirical methodology is not consistent across the experiments.

Clarity: Overall a nice reading, with a good structure. The first part of the experiments (CIFAR 10) can be better embedded in the overall paper context. *Some notations are used in the paper and explained in the appendix. *Minor typos: fucntions -> functions; that the the wide -> that the wide

Relation to Prior Work: There is a clear distinction between the Snake function and other activation functions. Still, a large range of activations functions are even not mentioned, such as SReLU, GELU, ELU, SELU, RReLU. Furthermore, the authors should be aware of the previous work done on the periodic and monotonic activation functions. J. M. Sopena, E. Romero and R. Alquezar, "Neural networks with periodic and monotonic activation functions: a comparative study in classification problems," ICANN, 1999.

Reproducibility: Yes

Additional Feedback: In my opinion, the paper will strongly benefit if *the claims are revised *expand the previous literature (see above) *clarify the experiments, and ensure their consistence