NeurIPS 2020

Revisiting the Sample Complexity of Sparse Spectrum Approximation of Gaussian Processes

Review 1

Summary and Contributions: This submission conducts sample complexity analysis for sparse spectrum Gaussian processes (SSGPs) when the input data exhibit certain clustering behavior. Further, a variational auto-encoder (VAE) based embedding algorithm is developed to project the raw input data on the hidden space where the desired regularity conditions are satisfied. Numerical tests validate that the proposed VAE-based SSGP exhibits improved performance relative to the conventional one.

Strengths: Commendable efforts to initiate sample complexity analysis for SSGPs under regularity conditions on the input data. Experiments demonstrate that the proposed SSGP method enjoys superior prediction performance relative to the original SSGP with the same number of spectral frequencies.

Weaknesses: 1. It is not clear how the VAE module couples with the SSGP for parameter training. It seems that the training objective for VAE (cf. Eq. (10)) relies on the parameters $\Theta$ of the GP. Further, is the dimension of the hidden variable $z$ the same as that of $x$? Further elaboration is due here. 2. Despite the claim that the cost of training the VAE is only linear in the number of data points, one would be interested to see the overall training time of the proposed SSGP relative to the plain SSGP. 3. In addition to RMSE-based evaluation, comparisons via the negative predictive log-likelihood that takes into account uncertainty would be recommended. 4. The results in this submission seem to hold for Gaussian kernels only. How about other shift-invariant GP kernels? A remark is expected in this aspect. 5. The result of sample complexity is only mentioned in the text after Theorem 2. Since this is a major result, it is recommended to state it in Theorem 2. 

Correctness: The claims and method seem correct. The proofs in the supplementary file have not been checked thoroughly.

Clarity: This paper is well written in general, but clarity could be improved. Please refer to the comments earlier.

Relation to Prior Work: Yes, differences from the most related works [3, 27, 28] have been discussed.

Reproducibility: Yes

Additional Feedback: ==update after rebuttal== The rebuttal letter has addressed most of my concerns regarding the NLL performance and training time. What is due for clarification is the explicit form of the VAE training objective; e.g., how about the additional term in (9), as mentioned also by Rev 4. Based on these considerations, I will increase my score from 6 to 7.

Review 2

Summary and Contributions: The paper proposes a novel approximation to sparse-spectrum Gaussian processes. The authors also provide a data partitioning algorithm with better sample complexity.

Strengths: The theoretical results seem sound. The experimental results are encouraging.

Weaknesses: Condition 1 seems very restrictive. (Please see my additional feedback below.)

Correctness: The main claims seem correct.

Clarity: The paper is clear.

Relation to Prior Work: Relation to prior work is partially addressed. (Please see my additional feedback below.)

Reproducibility: Yes

Additional Feedback: Some literature in the estimation of Gram matrices needs to be discussed: - Rahimi and Recht, "Random features for large-scale kernel machines", NeurIPS 2007. - Sriperumbudur and Szabo, "Optimal rates for random Fourier features", NeurIPS 2015. - Sutherland and Schneider, "On the error of random Fourier features", UAI 2015. Please discuss why Condition 1 is reasonable and not too restrictive. More specifically, please discuss the specific rates of the number of Gaussian components in the mixture b \in O(log n), the mixture weights \pi_i \approx 2^{i/2} as well as the variances \gamma_i \in O(1/\sqrt{d}). === AFTER REBUTTAL I am not too satisfied with the tangential answer given by the authors regarding Condition 1. I was specifically asking about the particular rates of b, \pi_i and \gamma_i. Given the above, I keep my initial evaluation of 6.

Review 3

Summary and Contributions: The paper exploits the clustering structure of the data to provide an approximation for the Gaussian Processes using mixtures of Gaussians, and develops a related algorithm based on finding the clustering of the data in a latent space using the variational auto-encoders techniques.

Strengths: Well-written, clear analysis, and interesting clustering algorithm

Weaknesses: It is unclear how frequently Conditions 1-3 are satisfied in practice

Correctness: Yes

Clarity: Yes

Relation to Prior Work: Yes

Reproducibility: Yes

Additional Feedback: The main question that I have that does not seem to be discussed is the complexity of the training step in the proposed scheme. Presented results were motivated by a high computational cost of learning and inference O(n^3) for the general Gaussian Processes; is the computational overhead introduced by VAEs to discover the latent space where the data may be clustered worth it for large n? What is the computational cost introduced? Is it possible that the resulting scheme will be even more expensive compared to the full GP procedure? ------------- Post-rebuttal: I am satisfied with the additional clarifications that the authors produced in their rebuttal. It would be beneficial to include these considerations in the final version of the paper. I think that the paper is a good study, while I still have some doubts about practicality meeting the conditions theoretically studied in the paper (no guarantees for a larger latent space); hence overall I don't see any particular reason to change my score, so I leave it as is (7 - accept).

Review 4

Summary and Contributions: The paper revisits the bounds for sparse gaussian processes, in the case where the data is distributed as a mixture of gaussian clusters. This theoretic analysis comes with very encouraging bounds that suggest a promising avenue for research. the paper also comes with the idea of training an auto-encoder that allows going from the input space to a latent space with the appropriate clustered structure.

Strengths: * The paper is very clear on both GP and prior work for sparse GP approximation. It flows naturally and provides adequate references * the motivations are quite clear * The contributions are at a theoretical level and do leave space for fruitful research. Basically, the authors show that bounds for GP inference on clustered data can be made super tight and computationally interesting. The idea to go from input space to a latent space where this clustered structure is preserved is interesting. * I appreciated the quality of the available github implementation

Weaknesses: * the experimental section is rather weak. Although results are shown as interesting, scalability is not demonstrated. Still, I would say that the contribution is rather at a theoretical level and that these experiments can be considered sufficient * I think there is a point where the paper somehow looses its initial rigour. This happens where the VAE scheme is introduced. it's ok to me, but I would suggest the authors acknowledge that that part is a baseline attempt at producing a clustered feature space. Indeed, it would make it easier for the reader to understand where room for improvement stands.

Correctness: I couldn't check everything, but the paper definitely looks correct to me.

Clarity: yes

Relation to Prior Work: A reference to the reparameterization trick could be beneficial

Reproducibility: Yes

Additional Feedback: • P2L68: although it's super common, it does not read obvious to me at first read that your further analysis is limited to this particular case of a gaussian (squared exponential SE) kernel • Related to the question above: although the SE kernel is common, do you think that the approach could be applicable to some that are more exotic ? For instance by seing them as sums of SE ? or some other trick ? • (9): how is the additional term computed in practice ? I can see the code is nicely available, it would be good to quickly advertise it here to reassure the anxious reader ? • I would say that the experimental section is a bit disappointing. The method is advertised for scaling up, but experiments report a few thousand data points only. I would have appreciated someting in the vein of millions. • P6L247-onwards: it looks to me that the choice of an auto-encoder is arguable. In a sense, I see this as a very conservative and pessimistic way to go for this clustering. Indeed, as far as I understand, the decoder is actually not used, but only as a regularizer ensuring that the trained clusters are indeed capturing enough to reconstruct the training data. It is actually likely that much less than this is required: maybe the GP to train only needs some super specific part of information, that would actually not need full reconstruction. put it otherwise: I wonder whether some joint training of the GP and the parameters would be feasible, with some cost function for the decoder that would be weaker than reconstruction. --- after rebuttal, I notice that the authors decided not to take my comments into account. lowered score in accordance: 1. they only consider toy datasets (thousands datapoints) everywhere , while.claiming.scalability 2. nothing IMO justifies the use of an auto-encoding criterion for the clustering procedure, and it loo's to me, again, like a conservative/pessimistic approach. This is not even discussed.