NeurIPS 2020

Kernel Methods Through the Roof: Handling Billions of Points Efficiently

Review 1

Summary and Contributions: In this paper, the authors introduce specialized and well engineered software for kernel ridge regression. The idea is to apply iterative numerical solvers to the Nystrom approximation, but with special care given to considerations like out of core computation, efficient memory usage and allocation, and so on.

Strengths: This is a solid machine learning systems paper, with appropriate care given to the various engineering challenges. I think engineering efforts like this are fantastic, and the empirical results for kernel ridge regression are impressive.

Weaknesses: In my opinion, comparing Nystrom for kernel ridge regression to variational GPs is apples to oranges in a lot of ways that are frankly unfair to variational GPs. In my view, a much more appropriate comparison would be a KeOps based implementation of SGPR or FITC with fixed inducing points. Variational GPs introduce a very large number of parameters in the form of the variational distribution and inducing point locations that require optimization and significantly increase the total amount of time spent in optimization. Methods that train GPs through the marginal likelihood with fixed inducing locations (e.g., as in Nystrom) may have as few as 3 parameters to fit. By contrast, SVGP learns (1) a variational distribution q(u) including a variational covariance matrix, and (2) the inducing point locations. These things result in a much more general version of the model, and are done to target test log likelihood, but increase the number of parameters from a handful to thousands or even millions in the full covariance S case. The effect of even subtle variations in these parameterizations have on wall clock efficiency both in terms of convergence rate and iteration speed is enormous and isn't considered. In fact, the effect of this can be seen in your own results -- you use GPyTorch 1.0.1, which is old enough to not have used a properly whitened parameterization of the variational parameters, while current versions of both GPFlow and GPyTorch do. These parameterization considerations can easily lead to several factors of speed up. The point of the above is not to nitpick specific package versions or various settings that could be used, but rather to argue that I don't believe SVGP is a fair baseline to compare a Nystrom based KRR solver to. More appropriate would be a standard KeOps based implementation of SGPR or FITC, with and without learning the inducing point locations, all of which are already supported by the relevant software.

Correctness: Again, my main point of concern is with the empirical methodology, specifically in the choice of SVGP as a baseline for Nystrom based KRR.

Clarity: The paper is very clearly written, and I have no comments here.

Relation to Prior Work: The authors do a good job of describing some relevant work, although my main source of criticism involves the focus on a comparison of Nystrom and SVGP, when I think there are more appropriate points of comparison.

Reproducibility: Yes

Additional Feedback: I want to emphasize here that I really do appreciate the sheer engineering effort that must have gone in to developing the software underlying this paper and am willing to consider accepting the paper regardless of my points above. From a scientific perspective, I just don't currently believe that Nystrom based KRR / predictive-mean-only GP and SVGP are at all comparable methods, although I am willing to be convinced otherwise. SVGP targets a much more general form of model, and pays a heavy parameterization price for it. I appreciate that you have at least used diagonally constrained covariance matrices for SVGP, since in the full variational covariance S case more than 99% of the parameters would be devoted to a predictive term that you aren't using. Nevertheless, I still think that approximate methods like FITC / SGPR trained with KeOps targeting the marginal likelihood would be much more appropriate points of comparison. I believe it is likely that your approach will significantly outperform a simple KeOps based solver for high dimensional datasets. --------- In the discussion period, I was in the minority for voting below the threshold. I am certainly not willing to argue strongly for the paper's rejection, as I mention above I quite like the methodology and appreciate the sheer engineering effort involved. I do strongly maintain my stance that using SVGP as a primary baseline is not useful. I appreciate the authors' point in the rebuttal that they point out the additional advantages of SVGP, but I feel that nevertheless using it as a primary point of comparison is still apples to oranges for the reasons I mention. Optimization considerations like whitening make enormous impacts on SVGP as a result of the sheer number of parameters involved -- particularly with a large number of inducing points -- and I would urge the authors to consider these factors more fairly than they currently do.

Review 2

Summary and Contributions: The computations involved in kernel methods must be done efficiently to scale kernel methods to large datasets. This paper proposes and implements a more computationally efficient version of FALKON, a kernel method by Rudi et al., 2017 that is practically statistically optimal based on the Nystrom method. This paper's main contribution is their highly-optimized implementation of FALKON that runs almost 20x faster than the version implemented by Rudi et al., 2017. The main algorithmic ideas of FALKON — subsampling points to make the Nystrom approximation, using iterative methods to reduce computational burden, using a carefully designed preconditioner that accelerates convergence, etc. — remain the same, but the authors carefully piece these ideas together to minimize memory usage and maximize computational throughput. This is done by using equivalent expressions that minimize the required memory, implementing a distributed, out-of-core, in-place Cholesky decomposition, and pipelining computation to overlap computation and data-transfer. The paper also derives a version of FALKON for loss functions beyond squared loss in the appendix. The paper compares their FALKON implementation with squared and logistic loss to other packages designed for large-scale kernel methods, including EigenPro, GPyTorch, and GPFlow. The results show large speedups despite modest hardware and demonstrating that proper engineering can make FALKON viable for even billions of data points. ************************** Post rebuttal: I thank the authors for their rebuttal. It is clear that the implementation is highly optimized. The rebuttal and promised edits addressed my concerns and I will raise my score to a 7. **************************

Strengths: The biggest strength of the paper is in the careful considerations of how to better implement FALKON by combining in-place Cholesky decompositions, GPU acceleration, state-of-the-art libraries like KeOps, preconditioning, and others. This allows the paper to use the Nystrom kernel approximation on over a billion data points with only 2 GPUs and a reasonable amount of RAM. The authors also took care to evaluate performance on a series of representative tasks, including sparse data and high-dimensional data. As large-scale machine learning pushes the boundaries of our computational limits, clever optimizations such as the ones presented in the paper enable practitioners to do more with the same computational resources, which is relevant for the community.

Weaknesses: Although the optimizations of the paper are sound and impressive, the empirical evaluations could be improved. Given that the main contribution of the paper is in building a highly optimized library, the presented benchmarks for efficiency should be as strong as possible. I appreciate the comparison to other packages designed for large scale kernel methods. However, Table 2 should clarify that the results for GPyTorch and GPFlow correspond to stochastic variational Gaussian processes (SVGPs). Currently, this is only mentioned in the main text away from the table. The same can be said about Figure 1. Also, does "time" column in Table 2 correspond to total training time, total prediction time, or both? Similarly, in Figure 1, what does the x-axis measure? Given that the training procedure for SVGP, stochastic optimization in minibatches, is drastically different from that of FALKON, solving a linear system in the case of squared loss, comparing training times without any qualifications would be potentially misleading. I would recommend comparing the training time for FALKON against just EigenPro, the most similar method, but comparing prediction times for FALKON against all the kernel methods. The authors should still report the training times for SVGPs but make clear how the training procedures differ from FALKON. If it is possible with the proposed implementation, the authors may want to include results on how the performance scales beyond 2 GPUs. Currently the appendix only shows the scaling with 1 and 2 GPUs.

Correctness: The theoretical claims are correct but can be hard to understand at places. However, the empirical methodology could be improved as I mention in the "Weaknesses" section.

Clarity: While the paper provides mostly clear exposition, the writing in the methodology sections can be improved. For example, much of section 3 mentions computing kernel matrices "in blocks", e.g. line 100, line 155, line 157, etc. However, it is not clear what this means. Are those matrices divided into blocks and computed one at a time, possibly in parallel? Similarly, the explanation around Figure 4 is also confusing. The allocated matrix is size m x m with the upper half storing an upper triangular T. However step (d) claims to store TT^\top, a m x m matrix, in the lower half of the allocated memory which is not enough for the m x m matrix.

Relation to Prior Work: Yes the paper makes clear that they are optimizing the computations required for a prior method.

Reproducibility: Yes

Additional Feedback: line 156 "in-place Cholesky decomposition is performed on the GPU" Which matrix is being decomposed? line 158: "Cholesky is performed on the lower triangle" Which matrix is being decomposed? line 171: "q x m" Based on figure 3, should this be "q x d"?

Review 3

Summary and Contributions: I have read the rebuttal and other reviewer's comments. I appreciate the authors' reply. I would like to keep my score on this work. ========================== This paper integrates several practical techniques to further scale the training of kernel machines. Its algorithm is based on a preconditioned conjugate gradient solver proposed in [43]. The major techniques proposed in this work are to relax memory constraint (of GPU) by performing low precision out-of-core computation and utilize more computation power by distributing computation among several GPUs. The final implementation shows satisfying speedup over several state-of-the-art kernel methods with highly competitive performance.

Strengths: I tend to accept this work as it presents solid algorithmic and system improvements for the state-of-the-art kernel methods.

Weaknesses: As space used by the preconditioner is quadratic in the number of Nystrom centers, it seems challenging to further increase the size of the kernel model, e.g., beyond 10^6 centers. What would be the possible solutions? Adopting float32 over float64 gives substantial savings on memory without much loss in performance. Do the authors think that the precision can be further dropped to float16 or even one byte (similar to what deep net uses nowadays)? If not, what are the major obstacles? I noticed that in the machine for experiments is equipped with two 12-core CPUs. So would CPU be the bottleneck when training with a consumer desktop (which typically has one cpu socket)? It would be nice if the authors can also present CPU and GPU utilization during training.

Correctness: Both are correct.

Clarity: It is very well organized and easy to follow.

Relation to Prior Work: Yes

Reproducibility: Yes

Additional Feedback:

Review 4

Summary and Contributions: Kernel methods provide an elegant and principled approach to nonparametric learning, but so far could hardly be used in large scale problems, since naïve implementations scale poorly with data size. Recent advances have shown the benefits of a number of algorithmic ideas, for example combining optimization, numerical linear algebra and random projections. Here, we push these efforts further to develop and test a solver that takes full advantage of GPU hardware. Towards this end, we designed a preconditioned gradient solver for kernel methods exploiting both GPU acceleration and parallelization with multiple GPUs, implementing out-of-core variants of common linear algebra operations to guarantee optimal hardware utilization. Further, we optimize the numerical precision of different operations and maximize efficiency of matrix-vector multiplications. As a result we can experimentally show dramatic speedups on datasets with billions of points, while still guaranteeing state of the art performance. Additionally, we make our software available as an easy to use library.

Strengths: The strength lies in the fact that a number of carefully design computational approaches specifically adapted to the GPU were developed for solving SVM using the Nystrom method. The resulting Falkon algorithm developed in the paper is efficient and the evaluation is sound. It is of extreme relevance to the NeurIPS community as this line of works stands in the trend of tweaking neural networks for performance gains, adding much needed diversity to the pool of thoughts.

Weaknesses: No much obvious weakness. Only nitpicking.

Correctness: The claims and methods are provably correct. The empirical methodology is well thought out.

Clarity: The paper is well written.

Relation to Prior Work: Kernel methods used for large scale problems are the most related line of work. In this paper, the preconditined conjugate gradient solvers that take advantage of GPUs were compared with EigenPro, GPytorch, and GPflow. The paper could use more description of the close relative of EigenPro.

Reproducibility: Yes

Additional Feedback: see sections above