Paper ID: | 7077 |
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Title: | Kernel quadrature with DPPs |

*** UPDATE *** Thank you for your helpful response. It is clear that there are shortcomings in the presentation to be addressed, but on the basis that these will be addressed in a revision of the manuscript I am prepared to increase my score from 5 to 6. This paper considers kernel quadrature (KQ) methods for numerical integration. The novelty comes from taking a sample from a determinental point process (DPP) as the point set. The paper focuses on a theoretical analysis, and includes a cursory empirical check that the method performs as expected. An interesting finding is that DPP-KQ empirically out-performs Bayesian Monte Carlo (BMC). The theoretical contribution looks carefully written and technically accomplished. However, I am sorry to have to say that I do not think the main result is useful. I will try to explain why: Let's take the Sobolev space of order s on [0,1], as indeed the authors also consider in the manuscript. In that context, it has been shown that the minimal worst-case error (over the unit ball of the Sobolev space) for a quadrature method based on a deterministic point set is O(N^(-s)). For a randomly selected point set, the minimal mean square error (MSE) (over the unit ball) is O(N^(-2s-1)). I believe such results can be traced back to Bakvalov and Suld'in in the 1950s Soviet literature, but more modern accounts can be found in the information-based complexity literature of Novak and Wozniakowski in the 1980s. The authors do not appear to be aware of these results and, more importantly, they do not appear aware that the optimal O(N^(-s)) rate for a deterministic point set can easily be established for Bayesian Quadrature (BQ) by taking a uniform grid as the point set and applying the fill-distance-type bounds from e.g. the book "Scattered Data Approximation" of Holger Wendland. See e.g. [BOGOS2019,KOS2018] and the references therein for details. Such "uniform grids are optimal" results cover a range of settings and kernels, which rather raises the question of why one would want to use a random point set. Indeed, in the BQ work of O'Hagan the point set was selected to minimise the worst-case error (c.f. posterior standard deviation of the integral). Of course, BMC uses a random point set, and in some cases - such as integration on manifolds - a random set can be justified since it may not be possible to easily construct a grid. But in simple situations like integration on [0,1], I do not see why randomness would be helpful. The authors justify their analysis on several occasions by the claim that BMC does not have theoretical guarantees. In fact, a rate O(N^(-2s)) for the MSE was established for BMC in the Euclidean context in [BOGOS2019] and in the manifold context in [EGO2019]. This is actually a better rate than what the authors have demonstrated for DPP-KQ, and this rather undermines the extent of the contribution. All this being said, the authors demonstrate that DPP-KQ out-performs BMC. This is a nice finding, but it is not demonstrated extensively enough to justify the paper in its own terms. For example, the experiments were limited to dimension d = 1 and also g = 1. Minor: l23: Higher-order QMC methods also exist, which achieve optimal Sobolev rates - see the work of Josef Dick and colleagues. l24: Reference did not compile. l74. O'Hagan was not the inventor of BQ, that can be traced back at least to [L1972]. l76. The greedy selection of points in BQ can be theoretically analyses as a special case of the work of Ronald DeVore on greedy approximation in Hilbert spaces, so it is not true to say that there are no theoretical guarantees. l79. The authors state that "implementing it [BQ] usually requires some heuristics". This rather overlooks the substantial contribution of [JH2018], [KS2018], who reduced the computational cost of BQ to near linear in N. l103. \mathbb{N}^* undefined. l110. "speaking, seeing" -> "speaking, the probability of seeing" l144. It is not clear in the main text that k is being defined as thr limit of (10). l144. The property being assumed is sometimes called "unisolvency of the point set". l147. There is a Corollary 1 in the main text and the appendix, but they are different. l155. The use of "quadrature error" is not appropriate - this is not a quadrature error, since there is no integrand. More precisely, it is the worst case error over the unit ball of the RKHS. l178. The decreasing nature of the \sigma_n needs to be explicitly assumed. l195. The bound in (21) seems quite loose, especially if the eigenvalues are geometrically or exponentially decreasing? l224. The authors should take care that when they write "BQ" they really mean "BMC". Otherwise, they need to explicitly explain in the main text how the points for BQ are being selected. [BOGOS2019] Briol F-X, Oates, CJ, Girolami, M, Osborne, MA, Sejdinovic, D. Probabilistic Integration: A Role in Statistical Computation? (with discussion and rejoinder) Statistical Science, 34(1):1-22. (Rejoinder on p38-42.) [EGO2019] Ehler M, Gräf M, Oates CJ. Optimal Monte Carlo Integration on Closed Manifolds. Statistics and Computing, to appear, 2019. [JH2018] R. Jagadeeswaran and F. J. Hickernell. Fast automatic Bayesian cubature using lattice sampling, 2018. [KS2018] Karvonen, T. and Sarkka, S., 2018. Fully symmetric kernel quadrature. SIAM Journal on Scientific Computing, 40(2), pp.A697-A720. [KOS2018] Karvonen T, Oates CJ, Särkkä S. A Bayes-Sard Cubature Method. Advances in Neural Information Processing Systems (NeurIPS 2018). [L1972] Larkin, F. M. (1972). Gaussian measure in Hilbert space and applications in numerical analysis. Rocky Mountain J. Math., 2:(3) 379–421.

+ Detailed comments on the methodological and empirical contributions: The experiments lack a simple baseline of using uniform grids as design points of kernel quadrature. It is known that this method attains the optimal rate of convergence for deterministic quadrature in the Sobolev setting: see Corollary 1 of the following paper: Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settings https://link.springer.com/article/10.1007/s10208-018-09407-7 Thus, I'm wondering whether kernel quadrature with DPPs can outperform such a simple baseline. If not, what is the advantage of the proposed approach? An obvious drawback of the use of uniform grids is that it suffers from the curse of dimensionality. Thus, another question would be whether the use of DPPs works for for modestly large dimensional problems. This point might need a discussion. + References for Bayesian / kernel quadrature are not up-to-date. For instance, the following paper, which has been on arXiv for several years and gained a number of citations, is one of the key references for Bayesian quadrature. This paper provides convergence analysis of Bayesian quadrature methods (or equivalent kernel quadrature). The authors will also find other papers on Bayesian / kernel quadrature that appeared in machine learning conferences such as NeurIPS and ICML. Probabilistic Integration: A Role in Statistical Computation? Statist. Sci. Volume 34, Number 1 (2019), 1-22. https://projecteuclid.org/euclid.ss/1555056025 The following paper is strongly related to the topic of the current paper, and needs a discussion. On the Sampling Problem for Kernel Quadrature ICML 2017 http://proceedings.mlr.press/v70/briol17a.html + How the design points are generated for "BQ" in the experiments? Did the authors use the approach of Huszár and Duvenaud [17]? If so, this should be explicitly mentioned. The authors mention that BQ does not have theoretical guarantees, but this is a bit confusing. As shown in the above papers, there are several theoretical guarantees for BQ methods. Minor comments: - The authors mention that Proposition 1 is Proposition 2 in Bach [3], but it seems that this result Proposition 1 in [3]. Also, in Eq. (7), the supremum over the weights w seems to be infimum in the original result of [3]. - The notation should be defined. For instance, where is \mathbb{N}^* defined? It seems that this is the set of positive integers, but I don't think this notation is standard in the literature. - The Sobolev spaces discussed in this paper are periodic Sobolev spaces (also known as Korobov spaces in the QMC literature), so the authors should mention this.

Paper summary: motivated by the need to improve convergence rates for quadrature rule for functions living in an RKHS, the paper proposes to sample quadrature nodes from a determinantal point process (DPP), and the weights are found by solving a least-squares problem. The paper analyzes the expected squared error of the proposed quadrature rule, bounding the convergence rate in terms of the spectrum of the kernel. The paper then empirically validates the proposed method with numerical simulation for functions in RKHSs associated with the Sobolev and the Gaussian kernel, showing faster convergence than kernel herding and leverage score sampling. Strengths: 1. The idea of sampling from a DPPs, whose kernel is associated with the kernel of the RKHS, is interesting and novel to the best of my knowledge. 2. The proposed quadrature rule gets explicit convergence rate, for example in the case of finite N (number of nodes) when lambda = 0, unlike that of Bach [3]. 3. Numerical simulation shows that the proposed method performs wells, often on par with Bayesian quadrature (but with convergence rate) and better than Monte Carlo, kernel herding, and leverage score sampling [3]. 4. The bound in expectation using the DPP (Section 4.2.2) is elegant. Weaknesses: 1. The proposed method only gets convergence rate in expectation (i.e. only variance bound), not with high probability. Though Chebyshev's inequality gives bound in probability from the variance bound, this is still weaker than that of Bach [3]. 2. The method description lacks necessary details and intuition: - It's not clear how to get/estimate the mean element mu_g for different kernel spaces. - It's not clear how to sample from the DPP if the eigenfunctions e_n's are inaccessible (Eq (10) line 130). This seems to be the same problem with sampling from the leverage score in [3], so I'm not sure how sampling from the DPP is easier than sampling from the leverage score. - There is no intuition why DPP with that particular repulsion kernel is better than other sampling schemes. 3. The empirical results are not presented clearly: - In Figure 1: what is "quadrature error"? Is it the sup of error over all possible integrand f in the RKHS, or for a specific f? If it's the sup over all f, how does one get that quantity for other methods such as Bayesian quadrature (which doesn't have theoretical guarantee). If it's for a specific f, which function is it, and why is the error on that specific f representative of other functions? Other comment: - Eq (18), definition of principal angle: seems to be missing absolute value on the right hand side, as it could be negative. Minor: - Reference for Kernel herding is missing [?] - Line 205: Getting of the product -> Getting rid of the product - Please ensure correct capitalization in the references (e.g., [1] tsp -> TSP, [39] rkhss -> RKHSs) [3] F. Bach. On the equivalence between kernel quadrature rules and random feature expansions. The Journal of Machine Learning Research, 18(1):714–751, 2017. ===== Update after rebuttal: My questions have been adequately addressed. The main comparison in the paper seems to be the results of F. Bach [3]. Compared to [3], I do think the theoretical contribution (better convergence rate) is significant. However, as the other reviews pointed out, the theoretical comparison with Bayesian quadrature is lacking. The authors have agreed to address this. Therefore, I'm increasing my score.