NeurIPS 2019
Sun Dec 8th through Sat the 14th, 2019 at Vancouver Convention Center
Paper ID:2126
Title:Principal Component Projection and Regression in Nearly Linear Time through Asymmetric SVRG

Reviewer 1

The main contribution relies on the AsySVRG to solve the squared ridge regression. This result is interesting and the properties is attractive. However, in the proof, it relies on a key result that $(M+M^T)/2 \succeq \mu I$ implies that $x^TMx \geq \mu x^T x$ (the equation line 475-476). This result is not obvious but detailed proof is missing.

Reviewer 2

## Update ## I'd like to thank the authors for their thorough response. I will maintain my initial score. I did review the discussion of rLanczos and sLanczos in Appendices F.1 and F.2 and agree that several additional sentences in the main paper should suffice to clarify the methods. ## Initial Review ## The authors propose a nearly linear-time randomized algorithm for principle component projection (PCP) and principle component regression (PCR). This is achieved using Zolotarev rational functions to approximate the sign function instead of the polynomial function approximations used in previous work. Computing the Zolotarev rational functions requires solving squared ridge-regression problems. The authors reduce solving squared ridge-regression problems to solving asymmetric linear systems and analyze convergence of a SVRG solver for generic asymmetric linear systems. The properties of the SVRG solver are further analyzed in the context of the specific asymmetric systems corresponding to these squared ridge-regression problems, which yields a tighter convergence (in high-probability) bound. Accelerated high-probability rates are proved in both the generic and specialized settings. The accelerated rate specialized to solving squared ridge-regression problems yields the proposed algorithm for PCP and, by reduction, PCR. The paper concludes with an experimental evaluation on several synthetic PCP problems. Variants of the proposed algorithm are shown to perform faster than methods based-on polynomial approximations and competitively with the Lanczos algorithm. Originality: The proposed algorithms are novel. Related work is appropriately cited. Clarity: The paper is clear and easy to follow. However, I suggest that some additional discussion of rLanczos and sLanczos be added to main text if possible. See minor comments below. Quality and Significance: This paper appears to resolve an significant outstanding problem with respect to randomized PCP and PCR algorithms. As a part of the analysis, several high-probability convergence bounds are proved for SVRG methods applied to solving asymmetric linear systems. This alone seems like an excellent contribution and a step towards faster linear algebra primitives via the modern optimization toolbox. Empirical investigations are provided to confirm the authors' theoretical conclusions. These experiments are small, but well-executed. Minor Comments: - Line 375: The title of Appendix C should read "Proofs for Results in Section 2". - Line 202 should read "We denote ... as *the* cost of *the* matrix-vector of ..." - Line 227 should read "... *the* data matrix A..." - Line 230 should read "In all figures below, *the* y-axis *denotes*..." - Section 2 might flow better if Figure 1 was placed at the top of the page, rather than embedded in the text. The same is true for Algorithm 1 and Figure 3.

Reviewer 3

This paper improves upon existing approaches to the principle component projection problem in an interesting way. The approach builds upon existing approaches by using more sophisticated techniques for matrix sign approximation, and for solving the resulting subproblem that arises. As a reviewer I'm knowledgable on SVRG and variants rather than the PCP problem, so I can't comment much on the novelty of the innovations specific to PCP. The application of SVRG here is interesting however. I'm not aware of previous applications of SVRG to asymmetric matrix solves as done here. They use a weighted sampling variant to get tighter bounds, which is interesting. Weighting sampling doesn't see much practical use as the sampling weights can be hard to determine, however for matrix solves there is a clear choice as used here. The experiments look well executed, however the focus on a single synthetic task is a definite weakness of the paper, it would be nice to see a real-world application. The paper is well written, although very dense. I think the description of the results is presented with sufficient formality. I don't have too many comments in that regard. In terms of significance this seems like a solid improvement in the research area. Smaller notes: Abstract: “had superlinear running times” should be sub-linear? I'm unsure. “directly applied to problem” should be “directly applied to this problem” “PCP and PCR to solving following “ should be “PCP and PCR to solving the following” “This is stated in formal in the” should be “This is stated formally in the” “compared with ones “ should be “compared with the ones “