NeurIPS 2020

Robust Recovery via Implicit Bias of Discrepant Learning Rates for Double Over-parameterization

Review 1

Summary and Contributions: The paper proposes a low-rank and sparse matrix recovery method which considers a double over-parameterized model solved by gradient descent with discrepant learning rates. The proposed method is provably correct, free of prior information, and scalable. Numerical experiments including low-rank matrix recovery and image denoising are provided to justify the empirical effectiveness of the proposed approach. ==post-rebuttal== The authors have answered some of my questions in the rebuttal. Although I'm still concerned about its limitations in real applications, I decide to raise my score to 7 considering its theoretical contributions.

Strengths: The idea of applying double over-parameterization to robust low-rank matrix recovery has certain novelty, and the convergence analysis via dynamic gradient flow together with learning rate discussion is interesting and insightful to other related works.

Weaknesses: 1. Natural images especially non-texture type of images may not have intrinsic low-rank structures would limit the application of the proposed method on image denoising. 2. Since only the L1-regularization is imposed on the noise component, the proposed method can handle the salt-and-pepper noise well but the denoising performance is unknown for other types of noise. 3. The paper still needs more practical guidance on the learning rate selection case-by-case and a brief discussion of the impact of sampling rate on the performance.

Correctness: The proposed method and theoretical discussions are fine, but the numerical experiments have certain issues. The paper in fact only considers the image denoising case with the salt-and-pepper noise, without any other types of image degradation. So some statements about image recovery sound over-claimed. More numerical comparisons with state of the art should be included. Computational time comparison and convergence behavior discussions are missing as well.

Clarity: The paper is mostly well written but not always. For example, Section 2.1 is confusing in terms of notation and organization. The representation of s in (4) is not well explained/defined before using. In (5), the assumption about linear measurements b=As is not consistent with that in (4).

Relation to Prior Work: Yes.

Reproducibility: No

Additional Feedback: More implementation details could be added. In Figure 3, PSNR values should be put either under the subfigures or in the caption.

Review 2

Summary and Contributions: This paper studies recovery of positive semidefinite low rank matrices from undersampled linear measurements that are corrupted by sparse noise. The authors introduce a least squares problem that uses a factorization of the matrix into a product UU^T and a certain quadratic representation of the noise with two vectors. (This is apparently the motivation of the term double over-parametrization.) They study the implicit bias that running gradient descent on this functional introduces. They show that under some assumptions gradient descent indeed converges to the minimizer of a certain functional that incorporates a nuclear norm term (promoting low rank) and an l_1-term (promoting sparse noise). They demonstrate the effectiveness of this approach via numerical experiments.

Strengths: The paper makes progress in the understanding of overparametrization and implicit bias of gradient descent. Moreover, it provides a new method for robust low rank matrix recovery. The paper incorporates some recent findings and techniques. The focus is clear and the flow is smooth. The experiments are illustrative. In the supplementary files the authors even provide the code for their experiments.

Weaknesses: The authors restrict to commuting measurements in their main result. While previous works have also made this restriction, it seems to rule out most cases of interest. It would be good if the authors could discuss at least 1-2 examples of commuting measurements, ideally appearing in practice. While the result itself appears to be novel and important, the proof seems to be quite similar to the reference [7] by Arora et al. It would be good if the authors could discuss commonalities and differences of their proof to [7].

Correctness: Yes. The methodologies for proof and experiments are reasonable.

Clarity: The content is clear and the paper is well-written.

Relation to Prior Work: The paper introduces a clever combination of two methods. The relation of previous contributions is discussed adequately.

Reproducibility: Yes

Additional Feedback: To be honest, I find the term "double overparametrization" a bit strange. I would still call it simply "overparametrization". Perhaps, the authors could think about this point and potentially adjust. I would suggest that the authors briefly discuss the following point which is sometimes overlooked when discussing implicit bias of gradient descent in the context of low rank matrix recovery. When additional restricting to positive semidefinite matrices it turns out that the original low rank matrix is often the UNIQUE solution to the linear equation y=A(X) that is positive semidefinite, see the paper "Implicit regularization and solution uniqueness in over-parameterized matrix sensing" by Geyer et al., arxiv:806.02046, for details. In such cases, it does not make much sense to speak about implicit bias and it is not surprising that gradient descent converges to this unique solution. (It is not completely clear to me, however, whether commutativity rules out such uniqueness.) In the context of the present paper, however, it seems that uniqueness of the solution does not hold, due to the additional terms representing the sparse noise. (However, if the noise terms are fixed, then due to positive definiteness, the solution may be unique in many cases of interest.) Another point: Section 2.2, last sentence: a reader might misinterpret this and the previous sentences in the sense that it is a weak point of (8) that one needs to choose the parameter lambda in the correct way. However, the right choice \lambda = 1/\sqrt{n} does not depend on unknown signal characteristics, so that this is fine. In the end, the gradient descent needs to choose the step size \alpha = 1/\lambda so it does not escape this problem, but again it is not a real problem because \alpha=1/\sqrt{n} is known to be the right choice. I would suggest to slightly adjust the text in order to avoid misunderstandings. ----------------- Comments after reading the author feedback and other reviews: I still think that this is a very good paper. However, my comment about the assumption of commuting measurements is not yet appropriately answered. I did not mean that they should specify a basically equivalent definition (joint diagonalization), but I was interested in a practical example. I am still not convinced that commuting measurements can be useful in practice. Also, I think that the commutativity assumption not only simplifies the analysis, but in fact "assumes away" most of the difficulties and the non-commuting case will actually be MUCH harder and possibly even much different.

Review 3

Summary and Contributions: The paper proposes an algorithmic regularization via the gradient descent (GD) (6) to solve the optimization problem (4) explicitly and the problem (8) implicitly in low-rank matrix recovery. The two problems are connected to each other by setting \alpha=1/\lambda, which is justified in Theorem 1. Numerical simulations show that the proposed algorithm (6) outperforms conventional algorithms in robust PCA and robust recovery of natural images.

Strengths: The main contributions are twofold: One is a doubly over-parameterized formulation (4). The other contribution is Theorem 1 claiming a connection between the two optimization problems (4) and (8). The proof of Theorem 1 needs to be improved, as pointed out in "correctness." Nonetheless, I believe that Theorem 1 is correct.

Weaknesses: The proposed algorithm requires a sufficiently small learning rate \tau in (6) in principle because Theorem 1 depends heavily on the continuum approximation in the limit \tau\to0. Nonetheless, Theorem 1 assumes that the GD (6) solves a global solution. These two assumptions might conflict with each other in practice.

Correctness: The current proof flow of Theorem 1 is confusing since the paper attempts to prove a global optimality by using a KKT "necessary" condition in Lemma 1. More precisely, we need a KKT sufficient condition: if a solution satisfies KKT conditions, then the solution is a global optimum. Nonetheless, I believe that this issue can be fixed by proving that the problem (8) is convex.

Clarity: Yes in the mainbody. No in the supplementary materials. The latter needs to be improved.

Relation to Prior Work: Yes.

Reproducibility: Yes

Additional Feedback: The authors addressed all comments I pointed out in Weaknesses. Thus, I keep my high score. ---- --Note the meaning of \circle in (4). Is it the Hadamard product? --Improve the presentation of the proof of Theorem 1. One option is to summarize the proof strategy of Theorem 1 at the beginning. Then, technical lemmas should be stated. Clarify where the lemmas are used. The proofs of the lemmas may be placed after the proof of Theorem 1. Finally, use the lemmas to prove Theorem 1.

Review 4

Summary and Contributions: This paper showed that how the implicit bias of gradient descent can be extended for robust low-rank matrix recovery, while avoiding overfitting. Authors proposed a double over-parameterization (DOP) for both the imposed model structures, namely the low-rank and sparse components. The proposed DOP formulation, algorithm, and application to robust recovery of natural images have been explained with a convergence analysis. Experimental results of (1) low-rank matrix robust recovery and (2) the robust recovery of images with varied salt-and-pepper noise (i.e., spatially sparse corruption) have been reported.

Strengths: The proposed double over-parameterization (DOP) method for robust low-rank matrix recovery aims to overcome the challenge of blindly estimation of the underlying rank and sparsity level of the oracle data, if such prior knowledge is unknown. While existing works proposed and demonstrated the use of implicit bias of gradient descent on over-parameterized model for blind (without knowledge of rankness) low-rank recovery (without sparse error), the reliable extension to robust recovery of low-rank matrix recovery is somewhat new. Authors presented the problem formulation, algorithm with the control of the implicit regularization, as well as convergence analysis. The results are shown to be practically useful for salt-and-pepper image blind denoising problems, and demonstrated relative strength over other unsupervised learning methods, such as DIP.

Weaknesses: Reviewer is not exactly in this field, thus may not be in a good position to comment all aspects. Novelty: Since existing works have already proposed the over-parameterization in low-rank recovery problem [5,6], it is natural to impose the similar over-parameterization to the sparse modeling (which is also proposed in [2,3]) when generalizing to robustness low-rank matrix recovery. The combination of [2,3] with [5,6] seems pretty straightforward, thus the novelty is that significant. Experiment: It is unclear why only the DIP based methods are chosen to be the competing methods in the experiment part. Even if the experiments are under the blind (i.e., unknown noise level) setting, there are other deep learning approaches for solving such problems, as well as running noise estimation methods explicitly. The experiment results seem less convincing by choosing only few selected baselines, such as DIP, whose results highly depends on the early stop which is hard to judge whether the algorithm has been tuned to be optimal.

Correctness: The description and explanation are overall correct.

Clarity: Yes, the presentation is clear, while the experiment part can be strengthen with more justification on the result evaluation and competing methods.

Relation to Prior Work: Yes, the related low-rank recovery and over-parameterization methods are discussed, such as [2,3], and [5,6].

Reproducibility: Yes

Additional Feedback: I am satisfied with the authors' rebuttal, and "up-scaled" my score.