NeurIPS 2020

Beyond Lazy Training for Over-parameterized Tensor Decomposition

Review 1

Summary and Contributions: The paper focuses on an over-parameterized tensor decomposition that is closely connected to over-parameterized neural networks and hence of interest to the NeurIPS community.

Strengths: Very strong theoretical result, probably worthy of a leading theory conference as well (SODA comes to mind, more than FOCS/STOC). Tensor decompositions are NP-hard (when the rank is unspecified). Assuming that the target rank is r (a priori know and fixed) the authors present a variant of a gradient descent algorithm that finds a decomposition with m components (informally speaking, rank-m) with m much, much smaller than prior work (depends on r^\ell as opposed to d^\ell, using the notation of the paper, d and \ell are the tensor dimensions and number of modes). This is a major and non-trivial improvement.

Weaknesses: No experimental component, but I would not count this as a weakness. This is a theory paper and NeurIPS should accept pure theory papers as well.

Correctness: Did not check. There are 40 pages of proofs that I cannot parse in the very short review period. The proof outline makes sense, but I did not check the details.

Clarity: Yes.

Relation to Prior Work: I am not the foremost expert in this area, but many relevant papers have been discussed.

Reproducibility: Yes

Additional Feedback: I had no real comments for the authors and I feel that their response addressed concerns raised by the other reviewers.

Review 2

Summary and Contributions: This paper addressed the problem of how to find a better over-parametrized rank of tensor decomposition. A variant of the gradient descent algorithm for tensor decomposition is proposed and plentiful theoretical results were provided to prove the proposed algorithm can approximate a tensor by a lower rank bound than the lazy training regime.

Strengths: Theoretically proving the existence of a lower approximation rank can be acquired by a novel gradient descent method is interesting and the results are promising. The proposed gradient descent method and theoretical contributions are potentially useful in increasing the performance of neural network training.

Weaknesses: Though strong theoretical proofs were given in the paper, the effectiveness of the proposed gradient descent method in neural network training still remains questionable. It is better to provide some numerical experiments to support the theoretical results of the lower bound and the improvement of the neural network training when applying the proposed gradient descent method.

Correctness: The theoretical results in this paper are plentiful and look sound.

Clarity: The paper is well-structured and the references are properly cited. The paper can explain the motivation and the proposed method well. Minor typos: Below Line 85, quote 1: effecitively --> effectively Line 165: min --> minimum Line 210: high level --> high-level Line 316: gradient based --> gradient-based

Relation to Prior Work: The paper addressed the related work of the lazy training regime and the existing problems and challenges of over-parametrization. The paper provided the clear comparison between the previous work and the proposed results.

Reproducibility: No

Additional Feedback:

Review 3

Summary and Contributions: This paper consider tensor factorization problem that decomposes a tensor into a set of rank-1 tensors. The main contribution of this paper is to propose a nonconvex optimization algorithm and to show it provably converges to a global minimum under over-parameterization setting, i.e., m = O(r^{2.5 \ell}), where m is the number of components in the optimization variable, r is the rank of the target tensor, and \ell is the order of the tensor.

Strengths: 1. This paper proposes a reparameterized nonconvex optimization formulation for tensor decomposition to avoid higher order saddles. 2. The authors then proposed a gradient descent based optimization algorithm and proved it converges to a global minimum with high probability.

Weaknesses: 1. The main concern is about the utilization of the over-parameterization in tensor decomposition. In other words, for a rank-r tensor, tensor decomposition aims to find the r components which have physical interpretation. However, the proposed approach instead finds m = O(r^{2.5 \ell}) components, which could be far away from the target r components. For example, even for third-order tensor, it finds m = O(r^7.5) components, much larger than r. 2. Perhaps the goal of this paper is to understand the effect of over-parameterization in tensor decomposition. However, if this is the case, the objective function is quite different to the classical one, and the algorithm is also different to simple gradient descent. The algorithm should also be numerically compared with existing algorithms.

Correctness: They are correct, but I didn't carefully check all the proofs in Appendix.

Clarity: Overall, the paper is well written and easy to follow. In Thm 3, does it require r < d or not? I guess there is no restriction on r, but it needs to be stated to make it clear, since in other place like Thm 2 it requires r<d. There are few grammar issue, e.g., line 20-21: we consider..., we aim

Relation to Prior Work: Relation to prior work is clearly discussed.

Reproducibility: Yes

Additional Feedback: ----------------------------------After rebuttal--------------------------------------------- I am still not sure how practical the current results are, but I think it will be more interesting if the results can be extended to other tasks such as tensor completion.

Review 4

Summary and Contributions: Consider a low-rank tensor decomposition problem: given a rank-r tensor T, find a rank-m tensor that approximates T with small l2 error. If m is as large as the number of elements in T, we can find a global optimum, but the setting is impractical. In this paper, a gradient descent algorithm along with a new tensor parameterization is proposed, which is theoretically guaranteed to minimize the l2 error arbitrary small with much smaller m.

Strengths: - The paper is clearly written and easy to read - Theoretically clear results - Interesting research direction that tries to connect tensor decomposition and neural tangent kernel

Weaknesses: - No empirical verification - Presented results are not ready to show the direct connection to neural tangent kernel (and the learning dynamics of neural networks)

Correctness: The obtained results seem correct (I did sanity check of the results but I didn't check the proofs)

Clarity: The paper is very clearly written.

Relation to Prior Work: Related work is sufficintly referenced.

Reproducibility: Yes

Additional Feedback: This is an interesting study that tries to go beyond the previous results in tensor decomposition methods and make some bridge to the learning dynamics of neural networks. Although I feel the bridging part is under development, the main contributions are still significant. To improve the quality of the paper, I have two suggestions. 1. It would be better to explain some intuitions for the new parameterization (the top equation in p5). I guess c_i is the normalizing term, but a_i looks mysterious to me. Could you explain why do we need a_i (or what would happen without a_i)? 2. It would be more convincing to have numerical results (e.g., comparing the error curve between normal gradient descent and the proposed algorithm).