NeurIPS 2020

### Review 1

Summary and Contributions: The paper studies the optimization trajectory in diagonal linear networks, when the data is linearly separable and exponential loss is used for training. In particular, the authors examine the relationship between the optimization solution and the initialization scale. They establish that in this setting, the trajectory can enter the "Rich Regime" only if the training loss is optimized below a certain threshold. Otherwise, the network will be in the kernel regime. In numerical section, the authors evaluate these thresholds for a few simple datasets. These thresholds end up being infinitesimally small. The authors conclude that more work is needed to understand the effective implicit bias in realistic settings.

Strengths: The paper provides an interesting analysis of a simple model that shows complex phenomena similar to what might be happening in real-life network. Further research in this direction can be very fruitful for the community.

Weaknesses: The paper studies a very limited model, under a very limited setting. In this scenario it is possible that some (or most) of the phenomena observed here are just simply by-products of this specific setting. It would have been nice to include a section to discuss the implications of these results on more general settings.

Correctness: On page (3), L(u) is defined as the average loss. But in various point through out the paper, it seems that the factor (1/N) has been discarded. For example, I believe the calculation of line 146 might be using L(u) = \sum_{n=1}^N \ell(y, y'). Similarly, in line 162, it is stated that \epsilon varies in the interval (0, N]. Should it be the case if the loss is normalized by a factor of N? This might be causing some issues. I believe equations (2), (3), and (4) are missing "\forall n" in their constraints.

Clarity: The paper is well-written for the most part. For a person who is not familiar with the work of Woodworth et al, this paper might be a difficult read as I believe some of the definitions and discussions are absent from this manuscript. For example, in section 3, there is no clear statement regarding the algebraic form of the multi-layer diagonal networks. There are minor typos on lines 125 and 155.

Relation to Prior Work: A substantial portion of the paper is spent on discussing previous work. In particular, the results of Chizat et al and Lyu and Li are discussed.

Reproducibility: Yes

Additional Feedback: In the mainstream results on NTK, overparameterization is key in achieving the kernel behavior. In the results presented in the paper, the role of the number of observations, N, seems to be rather limited. Would you mind commenting on that? The introduction compares and contrasts the results of this paper with the results of Chizat and Bach / Lyu and Li. Are these results proven under the same assumptions and settings? -------------- Comments After Authors' Feedback: Thanks for your comments and clarifications.

### Review 2

Summary and Contributions: This paper studies the asymptotic trajectories of gradient flow and their implicit bias for diagonal linear networks, a previously established toy model for deep nets. The authors in particular study the transition between the kernel (NTK) and non-kernel regimes and show the dependence of this transition on initialization scale, besides the training accuracy. Interestingly, much of this behavior happens at impractically small accuracies.

Strengths: The very exciting results\ is that the authors identified a transition boundary for the kernel regime and show transition of implicit bias from l_2 to that of l_{D/2} in the "rich" regime, building on previous work on implicit bias. The authors make very tight statements about the trajectories of the continuum version of gradient descent and verify them experimentally, showing exactly the behavior expected from their results. Perhaps even more importantly, the numerical results in section 5 confirm and give quantitative statements to the previous observation that convergence of gradient descent "in direction" happens over ridiculously long time scales.

Weaknesses: Main weakness of this work is simple - do these interesting results about diagonal linear nets extend in any way to deep nonlinear networks, either fully connected or convolutional? It is far from clear what the practical application of this work is. Nonetheless, this is a step in a good direction in understanding implicit bias of gradient descent. Another weakness is the fact that the results depend on the continuum approximation of gradient descent and it's clear that some of the crucial assumptions do not carry over, like for example the guarantee that the training loss is monotonically decreasing along the gradient trajectory (due to overshooting from finite steps). ---------------------------------------- After author response ---------------------------------------- Thank you for responding to my two concerns - while I understand that they still stand, thank you for pointing to Woodworth et al. 2020 [30] for the first one. Regarding the gradient flow point, I would have liked a response to my worry about the importance of monotonicity, but I appreciate the difficulty of this analysis.

Correctness: Despite attempting to, I have been unable to disprove any of the statements, which together with the numerical experiments gives me reasonable confidence of correctness of the results (though I would be surprised if all the numerical factors are correct).

Clarity: With basically all the proofs delegated to the supplementary material, the main text is very readable, with only some minor corrections necessary - see below.

Relation to Prior Work: Any theorems that are adapted are clearly marked and the discussion throughout the paper makes it clear what is a new result.

Reproducibility: Yes

Additional Feedback: Some comments: - In lines 162 and 164, you claim that \epsilon\in(0,N]. Could you clarify why that's the case? It seems to me that while \epsilon starts at 1, there is obviously no upper bound on the exponential loss (without the separability assumption). - (repeat) do you have any intuition about the discrete gradient descent case? - Your use of \mu on lines 195-211 is confusing, since you've previously used \mu=\alpha^D. This could use clarifying. - I think there's a typo in the \epsilon value in line 285. - I'm very impressed with your clever use of the Broader Impact statement. ---------------------------------------- After author response ---------------------------------------- Thank you for clarifying my confusions here. Overall, my opinion that this is a good paper has not changed.

### Review 3

Summary and Contributions: This paper studies the implicit bias of gradient flow for deep linear classification problems. In particular, this paper shows that different initialization scale will lead to different implicit bias. More specifically, this paper considers the transition between the rich regime and kernel regime, and precisely characterizes how the implicit bias varies when choosing different target accuracy epsilon and scaling parameter alpha. ####After reading authors' response. The authors have well addressed my concerns, thus I would like to increase my score to 7.

Strengths: This paper proves the implicit bias of gradient flow for deep linear classification under various ranges of the target accuracy epsilon and scaling parameter alpha. This paper is technically sound, the derived theory is interesting and helpful for further understanding of the role that gradient descent plays in deep learning.

Weaknesses: The network model considered in this paper is rather simple, and it seems that considering D-layer diagonal linear network is no more difficult than 2-layer diagonal linear network. The comparison to the related work [16, 30] has not been clearly stated

Correctness: yes

Clarity: yes

Relation to Prior Work: This part has not been clearly discussed in the paper. It remains unclear how the results in the paper compared with two relevant work [16, 30]

Reproducibility: Yes

Additional Feedback: Overall this is a nice work that proves the implicit bias of gradient flow in a broader range of regimes. My major concern is the novelty and impact of the results derived in this paper given existing works. Firstly, the theoretical results are mostly established based on a quite simple model (diagonal linear networks on separable data), which may be difficult to be generalized to more practical settings. Additionally, based on such a simple model, the effect caused by the depth may not be well characterized (currently it seems that this paper only leverages the fact that a D-layer network is D-homogeneous, which has already be well investigated in [19]). Secondly, as [16] has already shown that a deep linear network with fixed initialization scaling trained by gradient flow will finally converge to the max-margin solution (all weight matrices will be rank-1). How does the result proved in this paper compared to it? Moreover, whatâ€™s the difference and relationship between the theoretical results and [30], which shows the transition between the kernel regime and rich regime for nonlinear overparameterized models? The authors should precisely state the difference and relevance to those related work.

### Review 4

Summary and Contributions: This paper studies asymptotically how the scale of initialization and the training accuracy determine the solution trajectory of gradient descent enters the 'kernel regime' and 'rich regime' when optimize an exp loss in classification problems.

Strengths: As an followup work to [Woodworth 2019], this work extends the understanding of the transition between rich and kernel regime to the classification problem. It also provides some interesting understanding to the good performance of super deep network.

Weaknesses: 1, It considers the asymptotic setting, but this is fine for this line of research. 2. The theory is limited to linear diagonal network. Whether the conclusion can be generalized to nonlinear network remains unknown. 3.In Theorem 4, Lyu and Li [2020])'s result actually holds for every finite initialization. Why we need to make it go to infinity in equation 4? Moreover, in the main results of this paper, it still needs the initialization to be infinity. A finite initialization results may be more interested here in my opinion. 4. I suggest the author to highlight the technical difficulty of the paper compared to [Woodworth 2019] and [Chizat et al, 2020].

Correctness: The claim and proof looks correct to me.

Clarity: The paper is well written and easy to understand.

Relation to Prior Work: It highlights the difference between this paper and the previous works.

Reproducibility: Yes

Additional Feedback: