NeurIPS 2020

The Surprising Simplicity of the Early-Time Learning Dynamics of Neural Networks

Review 1

Summary and Contributions: This paper theoretically proves that gradient descent on a neural network can learn a NTK matrix which is close to a linear model in the early phase. First, several theoretical results are provided to prove that their proposal is valid on two-layer fully-connected neural networks. Then some analyses are given to show whether their proposal still holds on more complicated networks. This paper reveals that in the early learning of neural networks, it can be treated as a linear model, rather than as a black box that can be difficult to understand.

Strengths: This paper obtains their theoretical results by bounding the spectral norm of the difference between the Neural Tangent Kernel (NTK) and the kernel for a linear model. The proofs can withstand scrutiny. This paper proposes a direction to analyze the neural networks in early-time learning, which is relevant with the topics of NeurlPS.

Weaknesses: The theoretical results are not complete. Only the theoretical results and proofs of the two-layer fully-connected neural network are given, and the proof of extending the theoretical results to the deep neural network needs to be supplemented. The experimental results are not sufficient. This work only conducts experiments on single data set, and the experimental results could not fully verify the theoretical results.

Correctness: The theorems that have been proved in this paper are correct. However, the reasoning of deep neural networks is to be discussed. The theoretical results cannot be easily extended from two-layer fully-connected neural networks to deep neural networks. The analysis of the extended part in this paper needs further discussion.

Clarity: The logical relationship of the related work is not smooth enough. The writing of the proof by step in the theoretical proof section is clear. But the part of the theoretical results is not written well, since the experimental results appear among them.

Relation to Prior Work: The description of the related work is not sufficient, mainly because it does not explain the contribution of the work to the research of neural networks, but only shows the improvement compared with the related work.

Reproducibility: Yes

Additional Feedback: ------------------After rebuttal----------- Thank you for your response. I have carefully read the authors’ feedback and other reviews. Most of my initial concerns are erased. Thus I decide to improve my score.

Review 2

Summary and Contributions: This paper establishes a connection between the early-time learning dynamics of neural networks and the dynamics of a simple linear model on the input. The main contributions of this paper are as follows: -- It proves that the early-time dynamics of training neural networks and training a simple linear model of the input are coupled (close in function space). This is shown for training either or both layer in a wide two-layer neural network. -- When data is well conditioned, the paper shows in addition that the early-time interval is almost long enough for the dynamics to learn the *best* linear model that fits the training dataset. -- Experimental results verifying the agreement between neural network and the linear model in the early stage of training.

Strengths: -- The theoretical results presented in this paper are novel, and quite interesting to me in the following aspects: (1) The prior work of Nakkiran et al. 2019 showed empirically that the early time NN predictions are well explained by a linear model of the input, in an information theoretic sense. This paper establishes this connection theoretically. (2) The coupling result utilizes NTK theory, but unlike the NTK does not require the width to be a big polynomial of n. Indeed, the only requirement is m, n >= d^{1+\alpha}, and so the width can actually be smaller than n. (3) It is interesting to see training each of the two layers yields different linear models. In particular training W is coupled with an exact linear model of [x, 1] whereas training v is coupled with a linear model with additional features depending on ||x||^2. -- The proof builds on random matrix analyses of the NTK which could be of technical interest. -- The experiments are nice to see and support the theories quite well. In addition, though the paper is about the agreement between NN and the linear model, the experiments also demonstrated how NN deviates from the linear model after a certain time period. For example, in Figure 3(a) we see that the non-linear part of the error starts to decrease for both fully-connected and convolutional networks, and it seemed the convolutional network was reducing this error faster than the fully-connected network. These are complementary to the experiments in Nakkiran et al. and I can imagine more large-scale experiments of this type could be interesting.

Weaknesses: -- One minor concern I have about the theoretical result is the scaling regime, specifically why \alpha <= 1/4 is needed. What happens if \alpha > 1/4? (This may rather be a research question for the authors). Do you expect the NN to be coupled with a quadratic / higher-order kernel and thus learning a polynomial of the input? -- The extensions to the multi-layer and convolutional cases are only sketched out and not rigorously established.

Correctness: The proofs and empirical methodologies are correct upon my inspections.

Clarity: This paper is very well presented. I find it a quite enjoyable read.

Relation to Prior Work: The paper has sufficient discussions about prior work and the relationship between prior work and the present paper.

Reproducibility: Yes

Additional Feedback:

Review 3

Summary and Contributions: This paper proves that, if the input features are sampled from a well-behaved distribution, and if gradient descent is used for training, then on both the training set and test set, the output of a two-layer network is close to the output of a linear model (potentially with some norm-dependent features, as discussed in Section 3.4). Moreover, it is discussed how to extend the analysis to deep networks and convolutional networks, and some empirical verification is provided.

Strengths: I think the relationship between two-layer networks and linear models proved in this paper is interesting. Due to the NTK analysis, we already know that using gradient descent, the output of a wide network is close to the linear model with NTK features. However, this paper shows that for a well-behaved input distribution, the output of the network is close to the linear model with basically the original input features plus some norm-dependent features in the early phase of training.

Weaknesses: The main limitation in my opinion is the assumption on the input distribution, Assumption 3.1. Is it approximately satisfied by real-world data? Is there any way to transform a given dataset so that Assumption 3.1 holds, and does it improve the performance of the network?

Correctness: I do not see any correctness issue.

Clarity: Overall the paper is well-written. The empirical results described in Section 4.2 are interesting, and I have the following questions: 1. What exactly is the linear model? Did you calculate and include the constants xi, nu and the norm-dependent features? 2. How long does the early phase last if we use a large learning rate as in practice?

Relation to Prior Work: The discussion of prior work is thorough as far as I know.

Reproducibility: Yes

Additional Feedback: Reply to the feedback: Thanks for the response! I would like to increase my score, because I realize I missed an interesting point proved in this paper: as shown in Corollary 3.3, there exists a data set on which gradient descent first learns the optimal linear classifier, and then something more complicated (for example, the optimal solution in the NTK regime, when the width is large enough). Although people have had this intuition for a long time, I think it is nice to provide a concrete case and a rigorous proof. On the other hand, I think it would be nice to include more details of the experiments. I asked how long the early phase lasts with a large learning rate as in practice, but it seems that the authors did not answer this question. In Figure 3, did you train the networks until convergence? I feel that the convolutional network should be able to drive the loss lower than what is shown in Figure 3. I think it would be nice to see how long the early phase lasts in a typical training. Relatedly, in Appendix A it is mentioned that the learning rate is 0.01/||NTK||; how large is ||NTK|| exactly?

Review 4

Summary and Contributions: This paper studies the learning dynamics of a two-layer neural network and shows that in its early training phase, the network mimics behavior of a simple linear model on input data, which is assumed to be somewhat similar to Gaussian inputs. The analysis is based on NTK but only requires a modest number of hidden neurons. The theory is supported with experiments on synthetic and real data.

Strengths: To my knowledge, this work presents a new theoretical and empirical insight on the early-time training dynamics of neural networks. The analysis is intuitive and insightful. The requirement on width is reasonable, mainly because of the well-behaved data. Therefore, I recommend it for acceptance.

Weaknesses: There is a restrictive assumption on the data distribution, which behaves very much like Gaussian data. However, the experiment on CIFA10 does show a similar phenomenon.

Correctness: The theoretical results are sound and empirical studies are convincing. However, the claim “we show that these common perceptions can be completely false in the early phase of learning” is not accurate, for the reason that the results hold for a limited class of data distributions and two-layer networks. The phrase “early phase of learning” is not entirely clear.

Clarity: Yes

Relation to Prior Work: Yes

Reproducibility: Yes

Additional Feedback: I have some general questions: 1. Can the authors explain further the statement “we show that these common perceptions can be completely false”? 2. Any intuition/reason why the networks eventually escape the linear behavior? 3. Would one expect the same behavior for deep networks? Could the analysis be extended to deep network settings and more general data? 4. In Section 3.4, what if RELU was used instead of ERF? ---------- After rebuttal ---------- I am happy with the author response and keep the same score.