Eigenvalues of the Hessian in Deep Learning: Singularity and Beyond
We look at the eigenvalues of the Hessian of a loss function before and after training. The eigenvalue distribution is seen to be composed of two parts, the bulk which is concentrated around zero, and the edges which are scattered away from zero. We present empirical evidence for the bulk indicating how over-parametrized the system is, and for the edges that depend on the input data.
Studying the Hessian in deep learning, the experiments in this paper suggest that the eigenvalue distribution is concentrated around zero and the non zero eigenvalues are related to the complexity of the input data. I find most of the discussions and experiments to be interesting and insightful. However, the current paper could be significantly improved. Quality: It seems that the arguments in the paper could be enhanced by more effort and more comprehensive experiments. Performing some of the experiments discussed in the conclusion could certainly help a lot. Some other suggestions: 1- It would be very helpful to add other plots showing the distribution of eigenvalues for some other machine learning method for the purpose of comparison to deep learning. 2- There are some issues about the scaling of the weights and it make sense to normalize the weights each time before calculating the Hessian otherwise the result might be misleading. 3- It might worth trying to find a quantity that measures the singularity of Hessian because it is difficult to visually conclude something from the plots. 4- Adding some plots for the Hessian during the optimization is definitely needed because we mostly care about the Hessian during the optimization not after the convergence. Clarity: 1- There is no reference to figures in the main text which makes it confusing for the reading to know the context for each figure. For example, when looking at Figure 1, it is not clear that the Hessian is calculated at the beginning of optimization or after convergence. 2- The texts in the figures are very small and hard to read.
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
Eigenvalues of the Hessian in Deep Learning: Singularity and Beyond
We look at the eigenvalues of the Hessian of a loss function before and after training. The eigenvalue distribution is seen to be composed of two parts, the bulk which is concentrated around zero, and the edges which are scattered away from zero. We present empirical evidence for the bulk indicating how over-parametrized the system is, and for the edges that depend on the input data.
user
<think>Okay, to do a decent peer review, I should start with summarizing what the work is about. This includes novelty, contributions, clarity, and impact. I might include suggestions to issues that I find critical to the usefulness of the work. Starting with novelty, there are different types of novelty. The first aspect of novelty is scope and scale. Is the work covering a broader perspective than previous works? The second aspect is methodology. Is the proposed methodology novel compared to previous methods? Do they provide any significant advantage over previous methodologies? The third aspect is results. Are the results different compared previous work? Does this lead to any novel theories or an extension of current theories? Next is clarity, I will be reading the work in depth and analyze if any sentences can be potentially ambiguous or misleading. If any sentences are ambiguous, I will state them and give a suggestion on how to rephrase such sentence or how to make more clear to the audience. Lastly, impact is important. I will attempt to predict how such work will impact the scientific community. This might include the applications of such work and how the work might transform the way we study a certain topic. Finally, I will might only mention issues that are critical. This is to keep the review as concise and helpful as possible.</think>



Studying the Hessian in deep learning, the experiments in this paper suggest that the eigenvalue distribution is concentrated around zero and the non zero eigenvalues are related to the complexity of the input data. I find most of the discussions and experiments to be interesting and insightful. However, the current paper could be significantly improved. Quality: It seems that the arguments in the paper could be enhanced by more effort and more comprehensive experiments. Performing some of the experiments discussed in the conclusion could certainly help a lot. Some other suggestions: 1- It would be very helpful to add other plots showing the distribution of eigenvalues for some other machine learning method for the purpose of comparison to deep learning. 2- There are some issues about the scaling of the weights and it make sense to normalize the weights each time before calculating the Hessian otherwise the result might be misleading. 3- It might worth trying to find a quantity that measures the singularity of Hessian because it is difficult to visually conclude something from the plots. 4- Adding some plots for the Hessian during the optimization is definitely needed because we mostly care about the Hessian during the optimization not after the convergence. Clarity: 1- There is no reference to figures in the main text which makes it confusing for the reading to know the context for each figure. For example, when looking at Figure 1, it is not clear that the Hessian is calculated at the beginning of optimization or after convergence. 2- The texts in the figures are very small and hard to read.
assistant
Studying the Hessian in deep learning, the experiments in this paper suggest that the eigenvalue distribution is concentrated around zero and the non zero eigenvalues are related to the complexity of the input data. I find most of the discussions and experiments to be interesting and insightful. However, the current paper could be significantly improved. Quality: It seems that the arguments in the paper could be enhanced by more effort and more comprehensive experiments. Performing some of the experiments discussed in the conclusion could certainly help a lot. Some other suggestions: 1- It would be very helpful to add other plots showing the distribution of eigenvalues for some other machine learning method for the purpose of comparison to deep learning. 2- There are some issues about the scaling of the weights and it make sense to normalize the weights each time before calculating the Hessian otherwise the result might be misleading. 3- It might worth trying to find a quantity that measures the singularity of Hessian because it is difficult to visually conclude something from the plots. 4- Adding some plots for the Hessian during the optimization is definitely needed because we mostly care about the Hessian during the optimization not after the convergence. Clarity: 1- There is no reference to figures in the main text which makes it confusing for the reading to know the context for each figure. For example, when looking at Figure 1, it is not clear that the Hessian is calculated at the beginning of optimization or after convergence. 2- The texts in the figures are very small and hard to read.