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.
The paper analyzes the properties of the Hessian of the training objective for various neural networks and data distributions. The authors study in particular, the eigenspectrum of the Hessian, which relates to the difficulty and the local convexity of the optimization problem. While there are several interesting insights discussed in this paper such as the local flatness of the objective function, as well as the study of the relation between data distribution and Hessian, a somewhat lacking aspect of the paper is that most described effects are presented as general, while tested only in a specific setting, without control experiments, or mathematical analysis. For example, regarding the concentration of eigenvalues to zero in Figure 6, it is unclear whether the concentration effect is really caused by training (e.g. increasing insensitivity to local perturbations), or the consequence of a specific choice of scale for the initial parameters. In Figure 8, the complexity of the data is not defined. It is not clear whether two fully overlapping distributions (the Hessian would then become zero?) is considered as complex or simple data. Some of the plots legends (Fig. 1 and 2) and labels are unreadable in printed format. Plots of Figure 3 don*t have the same range for the x-axis. The image of Hessian matrix of Figure 1 does not render properly in printed format.
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.
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<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>



The paper analyzes the properties of the Hessian of the training objective for various neural networks and data distributions. The authors study in particular, the eigenspectrum of the Hessian, which relates to the difficulty and the local convexity of the optimization problem. While there are several interesting insights discussed in this paper such as the local flatness of the objective function, as well as the study of the relation between data distribution and Hessian, a somewhat lacking aspect of the paper is that most described effects are presented as general, while tested only in a specific setting, without control experiments, or mathematical analysis. For example, regarding the concentration of eigenvalues to zero in Figure 6, it is unclear whether the concentration effect is really caused by training (e.g. increasing insensitivity to local perturbations), or the consequence of a specific choice of scale for the initial parameters. In Figure 8, the complexity of the data is not defined. It is not clear whether two fully overlapping distributions (the Hessian would then become zero?) is considered as complex or simple data. Some of the plots legends (Fig. 1 and 2) and labels are unreadable in printed format. Plots of Figure 3 don*t have the same range for the x-axis. The image of Hessian matrix of Figure 1 does not render properly in printed format.
assistant
The paper analyzes the properties of the Hessian of the training objective for various neural networks and data distributions. The authors study in particular, the eigenspectrum of the Hessian, which relates to the difficulty and the local convexity of the optimization problem. While there are several interesting insights discussed in this paper such as the local flatness of the objective function, as well as the study of the relation between data distribution and Hessian, a somewhat lacking aspect of the paper is that most described effects are presented as general, while tested only in a specific setting, without control experiments, or mathematical analysis. For example, regarding the concentration of eigenvalues to zero in Figure 6, it is unclear whether the concentration effect is really caused by training (e.g. increasing insensitivity to local perturbations), or the consequence of a specific choice of scale for the initial parameters. In Figure 8, the complexity of the data is not defined. It is not clear whether two fully overlapping distributions (the Hessian would then become zero?) is considered as complex or simple data. Some of the plots legends (Fig. 1 and 2) and labels are unreadable in printed format. Plots of Figure 3 don*t have the same range for the x-axis. The image of Hessian matrix of Figure 1 does not render properly in printed format.