Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: The authors conduct a thorough hessian-related analysis of training dynamics in neural network training in several medium-sized tasks.  They analyze the eigenspectrum of the hessian throughout training, and investigate the effects of screening descent directions by various criteria related to properties of the hessian.  Additionally, they provide a bevy of theoretical results on their studied dynamics.
strengths_and_weaknesses: Originality: while studying hessian eigenspectra is not a _new_ idea, the approach considered by the authors does appear novel (i.e., screening of particular directions during training), and has yielded interesting results
Quality/Clarity: The paper is quite high quality and clear
Significance: I expect this paper to be significant, as it gestures at negative components of the hessian being more important than folks otherwise suspected.
questions: There is a sense in which components of this work are somewhat tautological, and other results might be slightly overstated.  As an example, the authors state:

```
In this paper, we have shown that chaotic dynamics of ANN training by SGD are linked to negative curvature.
```

The authors seem to have _defined_ “presence of chaotic dynamics” to be “local hessian has negative eigenvalues”.  Consider the following alternative: it could be the case that essentially parabolic regions of the loss landscape are stitched together by a small number of hyperbolic descent directions.  Restricting oneself to only the strictly positive eigenvalues then somewhat definitionally traps the performance of a model to within some locally-minimum basin, whereas restricting oneself to the negative-hessian directions is more akin to a kind of truncated BFGS iteration, where descent to the lowest minima is never truly ruled out.

It is not, then, anything particularly “chaotic” about the presence of these negative eigenvalues—only that they’re conduits between more locally quadratic regions.  This would likewise explain why learning trajectories tend to correlate more with the positive curative directions.

Do the authors have evidence that there is something more “chaotic” going on here, other than eigenvalues being negative?

As for the author’s definition of “chaos”:
```
Given a small deviation in the parameter state of a model at any point of the optimization, one should hope to obtain a similar solution at the end of the optimization procedure, and by extension a solution that performs similarly well both on training and validation data. Therefore, we use chaotic to refer to the sensitivity to initial conditions.
```
It might be that the community simply needs a different word to describe what is going on with neural network training.  A model whose training dynamics were purely diffusive would also seem to satisfy the author’s notion of “chaotic”.  I suppose we can identify “sensitive to initial conditions” with “chaotic”, but this might be overloading the word.

Regardless, stepping back for a moment and taking a more thousand foot view, here: it seems that the author’s primary result is that “performance of neural networks degrades when we restrict neural networks to not be able to descend along directions with negative hessian eigenvalues”.  Do the authors agree with this characterization?  Do the authors agree that restricting learning dynamics to positive-semi-definite subspaces of the hessian traps the learning dynamics near a local minimum / saddle point?
limitations: Beyond the comments above, many of the training curves in the paper seem not to be fully converged, but given the extreme numerical expense of running this analysis, I don’t think it’s the end of the world.
review_title: null

final_score: rating: 3: Reject: For instance, a paper with technical flaws, weak evaluation, inadequate reproducibility and incompletely addressed ethical considerations.
review_content: summary: This paper studies neural network training dynamics via the lens of the Hessian eigenspectrum. In particular, the authors show that the chaotic dynamics in neural network training are largely determined by the negative eigenvalue of the Hessian. Furthermore, the authors empirically demonstrate that the negative eigenvalue direction is important for making progress in training.
strengths_and_weaknesses: Strengths:
- The discussion about the role of negative eigenspectrum in optimization is interesting. 

Weaknesses:
- The theoretical model of Lyapunov exponent offers no further insights into the training dynamics compared to standard quadratic models or linearized models.
- The fact that negative eigenvalues would lead to chaotic dynamics is well known in the optimization community. Basically, I feel the whole section 2 is just background and I don't really know what's the contribution of this work.
- All experiments are done in very toy settings. The authors only conducted experiments on FashionMNIST and USPS, which I don't think very convincing.
questions: See the weaknesses mentioned above.

Other questions and suggestions:
- I suggest the authors clarify the main advantages of Lyapunov exponent analysis over existing analyses. If I understand correctly, most results derived in section 2 can be obtained in a toy quadratic model.
- To scale up to larger datasets, the authors could check out literature about the Lanczos algorithm and its stochastic version (see e.g. [1]).



References:    
1. An Investigation into Neural Net Optimization via Hessian Eigenvalue Density
limitations: The authors discussed the limitations at the end of the paper. I don't see any negative societal impact of this work as it is an "understanding" paper.
review_title: null

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: This manuscript mainly investigates a theoretical link between the curvature of the loss function and (local) chaotic dynamics in neural networks training through demonstrating the negative eigenvalue spectrum of the Hessian. Their empirical evidence verifies that the negative eigenspectrum and so the directions of local chaos cannot be removed from SGD without hurting training performance. 
strengths_and_weaknesses: **Strengths**

The main idea of the paper is interesting and it is nice to see a paper that is theoretically principled, and goes about formalizing some dynamical behavior arising in neural networks training. 

**Weaknesses**

 I checked most of the proofs in detail and spot some errors; for instance:

1) It seems that Theorem 2.1 cannot hold in a general case (as is), and some assumptions on the Hessian of the loss ($H$) are required to be considered. More clear, in the proof of this theorem (Appendix A.5, line 522) it is mentioned that "Since all eigenvalues of $(I − \gamma H)^2$ are positive, the matrix is positive definite". But, all eigenvalues of $(I − \gamma H)^2$ are not necessarily positive; e.g. when $\gamma H$ has some eigenvalues equal to $1$. I mean let $H $ be a matrix such that  $\det(I − \gamma H) = 0$, then $(I − \gamma H)^2$ can have zero eigenvalues. For example, let $H= diag(\frac{1}{\gamma}, b, c)$, then $(I − \gamma H)^2 $ obviously has at least one zero eigenvalue. This affects the rest of the proof.

2)  Appendix A.1: In Eq. (19), from right to left, the second term should be "$\frac{\partial \theta_i^{(t)}}{\partial \theta_j^{(t)}}$".

3) Appendix A.2: It seems that Eq. (22) is written for $t=0$? So it needs to be modified for $t=1$. Also is there the assumption $v(0)=0$ for obtaining Eq. (22)?

There are also some minor issues and typos: 

- The definition of chaotic systems (line 47) is for "smooth" dynamical systems (DS), but (it is well known that) it is no longer true for "non-smooth" DS such as RNNs with ReLU activation function (as piecewise linear DS). Especially, since piecewise linear DS can observe a specific kind of chaos called "Robust Chaos" in the absence of periodic windows and coexisting attractors; please see for instance the paper by Banerjee, Yorke and Grebogi: https://doi.org/10.1103/PhysRevLett.80.3049.

- Positive mLCE is a standard signature of chaos provided that the system's invariant set is **bounded** (as in this case two nearby orbits separate exponentially fast, but at the same time their mutual separation cannot go to infinity so that there are also folds). Otherwise a positive mLCE could be related to a diverging orbit which is not chaotic. Therefore, the authors should consider the assumption of "boundedness" and perhaps add it to Section 2.2.

- Is the reference " Robbins, H. E. (2007). A stochastic approximation method. Annals of Mathematical Statistics, 22:400–407" (line 369) correct? I couldn't find such a reference. 

- Line 89, A.3 ---> A.2
questions: Some effects of the local chaotic dynamics on the training performance is discussed through some experiments. However, I would like to see:  can the relation between local and long-term chaos be proven (mathematically) to demonstrate some problematic effects of local chaos on the training theoretically as well?

limitations: The authors have discussed the limitations of the work and there is no potential negative societal impact.
review_title: null

Meta-review: metareview: This paper studies the role of (local) chaos in determining the training dynamics of neural networks. The authors first introduce a standard global notion of chaos via the Lyapunov matrix and introduce a greedy local version which determines whether the dynamics are “locally chaotic”. The authors relate these dynamics to the eigenvalues of the hessian. The authors close with interesting experiments showing that the chaotic, negative, eigenvalues of the hessian are important for training.

This paper generated some debate and ultimately two reviewers favored acceptance, while one reviewer wanted to reject the paper. The two reviewers who wanted to accept the paper appreciated the new emphasis that the paper brought to the negative eigenvalues of the hessian.  The negative reviewer focused on weak experiments and thought the analysis was too similar to the noisy quadratic model. Ultimately, I do think the analysis done here might provide some insight not present in the NQM, since as the authors note, they consider the spectrum throughout training whereas the NQM considers a noise model on top of a fixed hessian. I share the positive reviewers’ feeling that the results about the negative eigenvalues seem quite interesting. At the same time, I do share the negative reviewer’s issue that the experiments might be too simplistic to draw interesting inferences from. 

Ultimately, I do think the pros probably outweigh the cons of accepting this paper. However, I would ask the authors whether the description in terms of chaos makes sense since, as they note, they generally only discuss the local behavior for which the notion of chaos doesn’t necessarily make sense.

confidence: Less certain
Decision: Accept