Kati8/ICLR_2023_2025 huggingface dataset:
authors: Zihan Gu
reviews: 
Comments: The authors propose a dynamical-systems view-point as a formal framework within which grokking can be understood. After discussing the use of general dynamical systems, specifically ODEs, in optimization, the manuscript proposes the study of a simple 3-D system as a reduced model capable of “identifying” grokking behavior. Several computational experiments are then discussed.
Strengths: The proposed idea of studying grokking as emergent behavior from a dynamical system is in itself very interesting, and formal guarantees that could be drawn from such an analysis would greatly enhance our theoretical understanding of the phenomenon.
Weaknesses: It is unclear how generalizable the analysis can be to arbitrary tasks, since it is applied to a very specific problem (even though the problem is, in some sense, canonical with respect to this phenomenon).

The manuscript offers a good formal description of the properties of the simple dynamical system. However, the connection between this projected system and its high-dimensional counterpart is not sufficiently motivated. This kind of discussion should be expanded.

Some minor concerns: 
There are several small typos throughout the text.
Definition 2 is not clear and should be expanded
The statement “If the model have learnt” should be clarified. What does learning mean in a mathematical sense?
Recommendation: 5
Reviewer confidence: 3
Soundness/correctness: 2
Clarity: 2
reviews: 
Comments: The paper presents an attempt an understanding and predicting 'Grokking", a curious phenomena in deep learning where test error rapidly decays long after training error appears to have converged. This paper models neural network parameter update as a dynamical system to 'reconstruct the analysis of grokking' and describes a metric which can 'predict' when grokking would occur during training.
Strengths: Viewing gradient descent updates to neural network parameters as the evolution of a dynamical system is an interesting (but not novel) perspective. Trying to relate generalisation phenomena to properties of this system is a cool approach. The paper seems to be comprised of two parts. The first is a 'theoretical' component, where the authors make some deductions regarding the property of the dynamical system, and the second is a discussion of their "Main embedding diff", a measure of to track grokking.
Weaknesses: I found this paper challenging to read. The language is unclear and the figures are repetitive and not particularly informative. While I understand the concept of modeling neural network updates as an ODE, it is not clear what is the exact dynamical system described in this paper. The lack of clarity is endemic to this work, as in all main equations in this paper (1-3, 4-6, and 7-9), it is left unexplained what their different components are. I believe that W & U are neural network parameters, but what are x, e, y, gamma, s, c, u, and w? The authors should make abundantly clear what are the roles of the different parameters in the system they model.
All this makes the "theoretical" part of this paper extremely hard to follow. While I could vaguely figure out the authors are attempting to make a claim regarding the existence of stable points relating to the neural network parameters, it is not obvious what that claim is, how they arrive at it, and why it is in any way relevant to grokking. The authors need to do a better job at explaining these extremely critical details.

In the second part, the authors introduce "Main embedding diff" (MED), a way of tracking grokking by measuring discrepancy between embeddings within a model. There seems to be no relationship between the first section and this one, and the paper is unclear as to what MED is, as the components laid out in eq 12 are not explained to the reader. The results consist of repetitions of the same figure presenting the grokking phenomena along with MED values during training for different model and task settings. Aside from MED being vaguely correlated with the grokking phenomena, it is not explained how it precisely relates to it. Furthermore, the paper only presents results for a single task (modular addition), while the original grokking work discusses several types of problems. 

Overall, while the paper attempts to make theoretical contributions to understanding grokking, its unclear presentation, disconnected sections and weak results make it difficult to assess its actual scientific merit or practical value.
Recommendation: 1
Reviewer confidence: 4
Soundness/correctness: 2
Clarity: 1
reviews: 
Comments: The paper aims to study "grokking," or a delayed generalization phenomenon, using ideas from dynamical systems. The authors define a "hidden progress measure" that correlates with grokking and then propose a method to accelerate grokking.
Strengths: The topic of delayed generalization is interesting and suitable to the ICLR audience.
Weaknesses: I found this manuscript very difficult to follow and understand. There are many typos and the ideas are not clearly expressed. To give just a single example, Theorem 4 seems to contain no falsifiable statement. This is not to say there are no valuable ideas in the work, but as it stands I believe the work is well below the standards of publication for ICLR.
Recommendation: 1
Reviewer confidence: 5
Soundness/correctness: 1
Clarity: 1