Are GATs Out of Balance?

While the expressive power and computational capabilities of graph neural networks (GNNs) have been theoretically studied, their optimization and learning dynamics, in general, remain largely unexplored. Our study undertakes the Graph Attention Network (GAT), a popular GNN architecture in which a node's neighborhood aggregation is weighted by parameterized attention coefficients. We derive a conservation law of GAT gradient flow dynamics, which explains why a high portion of parameters in GATs with standard initialization struggle to change during training. This effect is amplified in deeper GATs, which perform significantly worse than their shallow counterparts. To alleviate this problem, we devise an initialization scheme that balances the GAT network. Our approach i) allows more effective propagation of gradients and in turn enables trainability of deeper networks, and ii) attains a considerable speedup in training and convergence time in comparison to the standard initialization. Our main theorem serves as a stepping stone to studying the learning dynamics of positive homogeneous models with attention mechanisms.

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Peer review

Reviewer hDWR6/10 · confidence 3/52023-07-02

Summary

The work derives a conservation for the dynamics of the GAT gradient flow during training. The conservation law is used to explain why it is challenging to train deep GAT models and in particular why a large portion of parameters do not change much throughout training. A new initialization scheme is introduced to mitigate these issues.

Strengths

Training GATs is notoriously problematic and this is especially the case with many layers. The work takes an interesting angle of trying to further explain why this is the case by studying conservation laws induced by the gradient flow during training. The observation that weight gradients must be small in order to satisfy such a conservation law is very interesting and practically useful. The balancing algorithm proposed is also practically useful and seems to perform well especially with deep GATs.

Weaknesses

While the work focuses on GAT, it would be interesting to have a statement on more general classes of MPNNs. There are some faint connections in the paper to GCN for instance, but it would be interesting to have a more general result as well. Experimentally this might also be interesting to see if such an initialisation can be adapted to different types of MPNNs.

Questions

Would the proposed initialisation technique work well for other types of MPNNs (with appropriate modifications)? Would you have experiments/theory to back this up?

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

3 good

Presentation

3 good

Contribution

3 good

Limitations

The authors address the fact that the derived theory defines a conservation law that does not explain detailed dynamics but still remains relatively coarse. As such it is not able to explain some phenomena as why attention parameters change most on the first layer. I still believe that the work is a good step in helping explain phenomena in this direction.

Reviewer d9X86/10 · confidence 3/52023-07-04

Summary

This paper focuses on the parameter struggle training problem of the well-known GNN structure GAT. The authors propose to alleviate the issue via parameter norm balancedness. A conservation law is derived for GATs with positive homogeneous activation function as the theoretical support for the parameter norm balancedness based initialization method. The authors also conduct extensive experiments to prove its effectiveness and fast convergence property.

Strengths

- The problem is well identified. - The proposed theories are solid, and well serves the problem. - The proposed initialization method is simple yet effective in improving accuracy and speeding convergence.

Weaknesses

- Some explanation skips details and may raise confusions. For example, the equations in line 140-142. - Some formats could be improved for better illustration, such as larger figure size and fitted table style.

Questions

GAT mainly exploits the attention mechanism. Can this initialization be adapted to any other methods that utilize the attention mechanism?

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

4 excellent

Presentation

3 good

Contribution

3 good

Limitations

The author may address the limitation from the generalization of the proposed method beyond the GAT structure.

Reviewer yMby7/10 · confidence 4/52023-07-05

Summary

The authors propose a theoretical analysis of the initialization of GATs and their impact on the performance of such networks, focusing on the performance vs depth aspect. The authors propose an initialization algorithm, that starts from a random initialization and modifies the initial random weights to adhere to the findings of the theoretical analysis. The authors then show the impact of their proposed initialization on several node classification datasets. Specifically, they show that by initializing GATs with the proposed algorithm, deep GATs can be trained to achieve better than standard (Xavier) initialized GATs.

Strengths

The paper addresses a real issue with GATs - the degradation of performance when more layers are added. The theoretical analysis seems correct to me and is supported with experimental results as well as the inspection of actual training artifacts and details (e.g., the gradients of the GAT layers).

Weaknesses

The paper can be slightly better written, in terms of organization. I think that adding more paragraphs/subsections to better separate between the parts of the paper can help to ease the reader. The paper lacks a few relevant citations that also consider graph neural networks as gradient flow (see [1][2][3]). However, their focus is not on the initialization of GATs, and therefore the paper here is novel on its own. While the experimental results are compelling and show the benefit of the proposed method, I think that the authors should also include comparisons with other methods. [1] GRAND: Graph Neural Diffusion [2] Understanding convolution on graphs via energies [3] Improving Graph Neural Networks with Learnable Propagation Operators

Questions

1. Can the authors add results with more layers? For instance 64,80 layers? does the method still work with very deep network? 2. Regarding the initialization of graph neural networks, it is discussed in [3][4] that the GNN weights are initialized with identity matrices. Can the authors comment on this point? is it related to the proposed method? 3. Can the proposed initialization scheme be applied to other graph attention layers, such as superGAT[5] ? [4] PDE-GCN: Novel Architectures for Graph Neural Networks Motivated by Partial Differential Equations [5] How to Find Your Friendly Neighborhood: Graph Attention Design with Self-Supervision

Rating

7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.

Confidence

4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.

Soundness

3 good

Presentation

2 fair

Contribution

3 good

Limitations

Yes

Reviewer cXS57/10 · confidence 3/52023-07-07

Summary

This work proves a conservation law for GAT architectures, which is similar to conservation laws shown for fully connected networks. The conservation law shows a simple connection between the norms of weights of two consecutive layers. Using this law, an intuitive explanation is given for the trainability issues of deep GAT architectures. To mitigate this issue, a balanced initialization is suggested which shows improvements in generalization and training speed of deep GATs.

Strengths

1. Novel conservation law for GAT architectures. 2. The paper and technical details are well written.

Weaknesses

There are no major weaknesses. 1. The motivation for using the balanced initialization and why it should improve only for deep networks is not so clear. 2. The experimental results are not complete. See Questions for more details.

Questions

1. Does LLOrtho without balancing work well for deep networks? It is not clear if the cause for improvement is only balancing or also the initialization with orthogonal vectors. 2. The trainability issue is explained intuitively in pages 4-5: (a) What is the reason that the trainability issue is amplified with depth? This is not explained. (b) Why does this trainability issue occur only for GATs and not for fully connected networks? It seems that the same explanation holds for FC nets as well. (c) Suggestion: It would be helpful to explain why the balanced initialization solves the trainability issue (it is not explained).

Rating

7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

3 good

Presentation

3 good

Contribution

3 good

Limitations

Limitations are not discussed. Maybe it would be helpful to add a section on this.

Reviewer hDWR2023-08-11

Thank you for the response and the additional experiments with GCN. The smoothing results with GCN indeed seem quite promising. This is an interesting approach to a useful problem and I recommend accepting the work.

Reviewer yMby2023-08-16

Thank you for the detailed rebuttal

I would like to thank the authors for the detailed rebuttal. Your answers addressed all my questions and therefore I am happy to increase my score.

Program Chairsdecision2023-09-21

Decision

Accept (poster)

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