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Asymmetric Learning for Spectral Graph Neural Networks
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Proposes asymmetric learning, which preconditions gradients of spectral GNN graph-convolution parameters (Θ) and feature-transformation parameters (W) by norm ratios, and shows it reduces a newly defined block condition number of the Hessian, improving five spectral GNNs on 18 node-classification datasets.
Problem
Spectral GNN training is poorly conditioned because graph convolution parameters Θ (under 20 dims) and feature transformation parameters W (thousands of dims) differ in sensitivity, and this gap is worse on heterophilic graphs.
Approach
Introduces a block condition number of the Hessian (ratio of largest block eigenvalues for Θ and W) and an asymmetric preconditioner that rescales the gradients of Θ and W by parameter-to-gradient norm ratios, so both parameter groups update at the same relative rate; proves this does not increase the block condition number.
Key results
- On six small heterophilic datasets, asymmetric learning improves spectral GNNs by about 3.08% on average; ChebNet on Texas rises from 36.76 to 50.81 and on Chameleon from 25.65 to 38.19.
- On homophilic graphs the average gain is about 0.46%, e.g. ChebNet on Cora improves 80.26 to 81.02.
- Orthogonal-basis models (ChebNet, ChebNetII, JacobiConv) gain about 5.15% on small heterophilic datasets versus 1.69% for non-orthogonal ones (GPRGNN, BernNet).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add asymmetric learning to spectral GNN training | Spectral GNNs on small heterophilic graphs · Texas, Wisconsin, Actor, Chameleon, Squirrel, Cornell | test accuracy: 3.08% | Helped |
| Add asymmetric learning to spectral GNN training | Spectral GNNs on homophilic graphs · Citeseer, Pubmed, Cora, Computers, Photo, Coauthor-CS, Coauthor-Physics | test accuracy: 0.46% | Helped |
| Add asymmetric learning to ChebNet | ChebNet · Texas | test accuracy: +14.05 | Helped |
| Add asymmetric learning to ChebNet | ChebNet · Chameleon | test accuracy: +12.54 | Helped |
| Add asymmetric learning to spectral GNN training | Orthogonal spectral GNNs · Six small heterophilic datasets | test accuracy: 5.15% | Helped |
| Add asymmetric learning to spectral GNN training | Non-orthogonal spectral GNNs · Six small heterophilic datasets | test accuracy: 1.69% | Helped |
| Add asymmetric learning to ChebNet | ChebNet · Texas, Wisconsin, Actor, Chameleon, Squirrel, Cornell | test accuracy: +14.05; +0.04; +0.19; +12.54; +4.66; +0.40 | Mixed |
| Add asymmetric learning to ChebNet | ChebNet · Cora | test accuracy: +0.76 | Helped |
| Vary the parameter-norm EMA decay values βπΘ and βπW from 0 to 0.99 (ablation) | ChebNet with asymmetric learning · Node classification datasets | test accuracy | No clear effect |
Surprising findings
- Non-orthogonal spectral GNNs (GPRGNN, BernNet) benefit less than orthogonal ones, which the authors attribute to basis interference requiring a more sophisticated preconditioner than a uniform scaling.
- Some empirical checks of the theoretical assumptions fail on several datasets (e.g., GPNR ordering does not hold on Squirrel, Cornell, Actor), and the authors say the assumptions hold on most datasets only.
Limitations
- Theoretical guarantees rely on assumptions (point proximity, proportional GPNR, mild scaling) that the authors only check empirically and that hold on most but not all datasets.
- Analysis is restricted to polynomial spectral GNNs and node classification; not evaluated on other GNN families or tasks.
- Gains on large heterophilic and homophilic graphs are small (about 0.5% to 1.7% and 0.46%), and some configurations show decreases (e.g., ChebNetII on Questions, ChebNetII on Cora).
Experimental setup
Datasets: Texas Wisconsin Actor Chameleon Squirrel Cornell Roman_Empire Amazon_Ratings Minesweeper Tolokers Questions Citeseer
Benchmarks: Texas Wisconsin Actor Chameleon Squirrel Cornell Roman_Empire Amazon_Ratings Minesweeper Tolokers Questions Citeseer
Metrics: test accuracy ROC AUC
Baselines: ChebNet ChebNetII JacobiConv GPRGNN BernNet
Builds on: Adam Chebyshev polynomial spectral GNN GPRGNN BernNet JacobiConv
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