RandAlign: A Parameter-Free Method for Regularizing Graph Convolutional Networks

Haimin Zhang, Min Xu

Main figure for RandAlign: A Parameter-Free Method for Regularizing Graph Convolutional Networks

Abstract

Studies continually find that message-passing graph convolutional networks suffer from the over-smoothing issue. Basically, the issue of over-smoothing refers to the phenomenon that the learned embeddings for all nodes can become very similar to one another and therefore are uninformative after repeatedly applying message passing iterations. Intuitively, we can expect the generated embeddings become smooth asymptotically layerwisely, that is each layer of graph convolution generates a smoothed version of embeddings as compared to that generated by the previous layer. Based on this intuition, we propose RandAlign, a stochastic regularization method for graph convolutional networks. The idea of RandAlign is to randomly align the learned embedding for each node with that of the previous layer using randomly interpolation in each graph convolution layer. Through alignment, the smoothness of the generated embeddings is explicitly reduced. To better maintain the benefit yielded by the graph convolution, in the alignment step we introduce to first scale the embedding of the previous layer to the same norm as the generated embedding and then perform random interpolation for aligning the generated embedding. RandAlign is a parameter-free method and can be directly applied without introducing additional trainable weights or hyper-parameters. We experimentally evaluate RandAlign on different graph domain tasks on seven benchmark datasets. The experimental results show that RandAlign is a general method that improves the generalization performance of various graph convolutional network models and also improves the numerical stability of optimization, advancing the state of the art performance for graph representation learning.

Benchmark Results

Tested on 7 datasets 19 baseline values reported too low

Tables show presented results in global context (best 3 papers + GCN + MLP).

OGB

#ModelROC-AUC ↑Paper
1
Multi-RF Fusion with Multi-GNN Blending
0.8476 ±0.0002
2
HyperFusion
0.8475 ±0.0003
3
PAS+FPs
0.8420 ±0.0015
9
GatedGCN
0.8090 ±0.0160
22
GPS + RandAlign
0.8021 ±0.0305
This paper
37
DGN
0.7970 ±0.0097
This paper
38
GIN-AK+
0.7961 ±0.0119
This paper
62
PNA
0.7905 ±0.0132
This paper
69
GPS
0.7880 ±0.0101
This paper
78
DeeperGCN
actual #78 vs paper #78
0.7860 ±0.0120
best reported
actual #78 vs paper #78
paper 0.7858 ±0.0117

Difference within reported uncertainty; not counted as underperformance.

paper 0.7858 ±0.0117 Δ at pub 0.0002 Δ today 0.0002
102
ExpC
0.7799 ±0.0082
This paper
106
SAN
0.7785 ±0.2470
This paper
109
GIN
actual #109 vs paper #150 41 places worse
0.7780 ±0.0182
best reported
actual #109 vs paper #150 41 places worse
paper 0.7707 ±0.0149

Difference within reported uncertainty; not counted as underperformance.

paper 0.7707 ±0.0149 Δ at pub 0.0073
160
GraphSAGE
0.7690 ±0.0136
184
GCN
actual #184 vs paper #203 19 places worse
0.7630 ±0.0030
best reported
actual #184 vs paper #203 19 places worse
paper 0.7599 ±0.0119

Difference within reported uncertainty; not counted as underperformance.

paper 0.7599 ±0.0119 Δ at pub 0.0031

LRGB

#ModelAP ↑Paper
1
S^2GCN
0.7311 ±0.0066
2
S4G
0.7293 ±0.0004
3
BuNN
0.7276 ±0.0065
4
GCN
actual #4 vs paper #142 138 places worse
0.7261 ±0.0011
best reported
actual #4 vs paper #142 138 places worse
paper 0.5930 ±0.0023
paper 0.5930 ±0.0023 Δ at pub 0.0930 Δ today 0.1331
26
GIN
0.7059 ±0.0089
34
GatedGCN
actual #34 vs paper #149 115 places worse
0.7006 ±0.0033
best reported
actual #34 vs paper #149 115 places worse
paper 0.5864 ±0.0077
paper 0.5864 ±0.0077 Δ at pub 0.0901 Δ today 0.1142
73
GatedGCN+RWSE
actual #73 vs paper #134 61 places worse
0.6685 ±0.0062
best reported
actual #73 vs paper #134 61 places worse
paper 0.6069 ±0.0035
paper 0.6069 ±0.0035 Δ at pub 0.0616
76
GPS + RandAlign
0.6630 ±0.0005
This paper
78
GINE
actual #78 vs paper #162 84 places worse
0.6621 ±0.0067
best reported
actual #78 vs paper #162 84 places worse
paper 0.5498 ±0.0079
paper 0.5498 ±0.0079 Δ at pub 0.1123 Δ today 0.1123
89
SAN+RWSE
actual #89 vs paper #111 22 places worse
0.6562 ±0.0075
best reported
actual #89 vs paper #111 22 places worse
paper 0.6439 ±0.0075

Difference within reported uncertainty; not counted as underperformance.

paper 0.6439 ±0.0075 Δ at pub 0.0123
87
GPS
actual #87 vs paper #91 4 places worse
0.6562 ±0.0115
best reported
actual #87 vs paper #91 4 places worse
paper 0.6535 ±0.0041

Difference within reported uncertainty; not counted as underperformance.

paper 0.6535 ±0.0041 Δ at pub 0.0027
116
SAN+LapPE
0.6384 ±0.0121
This paper
118
Transformer+LapPE
0.6326 ±0.0126
This paper
181
MLP
0.4060 ±0.0021
#ModelF1 ↑Paper
1
NeuralWalker
0.4912 ±0.0042
2
PPGT
0.4641 ±0.0033
3
CRaWl
0.4588 ±0.0079
14
GPS
actual #14 vs paper #44 30 places worse
0.4440 ±0.0065
best reported
actual #14 vs paper #44 30 places worse
paper 0.3748 ±0.0109
paper 0.3748 ±0.0109 Δ at pub 0.0692 Δ today 0.0692
18
GPS + RandAlign
0.4288 ±0.0062
This paper
40
GatedGCN+RWSE
actual #40 vs paper #70 30 places worse
0.3882 ±0.0041
best reported
actual #40 vs paper #70 30 places worse
paper 0.2860 ±0.0085
paper 0.2860 ±0.0085 Δ today 0.1022
41
GatedGCN
actual #41 vs paper #70 29 places worse
0.3880 ±0.0040
best reported
actual #41 vs paper #70 29 places worse
paper 0.2873 ±0.0219
paper 0.2873 ±0.0219 Δ at pub 0.1007 Δ today 0.1007
42
Gated-GCN
0.3880 ±0.0040
67
SAN+LapPE
0.3230 ±0.0039
This paper
68
SAN+RWSE
0.3216 ±0.0027
This paper
75
GINE
actual #75 vs paper #92 17 places worse
0.2718 ±0.0054
best reported
actual #75 vs paper #92 17 places worse
paper 0.1265 ±0.0076
paper 0.1265 ±0.0076 Δ at pub 0.1453 Δ today 0.1453
76
Transformer+LapPE
0.2694 ±0.0098
This paper
81
GCN
actual #81 vs paper #92 11 places worse
0.2380 ±0.0160
best reported
actual #81 vs paper #92 11 places worse
paper 0.1268 ±0.0060
paper 0.1268 ±0.0060 Δ at pub 0.0810 Δ today 0.1112
93
MLP
0.1140 ±0.0230

GNNBenchmark

#ModelAccuracy ↑Paper
1
Schrödinger
0.9913 ±0.0004
2
CNN
0.9907 ±0.0007
3
MPNN
0.9895 ±0.0006
12
GatedGCN + RandAlign
0.9855 ±0.0002
This paper
11
GatedGCN
actual #11 vs paper #11
0.9855 ±0.0002
actual #11 vs paper #11
paper 0.9855 ±0.0002
paper 0.9855 ±0.0002 Δ at pub 0.0031
15
RandAlign
0.9851 ±0.0005
This paper
32
GIN
actual #32 vs paper #110 78 places worse
0.9833 ±0.0011
best reported
actual #32 vs paper #110 78 places worse
paper 0.9648 ±0.0025
best at pub: Graph-Coupled Oscillator Networks (ICML 2022)
paper 0.9648 ±0.0025 Δ at pub 0.0075 Δ today 0.0184
47
EGT
0.9817 ±0.0009
This paper
59
GPS
actual #59 vs paper #61 2 places worse
0.9808 ±0.0011
best reported
actual #59 vs paper #61 2 places worse
paper 0.9805 ±0.0013

Difference within reported uncertainty; not counted as underperformance.

paper 0.9805 ±0.0013 Δ today 0.0003
70
GatedGCN + SSFG
0.9798 ±0.0003
This paper
73
CRaWl
0.9794 ±0.0005
This paper
75
PNA
0.9794 ±0.0012
This paper
87
GAT + RandAlign
0.9755 ±0.0003
This paper
86
GAT
actual #86 vs paper #86
0.9755 ±0.0003
actual #86 vs paper #86
paper 0.9755 ±0.0003
paper 0.9755 ±0.0003 Δ at pub 0.0105
92
GatedGCN-16
0.9734 ±0.0014
This paper
95
GraphSAGE
0.9731 ±0.0010
This paper
97
SAGE
0.9731 ±0.0010
123
GAT-16
0.9554 ±0.0021
This paper
126
MLP
0.9534 ±0.0014
129
3WLGNN
0.9507 ±0.0096
This paper
136
GCN + RandAlign
0.9405 ±0.0005
This paper
135
GCN
actual #135 vs paper #135
0.9405 ±0.0005
actual #135 vs paper #135
paper 0.9405 ±0.0005
paper 0.9405 ±0.0005 Δ at pub 0.0136
137
MoNet
actual #137 vs paper #137
0.9111
best reported
actual #137 vs paper #137
paper 0.9081 ±0.0003
best at pub: Graph-Coupled Oscillator Networks (ICML 2022)
paper 0.9081 ±0.0003 Δ at pub 0.0030
141
GCNII
0.9067 ±0.0014
This paper
#ModelAccuracy ↑Paper
1
RandAlign
0.7942 ±0.0007
This paper
2
differential encoding
0.7907 ±0.0027
3
PPGT
0.7856 ±0.0070
9
GatedGCN + RandAlign
0.7640 ±0.0019
This paper
10
GatedGCN-16 + RandAlign
0.7640 ±0.0019
This paper
38
PNA
actual #38 vs paper #67 29 places worse
0.7311 ±0.0011
best reported
actual #38 vs paper #67 29 places worse
paper 0.7035 ±0.0063
paper 0.7035 ±0.0063 Δ at pub 0.0276
40
DGN
actual #40 vs paper #40
0.7284 ±0.0042
best reported
actual #40 vs paper #40
paper 0.7284 ±0.0042

Difference within reported uncertainty; not counted as underperformance.

paper 0.7284 ±0.0042 Δ at pub 0.0000
44
GIN-AK+
actual #44 vs paper #50 6 places worse
0.7239 ±0.0038
best reported
actual #44 vs paper #50 6 places worse
paper 0.7219 ±0.0013

Difference within reported uncertainty; not counted as underperformance.

paper 0.7219 ±0.0013 Δ at pub 0.0020
46
GPS
actual #46 vs paper #49 3 places worse
0.7236 ±0.0032
best reported
actual #46 vs paper #49 3 places worse
paper 0.7230 ±0.0036

Difference within reported uncertainty; not counted as underperformance.

paper 0.7230 ±0.0036 Δ today 0.0006
56
GatedGCN + SSFG
0.7194 ±0.0019
This paper
62
GatedGCN
0.7133 ±0.0039
69
GAT + RandAlign
0.6992 ±0.0008
This paper
70
GAT-16 + RandAlign
0.6992 ±0.0008
This paper
80
CRaWl
0.6901 ±0.0026
This paper
82
EGT
0.6870 ±0.0041
This paper
91
GatedGCN-16
0.6731 ±0.0031
This paper
92
GraphSAGE
actual #92 vs paper #104 12 places worse
0.6690 ±0.0020
best reported
actual #92 vs paper #104 12 places worse
paper 0.6577 ±0.0031
paper 0.6577 ±0.0031 Δ at pub 0.0113
103
MoNet
actual #103 vs paper #129 26 places worse
0.6591 ±0.0251
best reported
actual #103 vs paper #129 26 places worse
paper 0.5465 ±0.0052
paper 0.5465 ±0.0052 Δ at pub 0.1126
104
SAGE
0.6577 ±0.0031
109
GIN
actual #109 vs paper #129 20 places worse
0.6510 ±0.0050
best reported
actual #109 vs paper #129 20 places worse
paper 0.5525 ±0.0153
paper 0.5525 ±0.0153 Δ at pub 0.0985
114
GAT-16
0.6422 ±0.0046
This paper
123
3WLGNN
0.5917 ±0.0159
This paper
124
GCN
actual #124 vs paper #128 4 places worse
0.5839 ±0.0073
best reported
actual #124 vs paper #128 4 places worse
paper 0.5571 ±0.0038
paper 0.5571 ±0.0038 Δ at pub 0.0268
125
GCN + RandAlign
0.5774 ±0.0016
This paper
126
MLP
0.5634 ±0.0018
128
GCNII
0.5608 ±0.0020
This paper
#ModelAccuracy ↑Paper
1
GRPE-Small
0.8159 ±0.0019
2
TANGO-GPS
0.8011 ±0.0014
3
GRIT
0.8003 ±0.0028
10
GPS + RandAlign
0.7859 ±0.0005
This paper
22
GPS
0.7802 ±0.0018
This paper
23
K-Subgraph SAT
0.7786 ±0.0010
This paper
26
SAN + RandAlign
0.7785 ±0.0007
This paper
50
SAN
0.7669 ±0.0065
This paper
54
GatedGCN
actual #54 vs paper #62 8 places worse
0.7608 ±0.0020
best reported
actual #54 vs paper #62 8 places worse
paper 0.7384 ±0.0033
paper 0.7384 ±0.0033 Δ at pub 0.0224
57
GatedGCN + SSFG
0.7596 ±0.0002
This paper
67
GAT
0.7059 ±0.0045
This paper
70
GCN
actual #70 vs paper #71 1 place worse
0.6903 ±0.0137
best reported
actual #70 vs paper #71 1 place worse
paper 0.6850 ±0.0098

Difference within reported uncertainty; not counted as underperformance.

paper 0.6850 ±0.0098 Δ at pub 0.0053
76
MoNet
actual #76 vs paper #76
0.6641 ±0.0054
actual #76 vs paper #76
paper 0.6641 ±0.0054
paper 0.6641 ±0.0054 Δ at pub 0.0834
79
GIN
0.6472 ±0.0155
This paper
80
GraphSAGE
actual #80 vs paper #80
0.6388 ±0.0011
best reported
actual #80 vs paper #80
paper 0.6384 ±0.0011
best at pub: IJCAI--23 Formatting Instructions (IJCAI 2023)

Difference within reported uncertainty; not counted as underperformance.

paper 0.6384 ±0.0011 Δ at pub 0.0004
104
RingGNN
0.4242 ±0.2006
This paper
114
MLP
0.2097 ±0.0000
#ModelAccuracy ↑Paper
1
DeepGraph (48)
0.9066 ±0.0006
2
GRIT (our run)
0.9041 ±0.0023
3
GRIT 2-QiQRW
0.9016 ±0.0045
18
GPS + RandAlign
0.8686 ±0.0001
This paper
21
K-Subgraph SAT
0.8685 ±0.0004
This paper
31
SAN + RandAlign
0.8677 ±0.0007
This paper
50
GPS
0.8669 ±0.0006
This paper
58
SAN
0.8658 ±0.0004
This paper
66
GatedGCN
actual #66 vs paper #91 25 places worse
0.8651 ±0.0008
best reported
actual #66 vs paper #91 25 places worse
paper 0.8557 ±0.0009
best at pub: IJCAI--23 Formatting Instructions (IJCAI 2023)
paper 0.8557 ±0.0009 Δ at pub 0.0094
88
GCN
actual #88 vs paper #116 28 places worse
0.8561 ±0.0003
best reported
actual #88 vs paper #116 28 places worse
paper 0.7189 ±0.0033
paper 0.7189 ±0.0033 Δ at pub 0.1372
89
GIN
actual #89 vs paper #94 5 places worse
0.8559 ±0.0001
best reported
actual #89 vs paper #94 5 places worse
paper 0.8539 ±0.0014
best at pub: Factorizable Graph Convolutional Networks (NeurIPS 2020)
best today: GNNBenchmark (JMLR 2023)

Difference within reported uncertainty; not counted as underperformance.

paper 0.8539 ±0.0014 Δ at pub 0.0020 Δ today 0.0020
91
MoNet
actual #91 vs paper #91
0.8558 ±0.0004
actual #91 vs paper #91
paper 0.8558 ±0.0004
paper 0.8558 ±0.0004 Δ at pub 0.0010
107
GAT
0.7827 ±0.0019
This paper
123
MLP
0.5052
124
GraphSAGE
actual #124 vs paper #125 1 place worse
0.5052 ±0.0000
best reported
actual #124 vs paper #125 1 place worse
paper 0.5049 ±0.0000
best today: GNNBenchmark (JMLR 2023)
paper 0.5049 ±0.0000 Δ at pub 0.0002 Δ today 0.0002
SOTA performance (not significantly different from the best) Proposed method (this paper) Baseline / comparison (this paper) Global context (other papers) Best at pub — reported result worse than the best known when published (best reported in green) Best at pub — reported value improved over the best known when published Best today — OK at publication; today's leaderboard best is elsewhere