Machine Learning on Graph Data @ ICML 2019

Machine Learning on Graph Data
@ ICML 2019
亀澤諒亮 / DeNA
ICLR’19 / ICML’19 読み会 @DeNA

Words in Title @ ICML’17-‘19
4
2018 20192017

Words in Title @ ICML’17-‘19
5
2018 20192017

Graph ML @ ICML 2019
グラフデータを扱った論文はおよそ30本 (ICML’18ではおよそ15本) 
様々なタスクや手法をグラフに対しても適用できるように拡張したものが多い 
● Adversarial attacks
● Disentanglement
● Similarity measure
● Self attention, etc.
ニューラルネットワーク以外のアルゴリズムでグラフを扱うものも 
● Graph feature (kernel) / Gaussian process
応用 
● 回路設計の最適化 
● 楽譜からの演奏生成 
 
7

Outline
- Graph ML @ ICML 2019
- グラフについて
- 論文紹介
- A Persistent Weisfeiler–Lehman Procedure for Graph Classification
- Bastian Rieck, Christian Bock, Karsten Borgwardt / ETH Zurich
- Adversarial Attacks on Node Embeddings via Graph Poisoning
- Aleksandar Bojchevski, Stephan Günnemann / Technical University of Munich
- Simplifying Graph Convolutional Networks
- Felix Wu, Tianyi Zhang, Amauri Holanda de Souza Jr., Christopher Fifty, Tao Yu / Cornell
- Position-aware Graph Neural Networks
- Jiaxuan You, Rex Ying, Jure Leskovec / Stanford
8

Graph
- 頂点（node, vertex）と辺（link, edge）からなるデータ構造 
- 今回紹介するものは 
- 基本的に無向グラフ（辺の向きは考えない） 
- 各頂点が特徴ベクトルをもつ場合が多い 
 
- 具体的には隣接行列として表現 
- 隣接行列 
 
- 例 
- Web 
- SNSネットワーク 
- 引用ネットワーク 
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- タンパク質ネットワーク 
- 分子構造 
- etc.

A Persistent Weisfeiler-Lehman Procedure
for Graph Classification
Bastian Rieck, Christian Bock, Karsten Borgwardt / ETH Zurich
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http://proceedings.mlr.press/v97/rieck19a/rieck19a.pdf

Abstract
グラフ全体に対する特徴ベクトルを計算する手法を提案 
- WL Subtree [Shervashidze+, ‘11] は局所的な情報しか捉えない 
→ 閉路の情報を明示的に扱えるように拡張 
- 既存のGraph Feature/Kernel に比べて性能向上 
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Weisfeiler-Lehman Subtree (WL Subtree) [Shervashidze+, ‘11]
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http://www.jmlr.org/papers/volume12/shervashidze11a/shervashidze11a.pdf

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隣接するラベルを集める

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ラベル圧縮

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ラベルを更新

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戻って繰り返す

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ラベルの出現頻度をベクトル化

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WL subtree では全体的な構造に関する情報は含まれない

グラフの変化で構造（連結成分, 閉路, etc.) がどう変わるか？ 
- 辺の重みによってグラフを整列 
 
 
 
 
バーコード 
 
Persistent Homology
19
※Persistent homology における一般的な定義とは異なる https://www.math.upenn.edu/~ghrist/preprints/barcodes.pdf
連結成分の寿命 
CC persistence
閉路の寿命 
cycle persistence

 
 
 
 
バーコード 
 
Persistent Homology
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CC persistence
閉路の寿命 
cycle persistence

 
 
 
 
バーコード 
 
Persistent Homology
21
CC persistence
閉路の寿命 
cycle persistence

 
 
 
 
バーコード 
 
Persistent Homology
22
CC persistence
閉路の寿命 
cycle persistence

 
 
 
 
バーコード 
 
Persistent Homology
23
CC persistence
閉路の寿命 
cycle persistence

 
 
 
 
バーコード 
 
Persistent Homology
24
CC persistence
閉路の寿命 
cycle persistence

Persistent Homology
 
 
 
 
バーコード 
 
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CC persistence
閉路の寿命 
cycle persistence

Algorithm: Persistent subtree feature
グラフ G=(V, E), 反復数 H
Loop h = 1, …, H-1
隣接するラベルを集約 
辺の重み、PHを計算 
ラベル圧縮 
連結成分特徴 
 
閉路特徴 
 
 
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Loop h = 1, …, H-1
ラベル圧縮 
 
閉路特徴

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Loop h = 1, …, H-1
ラベル圧縮 
 
閉路特徴

Loop h = 1, …, H-1
ラベル圧縮 
 
閉路特徴 
 
 
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ラベル li
に対応するCC persistence の和

30
Loop h = 1, …, H-1
ラベル圧縮 
 
閉路特徴 
 
 
両辺どちらかの頂点のラベルが li
である
cycle persistence の和

Experiments
Benchmark datasets (classification)
- Graph feature / kernel の既存手法に比べて性能向上
31

Summary
Weisfeiler-Lehman subtree は明示的にトポロジカルな特徴を扱わない 
 
Persistent Weisfeiler-Lehman Procedure を提案 
- 明示的に閉路の情報を考慮 
- Persistent homology による表現を利用 
 
Graph feature / kernel の既存手法に比べて性能向上 
32

Adversarial Attacks on Node Embeddings
via Graph Poisoning
Aleksandar Bojchevski, Stephan Günnemann / Technical University of Munich
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http://proceedings.mlr.press/v97/bojchevski19a/bojchevski19a.pdf

Abstract
- DeepWalk（教師なしノード埋め込み）への攻撃を提案 
- 攻撃：グラフの辺の反転 
- 従来法[Zügner+, ICLR’19][Zügner+, KDD’18]は半教師あり学習に対する攻撃 
 
- 反転すべき辺を効率的に見つけるアルゴリズムを提案 
 
- 半教師あり学習でも攻撃によって性能劣化 
 
- 攻撃に用いたグラフはDeepWalk以外にも転用可能 
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DeepWalk [Perozzi+, KDD’14]
ノード埋め込みを計算する手法 
- サンプルされたrandom walk 上での skip-gram
- Random walkを文、ノードを単語とみなす 
- Skip-gramによってノード（単語）の埋め込みを計算 
実は特異値分解として解ける 
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Attack model
ノード埋め込みの目的関数に関する2段階の最適化 
 
 
 
 
 
36

Attack model
 
 
 
 
 
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隣接行列に対する制約

Attack model
 
 
 
 
 
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攻撃に対する制約

Attack model
 
 
 
 
 
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最適な埋め込みの計算

Attack model
 
 
 
 
 
40
損失を最大化する
隣接行列の計算

Attack model
 
 
 
 
DeepWalkの特異値分解としての解を使うことで 
内側の最適化(Zに関する損失の最小化)は閉じた形で求められる 
 
 
 
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Attack model
 
 
 
 
DeepWalkの特異値分解としての解を使うことで 
内側の最適化(Zに関する損失の最小化)は閉じた形で求められる 
 
 
このままでは外側の最適化は非効率 
→ 目的関数を近似 
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Approximation
- 勾配法 
- 特異値分解のためにナイーブには勾配を計算できない 
- Eigenvalue perturbationによって近似 
- 隣接行列の変化量は ±1のため誤差も大きくなる 
- 計算量  
 
 
 
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Approximation
- 勾配法 
- 特異値分解のためにナイーブには勾配を計算できない 
- Eigenvalue perturbationによって近似 
- 隣接行列の変化量は ±1のため誤差も大きくなる 
- 計算量  
 
- 貪欲法 
- 辺を反転した場合の損失の変化量を効率的に計算できるように近似 
- 辺の追加 
- 候補となる頂点対をサンプリングし、最適な辺を選択 
- 辺の削除 
- 存在する辺の中から最適な辺を選択 
 
 
44

Experiments
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勾配法貪欲法
横軸+ 辺の追加
横軸 - 辺の削除

Experiments
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転用可能性: DeepWalk以外のアルゴリズムでも性能劣化

Summary
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ノード埋め込みに対する攻撃を定式化、効率的なアルゴリズムを提案 
 
ノード埋め込みとそれに依存したタスク(ノード分類、リンク予測)でも性能劣化 
 
辺を反転する攻撃は他のアルゴリズム(node2vec, GCN, etc.)に対しても有効 
 
このような攻撃に対して頑健なアルゴリズムが必要

Simplifying Graph Convolutional Networks
Felix Wu, Tianyi Zhang, Amauri Holanda de Souza Jr., Christopher Fifty, Tao Yu / Cornell
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http://proceedings.mlr.press/v97/wu19e/wu19e.pdf

Abstract
- Graph Convolutional Network (GCN) [Kipf+, ICLR’17] を単純化したモデルでも 
ノード分類において大きな性能低下が見られないことを確認 
● Simple Graph Convolution (SGC)
 
- SGCがグラフスペクトル上のローパスフィルタであることを指摘 
 
 
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Graph Convolutional Network [Kipf+, ICLR’17]
 
 
 
 
 
 
正規化隣接行列 
50
(次数行列)

Simple Graph Convolution (SGC)
 
 
 
 
 
 
 
GCNの活性化関数(ReLU)を除くことで単純化 
特徴量の伝播は単純な行列積に帰着 
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Experiments
SGCと他のモデルで大きな変化は見られない 
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👍 特徴伝播
🤔 非線形性

Graph Signal Processing
正規化グラフラプラシアン  
ノード上の値（信号） 
グラフフーリエ変換 
逆グラフフーリエ変換 
 
 
 
 
 
 
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http://web.media.mit.edu/~xdong/presentation/CDT_GuestLecture_GSP.pdf
(固有値(周波数)小) (固有値(周波数)大)
固有値分解
グラフ上の固有ベクトル [Shuman+, ’13]
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.367.6064&rep=rep1&type=pdf

Low-pass Filter Effect
グラフフィルター(畳み込み) 
SGCの場合 
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各周波数成分にかかる係数

Summary
GCNの非線形層を取り除くことで単純化したモデルSGCを提案 
- 実験的に大きな性能劣化が見られないことを確認 
- GCNにおいて特徴伝播は有効だが、非線形性はあまり有効ではないことを示唆 
 
SGCにおける畳み込みはグラフスペクトル上のローパスフィルタであることを指摘 
- 多くのGNNはローパスフィルタと同等の性能 
 
 
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Position-aware Graph Neural Networks
Jiaxuan You, Rex Ying, Jure Leskovec / Stanford
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http://proceedings.mlr.press/v97/you19b/you19b.pdf

Abstract
従来のGNNだとノード埋め込みをした際に 
同じ局所構造をもつノードを区別できないことを指摘 
 
同じ局所構造であっても区別できるアーキテクチャを提案 
- P-GNN
 
実験的に既存法を上回る性能 
- Link prediction
- Pairwise node classification
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Motivation
既存のGNN(e.g. GCN)では同じ局所構造を持ったノードを区別できない 
 
 
 
 
 
 
 
 
→ 基準となるノードの集合 Anchor-setを設定し、そこから距離を考慮した特徴伝播を行う 
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Algorithm: P-GNN
1. Anchor-set S1
, ... ,Sk
のサンプリング 
2. 頂点v とAnchor-set Si
に対してメッセージMv
[i] を計算 
 
3. Mv
から出力ベクトルと次の層の入力ベクトルを計算 
 
 
 
 
 
 
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Algorithm: P-GNN
60
1. Anchor-set S1
, ... ,Sk
[i] を計算 
 
3. Mv
から出力ベクトルと次の層の入力ベクトルを計算

Algorithm: P-GNN
61
1. Anchor-set S1
, ... ,Sk
[i] を計算 
 
3. Mv
から出力ベクトルと次の層の入力ベクトルを計算

Anchor-Set
距離空間 
 
 
Bourgain Theorem [Bourgain, 1985]
距離空間 
 
 
具体的構成 
Anchor-set Si,j
は各頂点を確率1/2i
で独立にサンプリング 
 
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のdistortionがである  ⇔  
に対してdistortionが  となるような 
が存在する。

Anchor-Set
距離空間 
 
 
距離空間 
 
 
具体的構成 
Anchor-set Si,j
 
63
のdistortionがである 
が小さいほど距離を保つ
⇔  
が存在する。

Anchor-Set
距離空間 
 
 
距離空間 
 
 
具体的構成 
Anchor-set Si,j
 
64
が存在する。

Anchor-Set
距離空間 
 
 
距離空間 
 
 
具体的構成 
Anchor-set Si,j
→ Si,j
を使うことでグラフ上の距離を保ったまま特徴空間にマッピングできる 
 
65
が存在する。

Experiements
Link prediction
66

Experiments
Pairwise node classification
- 頂点対のラベルが同一かどうかを予測
67

Summary
既存のGNNでは局所構造が同じ頂点を区別できないことを指摘 
 
Anchor-set を導入することで局所構造が同じであっても区別できるようにGNNを拡張 
 
特定のタスクでは既存手法を大きく上回る性能 
- Link prediction
- Pairwise node classification
68

Whole Summary
- グラフを扱う論文は増えてきている (昨年からおよそ倍増) 
- タスクも手法も応用も多様化 
 
- 論文紹介 
- A Persistent Weisfeiler–Lehman Procedure for Graph Classification
- Adversarial Attacks on Node Embeddings via Graph Poisoning
- Simplifying Graph Convolutional Networks
- Position-aware Graph Neural Networks
 
69

References (ICML’19)
B. Rieck, C. Bock, and K. Borgwardt, “A Persistent Weisfeiler-Lehman Procedure for Graph Classification,” in International Conference on Machine Learning, 2019, pp. 5448–5458.
A. Bojchevski and S. Günnemann, “Adversarial Attacks on Node Embeddings via Graph Poisoning,” in International Conference on Machine Learning, 2019, pp. 695–704.
G. Zhang, H. He, and D. Katabi, “Circuit-GNN: Graph Neural Networks for Distributed Circuit Design,” in International Conference on Machine Learning, 2019, pp. 7364–7373.
Y. Yu, J. Chen, T. Gao, and M. Yu, “DAG-GNN: DAG Structure Learning with Graph Neural Networks,” in International Conference on Machine Learning, 2019, pp. 7154–7163.
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Machine Learning on Graph Data @ ICML 2019

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