# Flattened Graph Convolutional Networks For Recommendation

Yue Xu<EMAIL_ADDRESS>Alibaba Group. ,  Hao Chen
<EMAIL_ADDRESS>Tencent Inc. ,  Zengde Deng<EMAIL_ADDRESS>Cainiao Network. ,  Yuanchen Bei<EMAIL_ADDRESS>Zhejiang University.
and  Feiran Huang<EMAIL_ADDRESS>Jinan University.

###### Abstract.

Graph Convolutional Networks (GCNs) and their variants have achieved
significant performances on various recommendation tasks. However, many
existing GCN models tend to perform recursive aggregations among all related
nodes, which can arise severe computational burden to hinder their application
to large-scale recommendation tasks. To this end, this paper proposes the
flattened GCN (FlatGCN) model, which is able to achieve superior performance
with remarkably less complexity compared with existing models. Our main
contribution is three-fold. First, we propose a simplified but powerful GCN
architecture which aggregates the neighborhood information using one flattened
GCN layer, instead of recursively. The aggregation step in FlatGCN is
parameter-free such that it can be pre-computed with parallel computation to
save memory and computational cost. Second, we propose an informative
neighbor-infomax sampling method to select the most valuable neighbors by
measuring the correlation among neighboring nodes based on a principled
metric. Third, we propose a layer ensemble technique which improves the
expressiveness of the learned representations by assembling the layer-wise
neighborhood representations at the final layer. Extensive experiments on
three datasets verify that our proposed model outperforms existing GCN models
considerably and yields up to a few orders of magnitude speedup in training
efficiency.

††conference: 4th Workshop on Deep Learning Practice and Theory for High-
Dimensional Sparse and Imbalanced Data with KDD 2022; August 14, 2022; Virtual
Conference, Online††price: 15.00††isbn: 978-1-4503-XXXX-X/22/08

##  1\. Introduction

Graph Convolutional Networks (GCNs), which generalize the Convolutional Neural
Networks (CNNs) on graph-structured data (kipf2016semi, ), have achieved
impressive performance on various graph-based learning tasks
(velivckovic2017gat, ; hamilton2017representation, ; hamilton2017graphsage, ),
including recommendation (zhao2019intentgc, ). The core idea behind GCNs is to
iteratively aggregate information from locally nearby neighbors in a graph
using neural networks (chen2018fastgcn, ). Each node at one GCN layer performs
graph convolution operations to aggregate information from its nearby
neighbors at the previous layer. By stacking multiple GCN layers, the
information can be propagated across the far reaches of a graph, which makes
GCNs capable of learning from both content information as well as the graph
structure. As such, GCN-based models are widely adopted in recommendation
tasks (ying2018graph, ; zhao2019intentgc, ; fan2019meirec, ; wu2019dual, ;
chen2022neighbor, ; wang2019knowledge, ; Wang2019graph, ; chen2021non, ;
chen2022generative, ) which require learning from relational datasets.
However, although existing GCN-based recommendation models have set new
standards on various benchmark tasks (ying2018graph, ; zhao2019intentgc, ;
fan2019meirec, ; wang2019kgat, ; wang2019knowledge, ; chen2020label, ;
wang2019ngcf, ; he2020lightgcn, ; chen2020graph, ), many of them suffer from
two pitfalls.

First, many existing GCN models suffer from high computational complexity due
to the use of multi-layer architectures and complicated modeling techniques.
This may largely hinder their application to large-scale real-world
recommender systems. For example, the metapath-guided GCN models
(hu2018leveraging, ; fan2019meirec, ) propose to construct manifold metapaths
for neighborhood aggregation, which introduces high complexity on data pre-
processing and information aggregation. Meanwhile, the attention-based GCN
(GAT) models propose to generalize graph convolution with the attention
mechanism (velivckovic2017gat, ; wang2019kgat, ; wu2019dual, ), which,
however, introduces an excessive amount of additional parameters to fit. On
the other hand, the recent advances on simplified GCNs such as (wu2019sgc, ;
he2020lightgcn, ), indicate that it is feasible to remove certain components
from existing architectures while still maintaining comparable performances.
This motivates us to rethink the essential components of building an
expressive GCN model for recommendations.

Second, the recursive neighborhood aggregation among all nodes arises severe
computational burden but may have limited contribution in recommendation
tasks. Specifically, as pointed out in (li2018laplacian, ), the convolution in
the GCN model is indeed a special form of Laplacian smoothing, which makes the
feature of nodes within the same cluster similar to greatly ease the
classification or regression task. Therefore, it is critical for GCN models to
ensure that similar nodes have been grouped into the same cluster before
performing neighborhood aggregations. In homogeneous networks, it is highly
likely for two similar nodes to form a direct edge (mcpherson2001homophily, ).
However, the networks are heterogeneous in the context of recommendation. As
such, the difficulty of recognizing similar nodes arises since we need to
measure the correlation between any user-item pair based on their indirect
relationships. Existing models usually measure the correlation by counting the
historical interactions (fan2019meirec, ; Wang2019graph, ; wu2019dual, ;
wang2019knowledge, ). However, such intuitive measurements can be easily
dominated by the popular user/items. Besides, the correlations are unlikely to
scale linearly with the number of historical interactions. Therefore, it is
worth exploring the definition of a principled metric to quantitatively
evaluate the importance of neighbors in heterogeneous networks. This would
also pave the way for the design of an informative and efficient neighbor
sampling method for GCN models.

In this paper, we propose the flattened GCN (FlatGCN) model which has a much
lower complexity compared to existing GCN-based recommendation models but is
able to achieve superior performance. The main contributions are summarized as
follows. First, we propose the FlatGCN to simplify the recursive neighborhood
aggregation in standard GCNs as flattened layer aggregations which require
performing propagation only once over a single GCN layer. Moreover, the
aggregation step in FlatGCN is a parameter-free operation such that it can be
pre-computed to save vast memory and computational cost thereby easing the
training on large-scale graphs when applied to real-world recommender systems.
Second, we propose an informative and efficient sampling method named
neighbor-infomax to select the top informative neighbors for neighborhood
aggregation. Specifically, we propose a principled metric to explicitly
measure the correlation among neighboring nodes in a user-item bipartite graph
and then select the most informative neighbors according to the evaluation
results so as to improve the quality of neighborhood aggregation. Third, we
propose a layer ensemble technique which improves the expressiveness of the
learned representations by assembling the layer-wise neighborhood
representations at the final layer. Extensive experiments on two benchmarks
and one commercial dataset verify that our proposed model outperforms existing
GCN models considerably and yields up to a few orders of magnitude speedup in
training, in terms of recommendation tasks.

##  2\. METHODOLOGY

In this section, we first introduce the neighbor information measurement to
rank the neighbor set and select the most informative subset. Then we present
the efficient and powerful flattened layer aggregation to represent the
subgraph of a given user (item). Finally, we propose a novel layer ensemble
architecture to predict the relevance score. We assume that we have learned
users (items)’ embeddings with given collaborative filtering models such as
Meta2Vec (dong2017metapath2vec, ) or LightGCN (he2020lightgcn, ) which are set
as the node features for the users (items).

Figure 1. Sketch of FlatGCN. On the left, we present the flattened layer
aggregation, utilizing the neighbor information measurement to rank the
neighbors and select the top informative neighbors from each layer’s neighbor
set. For example, the sampling size of the middle layer is 2, and we select
the top-2 informative neighbors, while the sampling size for the bottom layer
is 4, and we select top-4 informative neighbors. The layer ensemble
architecture is shown on the right.

###  2.1. Neighbor Information Measurement

Starting from GraphSAGE (hamilton2017graphsage, ) and FastGCN
(chen2018fastgcn, ), GCN models downsample the neighbors to accelerate the
recursive message passing. Since this downsampling process would inevitably
lose information, it is critical to preserve as much information as possible
in the preserved neighbor, which could maximally represent the root node. To
this end, we introduce the mutual information measurement to evaluate to what
extent a neighbor can describe the given root node. Besides, we also propose a
simplified version of neighbor-root-node information measurement for efficient
computation.

Neighbor-Root-Node Mutual Information. Let $u$ denote the root node. We first
select an arbitrary node $v\in{\mathcal{N}}_{u}$. Let the random variable
${\bm{v}}$ be the feature of a randomly picked node in ${\mathcal{N}}_{u}$,
then the distribution of ${\bm{v}}$ is $P_{\bm{v}}=P({\bm{v}}={\bm{e}}_{v})$,
where ${\bm{e}}_{v}$ is the outcome feature of node $v$. Similarly, we assume
${\bm{u}}$ be the feature associated with the root node $u$, then the
distribution of ${\bm{u}}$ is $P_{\bm{u}}=P({\bm{u}}={\bm{e}}_{u})$, where
${\bm{e}}_{u}$ is the outcome feature of node $u$. After defining these
notations, their mutual information $I({\bm{v}},{\bm{u}})$ is the KL-
divergence between the joint distribution
$P_{{\bm{v}},{\bm{u}}}=P({\bm{v}}={\bm{e}}_{v},{\bm{u}}={\bm{e}}_{u})$ and the
product of marginal distributions $P_{\bm{v}}\otimes P_{\bm{u}}$:

(1)
$\begin{split}&I({\bm{v}},{\bm{u}})=D_{\text{KL}}(P_{{\bm{v}},{\bm{u}}}\parallel
P_{\bm{v}}\otimes P_{\bm{u}})\\\
\overset{(a)}{\geq}&\sup\limits_{T\in{\mathcal{T}}}\\{\mathbb{E}_{{\bm{e}}_{v},{\bm{e}}_{u}\thicksim
P_{{\bm{v}},{\bm{u}}}}[T({\bm{e}}_{v},{\bm{e}}_{u})]\\\
&-\mathbb{E}_{{\bm{e}}_{v}\thicksim P_{{\bm{v}},{\bm{u}}^{\prime}}\thicksim
P_{{\bm{u}}^{\prime}}}[e^{T({\bm{e}}_{v},{\bm{e}}_{u^{\prime}})-1}]\\},\end{split}$

where $(a)$ follows from $f$-divergence representation based on KL-divergence
(mohamed2018mutual, ); random variable ${\bm{u}}^{\prime}$ denotes the feature
associate with an arbitrary node $u^{\prime}\in{\bm{U}}\cup{\bm{I}}$;
$T\in{\mathcal{T}}$ is an arbitrary function that maps a pair of features to a
real value, which reflecting the correlation of two features. After replacing
the $f$-divergence in Eq. (1) with a GAN-like divergence, Eq. (1) is written
as follows,

(2)
$\begin{split}I_{\text{GAN}}({\bm{v}},{\bm{u}})\geq&\sup\limits_{T\in{\mathcal{T}}}\\{\mathbb{E}_{P_{{\bm{v}},{\bm{u}}}}[\text{log}\sigma(T({\bm{e}}_{v},{\bm{e}}_{u}))]\\\
&+\mathbb{E}_{P_{\bm{v}},P_{{\bm{u}}^{\prime}}}[\text{log}(1-\sigma(T({\bm{e}}_{v},{\bm{e}}_{u^{\prime}})))]\\},\end{split}$

where $\sigma(\cdot)$ is the sigmoid function.

Practical Simplification. In practice, we cannot go over the entire functional
space to evaluate the exact value of $I_{\text{GAN}}({\bm{v}},{\bm{u}})$.
Since the feature of nodes are learned collaborative filtering embeddings, we
parameterize $T({\bm{a}},{\bm{b}})$ by computing their inner product as
$T({\bm{a}},{\bm{b}})={\bm{a}}^{\top}\cdot{\bm{b}}$. Through the trained CF
embeddings, we obtain a lower bound of GAN-based mutual information as:

(3)
$\begin{split}I_{CF}({\bm{v}},{\bm{u}})=&\log\sigma({\bm{e}}_{v}^{\top}\cdot{\bm{e}}_{u})\\\
&+\frac{1}{N+M}\sum_{u^{\prime}\in{\bm{U}}\cup{\bm{I}}}\log(1-\sigma({\bm{e}}_{v}^{\top}\cdot{\bm{e}}_{u^{\prime}})),\end{split}$

where $N=|{\bm{U}}|$ and $M=|{\bm{I}}|$ denote the numbers of users and items,
respectively. In $I_{CF}({\bm{v}},{\bm{u}})$, the first term reflects the
correlation between node $v$ and the root node $u$, meaning to what extend
does the node $v$ describe the root node $u$. The second term evaluates the
difference between the sampling set and the whole user-item embeddings, which
estimates the particularity of the node $v$. Since $\log(1-\sigma(\cdot))$ is
a concave function, we can further simplify the second term of Eq. (3) with
Jensen’s inequality to obtain the following neighbor information evaluation
function,

(4)
$C(v,u)=\log\sigma({\bm{e}}_{v}^{\top}\cdot{\bm{e}}_{u})+\log(1-\sigma({\bm{e}}_{v}^{\top}\cdot\overline{{\bm{e}}})),$

where
$\overline{{\bm{e}}}=1/(N+M)\sum_{u^{\prime}\in{{\bm{U}}\cup{\bm{I}}}}{\bm{e}}_{u^{\prime}}$
denotes the average embedding of all the user-item embeddings. With this
neighbor information evaluation function Eq. (4), we propose the neighbor
infomax sampling and the flattened layer aggregation in the next subsection.

###  2.2. Flattened Layer Aggregation

Mathematically, let ${\mathcal{G}}({\bm{U}},{\bm{I}},{\bm{R}})$ denote the
user-item interaction graph. ${\bm{U}}=\\{u_{1},\cdots,u_{N}\\}$ denotes the
user set, and ${\bm{I}}=\\{i_{1},\cdots,i_{M}\\}$ denotes the item set, while
${\bm{R}}\in\mathbb{R}^{N\times M}$ describes the user-item interaction
matrix. After introducing the notations, the argument adjacent matrix for
users and items can be defined as follows,

(5) ${\bm{A}}=\left[\begin{matrix}\textbf{0}&{\bm{R}}\\\
{\bm{R}}^{\top}&\textbf{0}\end{matrix}\right]\in\mathbb{R}^{(N+M)\times(N+M)}.$

Therefore, we can define the multi-hop neighbors of a given node $u$ according
to the adjacent matrix ${\bm{A}}$. Specially, ${\mathcal{N}}_{u}^{0}=\\{u\\}$.
The first-order neighbor set is given by
${\mathcal{N}}_{u}^{1}=\\{v|{\bm{A}}_{uv}=1\\}$, while the second-order
neighbor set is denoted by ${\mathcal{N}}_{u}^{2}=\\{v|{\bm{A}}_{uv}^{2}\geq
1\\}$, so on and so forth.

As shown in Fig. 1, for a given root node $u$, we select the most informative
subset ${\mathcal{S}}_{u}^{k}$ from a given neighbor set
${\mathcal{N}}_{u}^{k}$ by greedily selecting the top-$S^{k}$ informative
neighbors, namely,

(6) ${\mathcal{S}}_{u}^{k}=\text{top-rank}\\{C(v,u)\ \text{for\
}v\in{\mathcal{N}}_{u}^{k},\ S^{k}\\},$

where top-rank is the function that returns the indices of the top-$S^{k}$
value and $S^{k}$ denotes the sampling size of the $k$-th layer. Based on the
definition of the neighbor infomax selection in Eq. (6), we can utilize flat
infomax aggregation to extract the information of different layers,

(7)
${\bm{h}}_{u}^{k}=\frac{1}{S^{k}}\sum_{v\in{\mathcal{S}}_{u}^{k}}{\bm{e}}_{v}.$

Noted that the aggregation step Eq. (6) and Eq. (7) is a parameter-free
operation such that it can be done in a pre-processing manner only once,
thereby saving vast memory and computational cost to ease the training on
large-scale graphs and application to real-world recommender systems.

###  2.3. Layer Ensemble Architecture

With Eq. (7), FlatGCN aggregates the information of a $K$-layer subgraph into
$K$ vectors. Thus for any give user-item pair $(u,i)$, we have
$\\{{\bm{h}}_{u}^{0},\cdots,{\bm{h}}_{u}^{K}\\}$ to represent the user
subgraph and $\\{{\bm{h}}_{i}^{0},\cdots,{\bm{h}}_{i}^{K}\\}$ to represent the
item subgraph.

The success of FM (guo2017deepfm, ; juan2016ffm, ) inspires that a hand-
crafted cross-interaction function may be better than brutally learning with
Multi-Layer Perception (MLP) function. As shown in Fig. 1, we first compute
the interactions between different layers combinations for every layer of
user’s and the item’s subgraphs. In such a way, we can easily take full
advantage of the aggregation embedding of each subgraph layer for both user
and item, and thus we model the complex interaction from user $u$ to item $v$
as,

(8)
$\hat{y}_{uv}=\text{MLP}[\parallel_{i=0}^{K}\parallel_{j=0}^{K}({\bm{h}}_{u}^{i})^{\top}\cdot{\bm{h}}_{v}^{j}],$

where $\parallel$ indicates the concatenation operation,
$({\bm{h}}_{u}^{i})^{\top}\cdot{\bm{h}}_{v}^{j}$ denotes the relevance score
between the $i$th-layer representation of the user subgraph and $j$th-layer of
the item subgraph. It is noteworthy that Eq. (8) actually summarize the
complicated relations between the user-item pair $(u,i)$ into a
$K^{2}$-dimension vectors, that greatly reduce the number of the parameters in
the followed MLP function.

Table 1. Overall Performance Comparison

|  | AmazonBook | Yelp2018 | WeChat
---|---|---|---|---
Embeddings | Methods | PRE | REC | NDCG | PRE | REC | NDCG | PRE | REC | NDCG
Meta2Vec | IntentGC | 0.0259 | 0.0517 | 0.0458 | 0.0274 | 0.0592 | 0.0490 | 0.0116 | 0.0908 | 0.0512
GCN | 0.0271 | 0.0543 | 0.0476 | 0.0299 | 0.0648 | 0.0527 | 0.0181 | 0.1425 | 0.0789
NGCF | 0.0384 | 0.0769 | 0.0678 | 0.0363 | 0.0769 | 0.0666 | 0.0189 | 0.1493 | 0.0837
LightGCN | 0.0448 | 0.0919 | 0.0833 | 0.0388 | 0.0842 | 0.0704 | 0.0191 | 0.1478 | 0.0940
FlatGCN | 0.0498 | 0.1014 | 0.0918 | 0.0430 | 0.0932 | 0.0807 | 0.0258 | 0.2019 | 0.1233
Improve | 11.16%* | 10.34%* | 10.20%* | 10.82%* | 10.69%* | 14.63%* | 35.08%* | 35.23%* | 31.17%*
LightGCN | IntentGC | 0.0307 | 0.0609 | 0.0540 | 0.0335 | 0.0723 | 0.0616 | 0.0230 | 0.1835 | 0.1057
GCN | 0.0317 | 0.0649 | 0.0576 | 0.0291 | 0.0625 | 0.0529 | 0.0233 | 0.1872 | 0.1079
NGCF | 0.0433 | 0.0880 | 0.0785 | 0.0401 | 0.0853 | 0.0737 | 0.0425 | 0.3533 | 0.2015
LightGCN | 0.0450 | 0.0919 | 0.0823 | 0.0410 | 0.0874 | 0.0752 | 0.0342 | 0.2831 | 0.1671
FlatGCN | 0.0485 | 0.0982 | 0.0884 | 0.0450 | 0.0954 | 0.0818 | 0.0709 | 0.5932 | 0.3543
Improve | 7.78%* | 6.86%* | 7.41%* | 9.76%* | 9.15%* | 8.78%* | 66.82%* | 67.90%* | 75.83%*

##  3\. Experiments

###  3.1. Experiment Setup

Datasets. We utilize two benchmark datasets and a commercial dataset to
evaluate the recommendation performance: AmazonBook and Yelp2018 are two
benchmark datasets used in (wang2019ngcf, ; he2020lightgcn, ). The WeChat
dataset contains users’ clicks on different articles recorded by the WeChat
platform. We use Meta2Vec (dong2017metapath2vec, ) and the state-of-the-art
(SOTA) LightGCN (he2020lightgcn, ) to produce the pre-trained embeddings of
users and items in datasets to feed into the GCN model as the raw features.

Evaluation Protocols. We randomly split the entire user-item recommendation
records of each dataset into an embedding training set, GCN model training
(validation) set, and a test set to evaluate the performance, where each of
them contains 65%, 15%, and 20% of the full records, respectively. Besides, we
apply the wildly used full-rank evaluation protocols (he2020lightgcn, ) to
evaluate the recommendation performance.

Comparison Methods. We compare the following SOTA baselines: I. GCN
(hamilton2017graphsage, ) applies 2-layer GCN to process input embeddings
following PinSAGE (ying2018graph, ), followed by an inner product function.
II. IntentGC (zhao2019intentgc, ) introduces a fast architecture named
IntentNet to avoid unnecessary feature interactions for efficient training.
III. NGCF* (wang2019ngcf, ) is an embedding learning method. We fix its inputs
as the given user-item embeddings and only train its propagation layers. We
denote it as NGCF*. IV. LightGCN* (he2020lightgcn, ) is the SOTA
recommendation model. Similar to NGCF*, we fix its inputs as the given user-
item embeddings and only train its propagation layers.

Parameter Settings. The optimal parameter settings for all comparison methods
are achieved by either empirical study or suggested settings by the original
papers. We consider the 3-layer subgraphs and set the sampling size $S_{k}=25$
for $k=1,2$. Especially, we adopt Adam (kingma2014adam, ) as the optimizer
with learning rate as $0.001$; the $L_{2}$ regularization weight as $10^{-5}$.

Table 2. Comparison of sampling policies.

Embeddings | Sampling | WeChat(PRE) | WeChat(REC) | WeChat(NDCG)
---|---|---|---|---
Meta2Vec | Random | 0.0237 | 0.1858 | 0.1104
Intuitive | 0.0241 | 0.1892 | 0.1115
Infomax | 0.0258 | 0.2019 | 0.1233
Improve | 7.05% | 6.71% | 10.58%
LightGCN | Random | 0.0659 | 0.5547 | 0.3268
Intuitive | 0.0685 | 0.5750 | 0.3388
Infomax | 0.0709 | 0.5932 | 0.3543
Improve | 3.50%* | 3.17%* | 4.57%*

###  3.2. Result Analysis

Performance Comparison. Table 1 reports the performance on the three datasets
w.r.t. precision (PRE@20), recall (REC@20), and ndcg (NCDG@20) metrics. The
improvement gives the relative enhancement of FlatGCN over the best baseline
(underlined). The superscript * denotes statistical significance against the
best baseline with $p<0.05$.

Overall, our proposed FlatGCN statically significantly outperforms the best
base baselines among all three datasets w.r.t. all evaluation metrics and both
Meta2Vec and LightGCN embeddings. These results verify the consistent
superiority of FlatGCN on characterizing the relevance between users and items
than SOTA GCN-based recommendation methods. Particularly for the commercial
WeChat dataset, FlatGCN leads to an enhanced performance by more than 30% on
Meta2Vec embeddings. With LightGCN embeddings, FlatGCN outperforms the best
baselines by over 60%.

Figure 2. Comparison of the training time. All methods are running on the same
GPU device.

Learning Efficiency. One main advantage of FlatGCN is the low training
complexity. In Fig. 2, we present the average training time of FlatGCN and the
other baselines. We also present the preprocessing time of FlatGCN. The
results in Fig. 2 shows that FlatGCN achieves superior performance with one or
two orders of magnitude speedup in training time. Notably, FlatGCN is $10$
times faster than the fastest baseline: NGCF. Moreover, the preprocessing time
of FlatGCN can be neglected, comparing with the training time of the
baselines.

Sampling Comparison. In Table 2, we compare three different neighbor sampling
methods on the WeChat dataset: 1) Random: Random walk-based sampling
(ying2018graph, ), which simulates random walks starting from each node and
computes the L1-normalized visit count of neighbors visited by the random
walk. 2) Intuitive: First-order proximity-based sampling (fan2019meirec, ;
Wang2019graph, ; wu2019dual, ), examining the neighborhood similarity based on
the edge weights (e.g., number of clicks). 3) Infomax: Our proposed neighbor
infomax selection. Overall, the random sampling method generates the worst
performance. Intuitive sampling outperforms random sampling since the edge
weights can represent the importance of an edge. Infomax sampling method
achieves the best performance, due to its ability to select the most
informative subset from the neighbor set.

##  4\. Conclusion

In this paper, we introduced the FlatGCN model which is able to achieve
superior performance along with a few orders of magnitude speedup in training
compared with existing models. Specifically, we proposed a flattened GCN
architecture which is able to perform neighborhood propagation only once over
a single GCN layer instead of recursively. Moreover, we proposed an
informative sampling method named neighbor-infomax to select the top
informative neighbors for neighborhood aggregation. Finally, we proposed a
layer ensemble technique to improve the model expressiveness. Extensive
experiments verified the superiority of the proposed FlatGCN on both learning
performance and training speed.

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