Revisiting Over-smoothing in BERT from the Perspective of Graph

1 INTRODUCTION . Over the past few years , Transformer ( Vaswani et al. , 2017 ) has been widely used in various natural language processing ( NLP ) tasks , including text classification ( Wang et al. , 2018a ) , text translation ( Ott et al. , 2018 ) , question answering ( Rajpurkar et al. , 2016 ; 2018 ) and text generation ( Brown et al. , 2020 ) . The recent application of Transformer in computer vision ( CV ) field also demonstrate the potential capacity of Transformer architecture . For instance , Transformer variants have been successfully used for image classification ( Dosovitskiy et al. , 2021 ) , object detection ( Carion et al. , 2020 ) and semantic segmentation ( Strudel et al. , 2021 ) . Three fundamental descendants from Transformer include BERT ( Devlin et al. , 2019 ) , RoBERTa ( Liu et al. , 2019 ) and ALBERT ( Lan et al. , 2020 ) , which achieve state-of-the-art performance on a wide range of NLP tasks . Recently , Dong et al . ( 2021 ) observes the “ token uniformity ” problem , which reduces the capacity of Transformer-based architectures by making all token representations identical . They claim that pure self-attention ( SAN ) modules cause token uniformity , but they do not discuss whether the token uniformity problem still exists in Transformer blocks . On the other hand , Gong et al . ( 2021 ) observe the “ over-smoothing ” problem for ViT ( Dosovitskiy et al. , 2021 ) , in that different input patches are mapped to a similar latent representation . To prevent loss of information , they introduce additional loss functions to encourage diversity and successfully improve model performance by suppressing over-smoothing . Moreover , “ overthinking ” phenomenon , indicating that shallow representations are better than deep representations , also be observed in ( Zhou et al. , 2020 ; Kaya et al. , 2019 ) . As discussed in Section 3 , this phenomenon has some inherent connection with over-smoothing . In this paper , we use “ over-smoothing ” to unify the above issues , and refer this as the phenomenon that the model performance is deteriorated because different inputs are mapped to a similar representation . As the over-smoothing problem is first studied in the graph neural network ( GNN ) literature ( Li et al. , 2018 ; Xu et al. , 2018 ; Zhao & Akoglu , 2020 ) , in this paper , we attempt to explore the cause of such ∗Equal contribution . problem by building a relationship between Transformer blocks and graphs . Specifically , we consider the self-attention matrix as the normalized adjacency matrix of a weighted graph , whose nodes are the tokens in a sentence . Furthermore , we consider the inherent connection between BERT and graph convolutional networks ( Kipf & Welling , 2017 ) . Inspired by the over-smoothing problem in GNN , we study over-smoothing in BERT from a theoretical view via matrix projection . As opposed to Dong et al . ( 2021 ) , where the authors claim that layer normalization is irrelevant to over-smoothing , we find that layer normalization ( Ba et al. , 2016 ) plays an important role in over-smoothing . Specifically , we theoretically prove that , if the standard deviation in layer normalization is sufficiently large , the outputs of the Transformer stacks will converge to a low-rank subspace , resulting in over-smoothing . Empirically , we verify that the conditions hold for a certain number of samples for a pre-trained and fine-tuned BERT model ( Devlin et al. , 2019 ) , which is consistent with our above observations . To alleviate the over-smoothing problem , we propose a hierarchical fusion strategy that adaptively fuses representations from different layers . Three fusion approaches are used : ( i ) Concat Fusion , ( ii ) Max Fusion , and ( iii ) Gate Fusion . The proposed method reduces the similarity between tokens and outperforms BERT baseline on the GLUE ( Wang et al. , 2018a ) , SWAG ( Zellers et al. , 2018 ) and SQuAD ( Rajpurkar et al. , 2016 ; 2018 ) data sets . In summary , the contributions of this paper are as follows : ( i ) We develop the relationship between self-attention and graph for a better understanding of over-smoothing in BERT . ( ii ) We provide theoretical analysis on over-smoothing in the BERT model , and empirically verify the theoretical results . ( iii ) We propose hierarchical fusion strategies that adaptively combine different layers to alleviate over-smoothing . Extensive experimental results verify our methods ’ effectiveness . 2 RELATED WORK . 2.1 TRANSFORMER BLOCK AND SELF-ATTENTION . Transformer block is a basic component in Transformer model ( Vaswani et al. , 2017 ) . Each Transformer block consists of a self-attention layer and a feed-forward layer . Let X ∈ Rn×d be the input to a Transformer block , where n is the number of input tokens and d is the embedding size . The self-attention layer output can be written as : Attn ( X ) = X + h∑ k=1 σ ( XWQk ( XW K k ) > ) XW Vk W O > k = X + h∑ k=1 ÂkXW V O k , ( 1 ) where h is the number of heads , σ is the softmax function , and WQk , W K k , W V k , W O k ∈ Rd×dh ( where dh = d/h is the dimension of a single-head output ) are weight matrices for the query , key , value , and output , respectively of the kth head . In particular , the self-attention matrix Â = σ ( XWQ ( XWK ) > ) = σ ( QK > ) ( 2 ) in ( 1 ) plays a key role in the self-attention layer ( Park et al. , 2019 ; Gong et al. , 2019 ; Kovaleva et al. , 2019 ) . As in ( Yun et al. , 2020 ; Shi et al. , 2021 ; Dong et al. , 2021 ) , we drop the scale product 1/ √ dh to simplify analysis . The feed-forward layer usually has two fully-connected ( FC ) layers with residual connection : FF ( X ) = Attn ( X ) +ReLU ( Attn ( X ) W1 + b1 ) W2 + b2 , where W1 ∈ Rd×dff , W2 ∈ Rdff×d ( dff is the size of the intermediate layer ) are the weight matrices , and b1 , b2 are the biases . Two layer normalization ( Ba et al. , 2016 ) operations are performed after the self-attention layer and fully-connected layer , respectively . 2.2 OVER-SMOOTHING . In graph neural networks , over-smoothing refers to the problem that the performance deteriorates as representations of all the nodes become similar ( Li et al. , 2018 ; Xu et al. , 2018 ; Huang et al. , 2020 ) . Its main cause is the stacked aggregation layer using the same adjacency matrix . Recently , several approaches have been proposed to alleviate the over-smoothing problem . Xu et al . ( 2018 ) propose a jumping knowledge network for better structure-aware representation , which flexibly leverages different neighborhood ranges . ResGCN ( Li et al. , 2019 ) adapts the residual connection and dilated convolution in the graph convolutional network ( GCN ) , and successfully scales the GCN to 56 layers . Zhao & Akoglu ( 2020 ) propose PairNorm , a novel normalization layer , that prevents node embeddings from becoming too similar . DropEdge ( Rong et al. , 2020 ; Huang et al. , 2020 ) randomly removes edges from the input graph at each training epoch , and reduces the effect of over-smoothing . Unlike graph neural networks , over-smoothing in Transformer-based architectures has not been discussed in detail . Dong et al . ( 2021 ) introduce the “ token-uniformity ” problem for self-attention , and show that skip connections and multi-layer perceptron can mitigate this problem . However , Gong et al . ( 2021 ) still observe over-smoothing on the Vision Transformers ( Dosovitskiy et al. , 2021 ) . 3 DOES OVER-SMOOTHING EXIST IN BERT ? . In this section , we first explore the existence of over-smoothing in BERT , by measuring the similarity between tokens in each Transformer layer . Specifically , we use the token-wise cosine similarity ( Gong et al. , 2021 ) as our similarity measure : CosSim = 1 n ( n− 1 ) ∑ i 6=j h > i hj ‖hi‖2‖hj‖2 , where n is the number of tokens , hi and hj are two representations of different tokens , and ‖ · ‖2 is the Euclidean norm . Following Dong et al . ( 2021 ) , we use WikiBio ( Lebret et al. , 2016 ) as input to the following Transformer-based models fine-tuned on the SQuAD data set ( Rajpurkar et al. , 2018 ) : ( i ) BERT ( Devlin et al. , 2019 ) , ( ii ) RoBERTa ( Liu et al. , 2019 ) and ( iii ) ALBERT ( Lan et al. , 2020 ) .1 For comparison , all three models are stacked with 12 blocks . We calculate each CosSim for each data sample and show the average and standard derivation of CosSim values over all WikiBio data . In the figures , layer 0 represents original input token representation , and layer 1-12 represents the corresponding transformer layers . As shown in Figure 1 ( a ) , the original token representations are different from each other , while token similarities are high in the last layer . For instance , the average token-wise cosine similarity of the last layer of ALBERT and RoBERTa are both larger than 90 % . To illustrate the relationship between “ over-thinking ” and “ over-smoothing ” , we compare the tokenwise cosine similarity at each layer with the corresponding error rate . As for the corresponding error rate of layer i , we use the representations from layer i as the final output and fine-tune the classifier . Following Zhou et al . ( 2020 ) , we experiment with ALBERT ( Lan et al. , 2020 ) fine-tuned on the MRPC data set ( Dolan & Brockett , 2005 ) and use their error rate results for convenience . As shown in Figure 1 ( b ) , layer 10 has the lowest cosine similarity and error rate . At layers 11 and 12 , the tokens have larger cosine similarities , making them harder to distinguish and resulting in the performance drop . Thus , “ over-thinking ” can be explained by “ over-smoothing ” . A direct consequence of over-smoothing is that the performance can not be improved when the model gets deeper , since the individual tokens are no longer distinguishable . To illustrate this , we increase the number of layers in BERT to 24 while keeping the other settings . As shown in Figure 1 ( c ) , the 1Our implementation is based on the HuggingFace ’ s Transformers library ( Wolf et al. , 2020 ) . performance of vanilla BERT can not improve as the model gets deeper . In contrast , the proposed hierarchical fusion ( as will be discussed in Section 6 ) consistently outperforms the baseline , and has better and better performance as the model gets deeper . Based on these observations , we conclude that the over-smoothing problem still exists in BERT . 4 RELATIONSHIP BETWEEN SELF-ATTENTION AND GRAPH . Since over-smoothing is first discussed in the graph neural network literature ( Li et al. , 2018 ; Zhao & Akoglu , 2020 ) , we attempt to understand its cause from a graph perspective in this section .

Summary: This paper tries to understand the over-smoothing phenomenon of Transformer-based models such as BERT. The analysis is from the perspective of viewing BERT and Transformer as graph neural networks. By analogy and analysis of graph neural networks, authors show some theoretical analysis and find that layer normalization plays a key role in the over-smoothing issue. Specifically, if the standard deviation of layer normalization is sufficiently large, the output of Transformer stacks will converge to a specific low-rank subspace and results in over-smoothing. Then to alleviate the over-smoothing problem, authors consider hierarchical fusion strategies, which combine the representations from different layers adaptively to make the output more diverse.