# FastRE: Towards Fast Relation Extraction with Convolutional Encoder and
Improved Cascade Binary Tagging Framework

Guozheng Li1 Xu Chen4 Peng Wang${{}^{1}}{{}^{,}}{{}^{2}}{{}^{,}}{{}^{3}}$
Corresponding author Jiafeng Xie1 &Qiqing Luo${{}^{1}}{{}^{,}}{{}^{3}}$
1School of Computer Science and Engineering, Southeast University
2School of Cyber Science and Engineering, Southeast University
3School of Artificial Intelligence, Southeast University
4Tencent Inc.
{gzli, pwang, xjf<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Recent work for extracting relations from texts has achieved excellent
performance. However, most existing methods pay less attention to the
efficiency, making it still challenging to quickly extract relations from
massive or streaming text data in realistic scenarios. The main efficiency
bottleneck is that these methods use a Transformer-based pre-trained language
model for encoding, which heavily affects the training speed and inference
speed. To address this issue, we propose a fast relation extraction model
(FastRE) based on convolutional encoder and improved cascade binary tagging
framework. Compared to previous work, FastRE employs several innovations to
improve efficiency while also keeping promising performance. Concretely,
FastRE adopts a novel convolutional encoder architecture combined with dilated
convolution, gated unit and residual connection, which significantly reduces
the computation cost of training and inference, while maintaining the
satisfactory performance. Moreover, to improve the cascade binary tagging
framework, FastRE first introduces a type-relation mapping mechanism to
accelerate tagging efficiency and alleviate relation redundancy, and then
utilizes a position-dependent adaptive thresholding strategy to obtain higher
tagging accuracy and better model generalization. Experimental results
demonstrate that FastRE is well balanced between efficiency and performance,
and achieves 3-10$\times$ training speed, 7-15$\times$ inference speed faster,
and 1/100 parameters compared to the state-of-the-art models, while the
performance is still competitive. Our code is available at
https://github.com/seukgcode/FastRE.

##  1 Introduction

Relation extraction (RE) aims to identify the relationships between entities
in texts, and plays an important role in natural language processing
applications, knowledge graph construction, and question answering. Recent RE
studies have made great progress in pursuing excellent performance Wei et al.
(2020); Ren et al. (2021). However, in realistic scenarios, RE models are
often expected to keep high performance and fast speed simultaneously. For
example, financial investors require extracting relational triples effectively
from massive and streaming real-time news, financial and political data to
build knowledge graphs Dong et al. (2014) to help make decisions.
Unfortunately, most existing methods lack considering time consumed in model
training and inference and cannot efficiently extract relations from texts.
Early RE research applied approximate frequency counting and dimension
reduction to speed up similarity computation in unsupervised RE Takase et al.
(2015), and adopted neural metric learning methods to accelerate RE Tran and
Kavuluru (2019). Various CNN based approaches Zeng et al. (2014); dos Santos
et al. (2015) had also shown effective in tackling this problem. However,
these efforts fail to address the challenges of RE on both performance and
efficiency.

Recent advances on RE performance Wei et al. (2020); Wang et al. (2020); Sui
et al. (2021); Zheng et al. (2021); Ren et al. (2021); Huguet Cabot and
Navigli (2021) mainly due to adopting the encoders with the Transformer
Vaswani et al. (2017) based pre-trained language models (PLMs) such as BERT
Devlin et al. (2019) and BART Lewis et al. (2020), which have powerful
abilities to capture long-distance dependencies and context semantic features.
However, the token-pair based attention operation in Transformer requires much
time consumption and memory consumption within GPUs. Moreover, the memory
consumption of PLMs restricts the batch size during model training and
inference, which means that models are restricted to set a relatively small
batch size within limited computing resources. Although it is not a serious
problem in training, it limits the parallel processing ability in the
inference.

Instead of using the Transformer, we design a novel convolutional structure to
tackle the computational efficiency issue in encoder. It significantly
accelerates the training and inference speed with dilated convolution Yu and
Koltun (2016), gated unit Dauphin et al. (2017) and residual connection He et
al. (2016). First, the dilated convolution increases the receptive region of
network outputs exponentially with respect to the network depth, which results
in obtaining drastically shortened computation paths and capturing arbitrarily
long-distance dependencies. In other words, using dilated convolution can
achieve the high efficiency of vanilla convolution with fewer layers. Second,
gated unit is used to control what information should be propagated through
the hierarchy of layers. Then residual connection is used for avoiding
gradient vanishing to enable deep convolutional networks. Our convolutional
encoder not only greatly reduces the model training and inference consumed
time, but also ensures the competitive performance in RE.

On the other side, recent work also shows the effectiveness of the cascade
binary tagging framework Wei et al. (2020) for solving the overlapping RE.
However, this framework suffers from two shortcomings: relation redundancy and
poor generalization Zheng et al. (2021). Relation redundancy, namely,
extracting tail entity extraction to all relations, would cause plenty of
meaningless computations. Besides, due to that the cascade binary tagging
framework reduces the multi-label problem to a binary classification problem,
it needs heuristic threshold tuning and introduces boundary decision errors.

To solve above issues, we introduce an entity type to predefined relation
(type-relation) mapping mechanism and a position-dependent adaptive
thresholding strategy to improve the cascade binary tagging framework.
Specifically, the mapping between head entity type and predefined relation is
maintained, when the head entity type is determined, its potential
corresponding relations are also determined. Under this mechanism, it avoids
traversing for all the relations when predicting overlapping relations. In
addition, incorporating entity type information into RE can improve
performance Zhong and Chen (2021). The position-dependent adaptive
thresholding replaces the global threshold with learnable thresholds for
different positions in a sentence when performing binary tagging. The
thresholds are learned with a rank-based loss function Zhou et al. (2021),
that pushes positive classes scores above the thresholds and pulls negative
classes scores below in training. And the tagger sets a position to 1 that has
the higher score than its position-dependent threshold or set to 0 that has
the lower score. This simple strategy avoids threshold tuning, and makes the
threshold adjustable to different tagging positions, which leads to better
generalization.

In this paper, we propose a simple and fast relation extraction model, FastRE,
to significantly reduce training and inference time but retains competitive
performance. To the best of our knowledge, FastRE is the first effort to
address the balance between efficiency and performance in RE. In summary, the
main contributions of this paper are three-fold:

  * •

We propose a novel convolutional encoder combined with dilated convolution,
gated unit and residual connection for RE, which significantly reduces the
computation cost but maintaining the satisfactory performance.

  * •

We improve the cascade binary tagging framework with type-relation mapping
mechanism and position-dependent adaptive thresholding, which solves the
relation redundancy and poor generalization issues.

  * •

Experimental results on public datasets show that FastRE achieves 3-10$\times$
training speed, 7-15$\times$ inference speed faster, and 1/100 parameters
compared to the state-of-the-art models, while the performance still
competitive, namely, it is even slightly improved on NYT10 and NYT11, and only
reduced by less than 4% on NYT24.

##  2 Methodology

###  2.1 Problem Formulation and Model Overview

Given the predefined relation set $\mathcal{R}=\\{r_{1},r_{2},...,r_{N}\\}$,
for any sentence $\mathcal{S}$ containing the entity set
$\mathcal{E}=\\{e_{1},e_{2},...,e_{M}\\}$ with the entity type set
$\mathcal{T}=\\{t_{1},t_{2},...,t_{K}\\}$, the relation extraction task aims
to extract the relational triples in $\mathcal{Z}=\\{(e_{i},r_{k},e_{j})\\}$,
where $N$ denotes the number of relations, $M$ represents the number of
entities, $K$ is the number of entity types, and
$e_{i},e_{j}\in\mathcal{E},r_{k}\in\mathcal{R}$.

The principle of FastRE is shown in Figure 1. FastRE consists of a
convolutional encoder and a improved cascade binary tagging framework. The
convolutional encoder contains $L$ stacked $\rm{Block}(\cdot)$ which consists
of two dilated convolutions: a gated unit and a residual connection. Compared
with the original one, the improved cascade binary tagging framework not only
adds type-relation mapping mechanism to determine potential relations based on
entity types, but also adopts position-dependent adaptive thresholding
strategy for different positions. First, the convolutional encoder efficiently
converts the input sentence $\mathbf{X}$ composed of word embedding
$\mathbf{X_{g}}$ and position embedding $\mathbf{X_{p}}$ into sentence
representation $\mathbf{H}$. Subsequently, two separate auxiliary features
$\mathbf{H}_{h}$ and $\mathbf{H}_{t}$ are generated based on $\mathbf{H}$ via
two different multi-head self-attention layers. Then FastRE utilizes a feed
forward network (FFN) to obtain all the head entities and their types.
Concretely, sentence representation $\mathbf{H}$ and head entity auxiliary
features $\mathbf{H}_{h}$ are concatenated and feed to the FNN. Then, for each
entity type, FastRE calculates the score of every token as the start and end
position of a head entity. With adaptive thresholding, the position that has
higher score than its position-dependent threshold is set to 1. Through the
mapping mechanism, FastRE determines the potential relations corresponding to
the current head entity type. Finally, similar to head entity tagging, with
the sentence representation $\mathbf{H}$, tail entity auxiliary features
$\mathbf{H}_{t}$ and head entity features $\mathbf{F}_{h}$, FastRE utilizes
another FFN to obtain all the tail entities and forms all the relational
triples in $\mathcal{Z}$.

Figure 1: The overall structure of FastRE.

###  2.2 Convolutional Encoder

Theoretically, we can achieve the aim of modeling arbitrarily long-distance
dependencies similar to Transformer Vaswani et al. (2017) by stacking enough
vanilla convolutional layers. But too many parameters will bring too much
computation. Therefore, we choose the dilated convolution Yu and Koltun (2016)
instead because increasing the receptive region in convolution can achieve the
high efficiency of vanilla convolution with fewer layers. However, the pre-
determined receptive region prevents subsequent layers from examining previous
information in detail. We alleviate this issue by using the gated unit Dauphin
et al. (2017) in convolutions to select important features of low-level tokens
and phrases. To enable deep networks, we adopt the residual connection He et
al. (2016) to avoid gradient vanishing.

Let $\mathbf{X}=[\mathbf{x}_{1},\mathbf{x}_{2},...,\mathbf{x}_{n}]$ represent
the input sentence, where $\mathbf{x}_{i}\in\mathbb{R}^{d}$ is the $i$-th
token embedding with dimension $d$, composed of GloVe embedding Pennington et
al. (2014) $\mathbf{X_{g}}$ and trainable position embedding $\mathbf{X_{p}}$.
The encoder contains $L$ stacked $\rm{Block}(\cdot)$. The sentence
representation $\mathbf{H}=[\mathbf{w}_{1},\mathbf{w}_{2},...,\mathbf{w}_{n}]$
is obtained by:

$\mathbf{H}=\rm{Block}(\cdot\cdot\cdot(\rm{Block}(\mathbf{X})))$ (1)

where $\mathbf{H}\in\mathbb{R}^{n\times d}$ and
$\mathbf{w}_{i}\in\mathbb{R}^{d}$ is the encoded contextual representation of
$i$-th token. The $i$-th $\rm{Block}(\cdot)$ in the encoder contains two
dilated convolutions with dilation rate $d_{i}$, a gated unit, and a residual
connection. It contains $d$ convolution kernels with size $k_{s}$. We denote
dilated convolution as $\rm{DilatedConv}(\cdot)$ to map the input
$\mathbf{X}\in\mathbb{R}^{n\times d}$ to the output
$\mathbf{Y}_{a,b}\in\mathbb{R}^{n\times d}$:

$\displaystyle\mathbf{Y}_{a}$ $\displaystyle=$
$\displaystyle{\rm{DilatedConv}}_{a}(\mathbf{X})$ (2)
$\displaystyle\mathbf{Y}_{b}$ $\displaystyle=$
$\displaystyle{\rm{DilatedConv}}_{b}(\mathbf{X})$ (3)

For all the $\rm{Block}(\cdot)$, we ensure the output dimension matches the
input dimension by padding. Similar to Gehring et al. (2017), we implement a
gated unit over $\mathbf{Y}_{a},\mathbf{Y}_{b}$ and add a residual connection
from the $\rm{Block}(\cdot)$ input to its output:

$\mathbf{Y}_{i}=\mathbf{Y}_{a}\otimes{\rm{sigmoid}}(\mathbf{Y}_{b})+\mathbf{X}$
(4)

where $\otimes$ denotes the element-wise multiplication, $\mathbf{Y}_{i}$ is
the output of $i$-th block and the input of $(i+1)$-th block. Obviously, the
sentence representation $\mathbf{H}$ is equal to $\mathbf{Y}_{L}$, which is
the output of the last $\rm{Block}(\cdot)$.

###  2.3 Improved Cascade Binary Tagger

FastRE first tags all the head entities, then tags their corresponding
relations and tail entities, which is a typical cascade binary tagging
framework Wei et al. (2020). Furthermore, FastRE introduces the type-relation
mapping mechanism and position-dependent adaptive thresholding to overcome the
drawbacks (i.e., relation redundancy and poor generalization Zheng et al.
(2021)) of cascade binary tagging framework.

Because the convolutional encoder shares the most parameters, it treats every
token in the sentence equally and pays less attention to the most important
parts of the sentence. Therefore, FastRE generates two separate auxiliary
features via two multi-head self-attention Vaswani et al. (2017) layers for
head entity tagging and tail entity tagging, respectively. Take the head
entity auxiliary features $\mathbf{H}_{h}$ as the example:

$\displaystyle\mathbf{H}_{h}$ $\displaystyle=$
$\displaystyle{\rm{softmax}}(\frac{\mathbf{Q}\mathbf{K^{\top}}}{\sqrt{d_{k}}})\mathbf{V}$
(5) $\displaystyle\mathbf{Q}$ $\displaystyle=$
$\displaystyle\mathbf{W}_{q}\cdot\mathbf{H}+\mathbf{b}_{q}$ (6)
$\displaystyle\mathbf{K}$ $\displaystyle=$
$\displaystyle\mathbf{W}_{k}\cdot\mathbf{H}+\mathbf{b}_{k}$ (7)
$\displaystyle\mathbf{V}$ $\displaystyle=$
$\displaystyle\mathbf{W}_{v}\cdot\mathbf{H}+\mathbf{b}_{v}$ (8)

where $d_{k}=d$ is the dimension of the attention key. The
$\mathbf{W}_{q},\mathbf{W}_{k},\mathbf{W}_{v}\in\mathbb{R}^{d\times
d},\mathbf{b}_{q},\mathbf{b}_{k},\mathbf{b}_{v}\in\mathbb{R}^{d}$ are the
weights and biases for querys, keys and values to obtain representations
$\mathbf{Q},\mathbf{K},\mathbf{V}\in\mathbb{R}^{n\times d}$, respectively.

For head entity tagging, we concatenate the sentence representation
$\mathbf{H}$ and head entity auxiliary features $\mathbf{H_{h}}$ as
$[[\mathbf{w}_{1},\mathbf{w}_{1}^{h}],...,[\mathbf{w}_{n},\mathbf{w}_{n}^{h}]]$.
Then we calculate the score of the $i$-th token as the start and end position
of a head entity with type $t_{j}\in\mathcal{T}$, respectively:

$\displaystyle o_{ij}^{h_{s}}$ $\displaystyle=$
$\displaystyle\mathbf{W}_{ij}^{h_{s}}\cdot[\mathbf{w}_{i},\mathbf{w}_{i}^{h}]+\mathbf{b}_{ij}^{h_{s}}$
(9) $\displaystyle o_{ij}^{h_{e}}$ $\displaystyle=$
$\displaystyle\mathbf{W}_{ij}^{h_{e}}\cdot[\mathbf{w}_{i},\mathbf{w}_{i}^{h}]+\mathbf{b}_{ij}^{h_{e}}$
(10)

where $o_{ij}^{h_{s}}$ represents the score that the $i$-th token is the start
of the head entity with type $t_{j}$.

Intuitively, different positions should have different thresholds, e.g., the
threshold at entity boundary could be different from the boundary of other
words. And tuning a global threshold leads to suboptimal results because of
the dependent on validation set data distribution, which leads to
unsatisfactory test results and poor generalization. To improve the tagging
accuracy and model generalization, we use the adaptive thresholding strategy
to learn thresholds automatically. The tagger sets a position to 1 only when
it has a higher score than its position-dependent threshold. For all start
positions $p_{ij}^{s}$ of head entities with their types $t_{j}\in\mathcal{T}$
at position $i\in[1,n]$, we denote the positive classes (i.e. the positions
expected to be tag to 1) as $\mathcal{P}_{i}$, negative classes as
$\mathcal{N}_{i}$. And we set a ${AT}$ class to store all the start positions
with respect to ${AT}$ type, where the position $i$ is denoted as
$p_{i;{AT}}$. During training, the start tagging loss is as follows:

$\begin{split}\mathcal{L}_{h}^{s}=&-\sum_{i=1}^{n}\sum_{{p_{ij}^{s}}\in\mathcal{P}_{i}}\log(\frac{\exp{(o_{ij})}}{\sum_{p_{ik}^{s}\in\mathcal{P}_{i}\cup\\{p_{i;{AT}}\\}}\exp{(o_{ik})}})\\\
&-\sum_{i=1}^{n}\,\,\log(\frac{\exp{(o_{i;{AT}})}}{\sum_{p_{ik}^{s}\in\mathcal{N}_{i}\cup\\{p_{i;{AT}}\\}}\exp{(o_{ik})}})\end{split}$
(11)

where the $o_{i;{AT}}$ represents the threshold at position $i$. Note that the
start and end tagging requires different thresholds and follow the same loss
function. The head entity tagging loss is the sum of start tagging loss
$\mathcal{L}_{h}^{s}$ and end tagging loss $\mathcal{L}_{h}^{e}$:

$\mathcal{L}_{h}=\mathcal{L}_{h}^{s}+\mathcal{L}_{h}^{e}$ (12)

Finally, we need to identify the relations and tail entities. We set an entity
type embedding layer $\mathbf{V}_{t}\in\mathbb{R}^{K\times d_{t}}$ and a
relative position embedding layer $\mathbf{V}_{p}\in\mathbb{R}^{n\times d}$.
First, we obtain the head entity start and end token features
$\mathbf{w}_{a}$, $\mathbf{w}_{b}$ from $\mathbf{H}$, derive the head entity
type feature $\mathbf{w}^{t}$ from $\mathbf{V}_{t}$ and the relative position
features $\mathbf{w}_{a}^{p}$, $\mathbf{w}_{b}^{p}$ from $\mathbf{V}_{p}$.
Then the $(\mathbf{w}_{a}+\mathbf{w}_{a}^{p})$,
$(\mathbf{w}_{b}+\mathbf{w}_{b}^{p})$ and $\mathbf{w}^{t}$ are concatenated to
form the head entity features $\mathbf{w}^{h}$. Through the type-relation
mapping mechanism, we determine the potential relation set
$\mathcal{R}^{{}^{\prime}}$ based on current entity type, aiming to help
identify the tail entity more accurately by incorporating entity type
information. We concatenate the sentence representation $\mathbf{H}$, tail
entity auxiliary features $\mathbf{H}_{t}$ and head entity features
$\mathbf{w}^{h}$ as
$[[\mathbf{w}_{1},\mathbf{w}_{1}^{t},\mathbf{w}^{h}],...,[\mathbf{w}_{n},\mathbf{w}_{n}^{t},\mathbf{w}^{h}]]$.
Consequently, we calculate the score of the $i$-th token as the start and end
position of a tail entity with potential relation
$r_{j}\in\mathcal{R}^{{}^{\prime}}\subset\mathcal{R}$, respectively:

$\displaystyle o_{ij}^{t_{s}}$ $\displaystyle=$
$\displaystyle\mathbf{W}_{ij}^{t_{s}}\cdot[\mathbf{w}_{i},\mathbf{w}_{i}^{t},\mathbf{w}^{h}]+\mathbf{b}_{ij}^{t_{s}}$
(13) $\displaystyle o_{ij}^{t_{e}}$ $\displaystyle=$
$\displaystyle\mathbf{W}_{ij}^{t_{e}}\cdot[\mathbf{w}_{i},\mathbf{w}_{i}^{t},\mathbf{w}^{h}]+\mathbf{b}_{ij}^{t_{e}}$
(14)

where $o_{ij}^{t_{s}}$ represents the score that the $i$-th token is the start
of the tail entity with relation $r_{j}$. Similar to head entity tagging loss,
we directly calculate the tail entity tagging loss $\mathcal{L}_{t}$ according
to Equation (12).

Let $\mathcal{D}$ denote all the sentences, and $\mathcal{Z}_{i}$ denote all
the relational triples in sentence $\mathcal{S}_{i}$. The loss function
$\mathcal{L}$ is composed of two parts as follow:

$\mathcal{L}=\frac{1}{|\mathcal{D}|}\,(\sum_{\mathcal{S}_{i}\in\mathcal{D}}\sum_{h_{j}\in\mathcal{Z}_{i}}\mathcal{L}_{h_{j}}+\sum_{\mathcal{S}_{i}\in\mathcal{D}}\sum_{t_{j}\in\mathcal{Z}_{i}}\mathcal{L}_{t_{j}|h})$
(15)

##  3 Experiments

###  3.1 Datasets, Baselines, and Settings

We evaluate FastRE on three widely used public relation extraction datasets:
NYT10 Takanobu et al. (2019), NYT11 Takanobu et al. (2019) and NYT24 Zeng et
al. (2018). The statistics of these datasets are shown in Table 1. We use
[PER], [LOC], [ORG] and [OTH] to indicate the four entity types and determine
the mapping mechanism between entity types and predefined relations based on
the semantics of each relation.

We compare FastRE with the strong state-of-the-art relation extraction models:
(1) Tagging based models have been widely explored, such as NovelTagging Zheng
et al. (2017), CasRel Wei et al. (2020), TPLinker Wang et al. (2020) and PRGC
Zheng et al. (2021). (2) Generating based models convert the relation
extraction task into a generation task, such as CopyRE Zeng et al. (2018),
WDec Nayak and Ng (2020), and SPN Sui et al. (2021). (3) Other strong
baselines cast the relation extraction task in a reinforcement learning
perspective such as HRL Takanobu et al. (2019), or table filling framework
like GRTE Ren et al. (2021).

The standard micro precision, recall, and F1 score are used to evaluate the
results. To fairly compare with existing models, we follow previous work
Takanobu et al. (2019); Wei et al. (2020) and use _Partial Match_ on all three
datasets.

To evaluate the robustness and stability of the models, we use the same hyper-
parameter setting on all datasets. Specifically, FastRE uses the
300-demensional GloVe embeddings Pennington et al. (2014) as the initial word
vectors with $d$=128 hidden dimension. The kernel size $k_{s}$ and type
embedding dimension $d_{t}$ is 3 and 64, respectively. FastRE applies 6
stacked $\rm{Block}(\cdot)$ and the dilation rate in each layer are 1, 2, 4,
1, 1, 1. Our model is optimized with AdamW Loshchilov and Hutter (2019) using
learning rate 1e-3 with a linear warm up Goyal et al. (2017) for the first 6%
steps followed by a linear decay to 0. We apply dropout Srivastava et al.
(2014) between convolutional layers with a rate 0.1. All the hyper-parameters
reported in this work are based on the results on the validation sets. Other
parameters are randomly initialized. Following CasRel Wei et al. (2020),
TPLinker Wang et al. (2020) and GRTE Ren et al. (2021), the max length of
input sentences is set to 100. We train FastRE for 60 epochs with batch size
32, and all experiments are conducted on a NVIDIA RTX 2080Ti GPU.

Dataset | #Relation | Train | Valid | Test
---|---|---|---|---
NYT10 | 29 | 70,339 | - | 4006
NYT11 | 12 | 62,648 | - | 369
NYT24 | 24 | 56,196 | 5,000 | 5,000
Table 1: Statistics of datasets. Models | NYT10 | NYT11 | NYT24
---|---|---|---
Prec. | Rec. | F1 | Train. | Infer. | Prec. | Rec. | F1 | Train. | Infer. | Prec. | Rec. | F1 | Train. | Infer.
NovelTagging Zheng et al. (2017) | 59.3 | 38.1 | 46.4 | - | - | 46.9 | 48.9 | 47.9 | - | - | 62.4 | 31.7 | 42.0 | - | -
CasRel Wei et al. (2020)$\dagger$ | 78.0 | 69.0 | 73.2 | 521 | 266 | 50.3 | 58.1 | 53.9 | 448 | 24 | 89.9 | 89.1 | 89.5 | 420 | 327
TPLinker Wang et al. (2020)$\dagger$ | 80.1 | 66.4 | 72.6 | 984 | 191 | 56.2 | 55.1 | 55.7 | 973 | 16 | 91.0 | 91.8 | 91.4 | 885 | 235
PRGC Zheng et al. (2021)$\dagger$ | 80.2 | 66.5 | 72.7 | 290 | 134 | 54.4 | 56.3 | 55.3 | 267 | 12 | 89.9 | 90.9 | 90.4 | 272 | 161
CopyRE Zeng et al. (2018) | 56.9 | 45.2 | 50.4 | - | - | 34.7 | 53.4 | 42.1 | - | - | 61.0 | 56.6 | 58.7 | - | -
WDec Nayak and Ng (2020) | 84.6 | 62.1 | 71.6 | - | - | - | - | - | - | - | 94.5 | 76.2 | 84.4 | - | -
SPN Sui et al. (2021)$\dagger$ | 79.5 | 67.1 | 72.8 | 516 | 202 | 52.7 | 55.4 | 54.0 | 380 | 19 | 93.3 | 91.8 | 92.5 | 473 | 254
HRL Takanobu et al. (2019) | 71.4 | 58.6 | 64.4 | - | - | 53.8 | 53.8 | 53.8 | - | - | - | - | - | - | -
GRTE Ren et al. (2021)$\dagger$ | 79.8 | 67.6 | 73.2 | 890 | 176 | 53.6 | 58.2 | 55.8 | 843 | 17 | 92.5 | 92.7 | 92.6 | 795 | 221
FastRE (Ours) | 78.0 | 70.1 | 73.8 | 100 | 18.4 | 54.1 | 58.7 | 56.3 | 87 | 1.6 | 89.6 | 86.3 | 87.9 | 96 | 23.0
Table 2: Main Results. Baselines with $\dagger$ are produced with the source code provided in the original papers. The rest results of the baselines are retrieved from the original papers. Results of all the models are the average of random three times. Train. and Infer. represent the total training (min) and inference time (s) on each dataset, respectively.  Models | Complexity | NYT10 | NYT11 | NYT24
---|---|---|---|---
Param. | Infer. (1 / 8) | Param. | Infer. (1 / 8) | Param. | Infer. (1 / 8)
CasRel Wei et al. (2020) | $O(kn)$ | 107,729 K | 66.2 / - | 107,698 K | 62.9 / - | 107,720 K | 65.3 / -
TPLinker Wang et al. (2020) | $O(kn^{2})$ | 109,626 K | 104.7 / 47.5 | 109,548 K | 97.5 / 42.5 | 109,603 K | 103.2 / 47.0
SPN Sui et al. (2021) | $O(cn)$ | 141,754 K | 197.4 / 50.2 | 140,648 K | 199.2 / 51.6 | 141,429 K | 198.9 / 50.7
PRGC Zheng et al. (2021) | $O(n^{2})$ | 108,931 K | 132.6 / 33.3 | 108,891 K | 127.1 / 30.6 | 108,919 K | 130.7 / 32.1
GRTE Ren et al. (2021) | $O(kn^{2})$ | 119,450 K | 97.5 / 43.8 | 119,166 K | 98.8 / 44.2 | 119,387 K | 98.3 / 44.1
FastRE (Ours) | $O(mn)\to O(mn+kn)$ | 1,096 K |  12.9 / 5.9 / 4.6 | 1,092 K |  11.7 / 5.5 / 4.4 | 1,095 K |  12.8 / 5.8 / 4.6
Table 3: Comparison of inference efficiency. Complexity are the theoretical
decoding complexity with respect to sequence length $n$ and relation set size
$k$. In SPN, $c$ is a constant independent of $k$ and typically less than $k$.
In FastRE, $m$ denotes the number of entity types. Param. denotes the number
of model parameters obtained by the official implementation with default
configuration. Infer. (1 / 8) denotes the inference time (ms) per instance
with the batch size of 1 and 8. Results marked with bold denote the inference
batch size is set to 128.

###  3.2 Main Results

According to the main results shown in Table 2, FastRE is very effective
compared with the state-of-the-art models based on BiLSTM and BERT, and
achieves excellent performance on NYT10 and NYT11. Especially, FastRE
outperforms the existing models in terms of training and inference speed on
all datasets, which demonstrates the advantages of our convolutional encoder
in efficiency. For example, the inference speed of FastRE is more than 7
$\times$ as PRGC, 15 $\times$ as CasRel and more than 10 $\times$ as other
BERT based models. And the training speed of FastRE is 3 $\times$ as PRGC and
more than 10 $\times$ as TPLinker. Note that except PRGC, the training time of
other BERT based models varies greatly on different datasets. The training
time required by FastRE on all datasets is close, although these datasets have
different relation numbers. This is mainly because of introducing the mapping
mechanism. From the results in PRGC which predicts relations first and
extracts entity later, alleviating relation redundancy typically leads a
relatively stable training time. In addition, FastRE achieves significantly
better performance on NYT10 and NYT11 compared to CasRel, indicating the
effectiveness of the improved cascade binary tagging framework.

We observe that the performance on NYT24 of FastRE is not good as the models
based on the BERT. We believe that there exists two potential reasons for this
phenomenon: (1) insufficient expression ability of FastRE, and (2)
irrationality of the NYT24. First, FastRE employs a simple and efficient
structure to approximate large Transformer based structures, so as to
accelerate model training and inference speed, which limits its expression
ability. Nonetheless, FastRE still outperforms BiLSTM based models such as
WDec and closes to CasRel, means that the convolution based structure achieves
comparable expression ability with pre-trained model BERT on RE task. In
addition, the irrationality of the dataset would greatly impact the model
performance. As Zhang et al. (2020) pointed out, over 90% triplets in the
NYT24 test set reoccurred in its training set, which easily overfitted by
existing models and significantly limits the ability to verify the model
generalization. We believe that these BERT based models gain a high score in
NYT24 via memorizing the frequently reoccurred training set triplets because
of the strong expression and fitting ability of BERT, which causes the
overfitting instead of better generalization.

###  3.3 Inference Efficiency

According to the results shown in Table 3, the total parameters in FastRE are
almost $1/100$ of other baselines, that ensures the faster forward calculation
speed and back propagation update speed. Consequently, very few parameters can
help FastRE achieve fast inference. Moreover, FastRE has lower theoretical
decoding complexity compared to most models. When there are fewer entity types
($m<k$) and fewer average number of corresponding relations, FastRE has lower
decoding complexity than CasRel. Although SPN has the lowest decoding
complexity, the BERT based encoder and the non-autoregressive decoder become
the most consuming parts and limit its inference efficiency.

We set the batch size of all BERT based models to 8 for evaluating parallel
processing ability. Consistent with the results obtained by Ren et al. (2021),
TPLinker, SPN and GRTE have similar inference speed. FastRE is significantly
faster than other models during inference. The results also indicate that the
single-thread speed of FastRE is still much faster than other models.
Similarly, although CasRel takes the longest time in whole inference stage, it
is significantly faster than other BERT based models when processing a single
instance, which shows the efficiency of cascade binary tagging framework. When
the batch size of FastRE is 8, the speed of FastRE is about 5.5$\times$ as
PRGC. And we can increase the batch size to 128 because of the simple model
structure and the small parameter scale to obtain faster inference speed,
which shows the superior parallel processing ability.

###  3.4 Convergence Efficiency

To evaluate the model convergence efficiency, we analyze the iteration time
and total iterations of different models on NYT11 as shown in Figure 2. As
different models require different batch sizes for training, we use the
official implementation with default batch size configuration. Note that some
models such as CasRel and TPLinker set the total epochs to 100 but could
perform early stopping when the model performance no longer improving. For
example, CasRel typically converges within 30 epochs. Therefore, we only count
the time when a model converge, not the total number of epochs, which is quite
different with another analysis Ren et al. (2021). We believe that our
evaluation can more accurately reflect the real convergence efficiency.

The results show that FastRE outperforms other models in both aspects. FastRE
consumes the least iteration time because it requires the least computation
with least parameters. Note that the iteration time in SPN is less than other
BERT based models, but it requires much more iterations to converge as it
casts the RE task to a complex bipartite matching problem and contains the
most parameters compared to other models. Similar drawbacks occur in GRTE,
which performs fussy table-filling with an iterative way and requires the most
iterations among all the models. Intuitively, FastRE should have few
iterations for convergence with far less parameters than the BERT based model.
However, the results show that FastRE has less advantage in the number of
total iterations compared with PRGC. This phenomenon motivates us to adopt
more effective training strategies in the future.

Figure 2: Iteration time (ms) and total iterations on NYT11.

###  3.5 Structure Search

We conduct structure search experiment as shown in Table 4. Setting the
stacked block numbers to 3 greatly accelerates the model speed, but brings
serious performance degradation. By changing the dilation rate of different
blocks, we observe that the appropriate dilation rate plays an important role
in improving the final performance. For example, when the dilated convolution
is not used (the dilation rate is set to 1) or the dilation rate is set too
large, the performance suffers grave decline. This is because dilated
convolution is hard to capture long-distance dependencies by simply increasing
dilation rate exponentially. The few parameters of CNN fail to capture so much
information and blindly increasing the receptive region will lead to worse
results. Therefore, after a certain number of dilated convolutional layers, we
replace the rest layers with vanilla convolutional layers to narrow the
receptive region for fine tuning. In addition, continue increasing the number
of stacked blocks only brings little performance improvement, but greatly
slows down the model convergence speed and inference speed. Therefore, FastRE
adopts 6 stacked blocks and sets the dilation rate to 1, 2, 4, 1, 1, 1, which
is a structure that balances speed and performance.

Models | Param. | Train. | Infer. | F1
---|---|---|---|---
FastRE ($L$=6, $d_{i}$=1, 2, 4, 1, 1, 1) | 1096 K | 100 | 18.4 | 73.8
$L$=3, $d_{i}$=1, 2, 4 | 800 K | 50 | 13.7 | 68.6
$L$=6, $d_{i}$=1, 1, 1, 1, 1, 1 | 1096 K | 100 | 17.5 | 71.5
$L$=6, $d_{i}$=1, 2, 4, 8, 16, 32 | 1096 K | 100 | 18.9 | 70.9
$L$=9, $d_{i}$=1, 2, 4, 1, 2, 4, 1, 1, 1 | 1392 K | 150 | 28.3 | 74.1
Table 4: Structure search on NYT10. Param. denotes the number of model parameters. Train. denotes the total training time (min) until convergence, and Infer. denotes the total inference time (s). For example, $L$=3, $d_{i}$=1, 2, 4 represents the encoder has 3 stacked blocks and the dilation rate is 1, 2, 4, respectively. Models | NYT10 | NYT11 | NYT24
---|---|---|---
FastRE | 73.8 | 56.3 | 87.9
\- Dilated Convolution | 71.5 | 54.8 | 86.3
\- Gated Unit | 69.4 | 53.2 | 84.2
\- Residual Connection | 70.8 | 54.4 | 85.9
\- Mapping Mechanism | 72.2 | 54.4 | 86.4
\- Adaptive Thresholding | 73.0 | 55.5 | 87.0
Table 5: Ablation study of components in FastRE.

###  3.6 Ablation Study

We conduct ablation experiments to demonstrate the effectiveness of each
component in FastRE with F1 scores (%) reported in Table 5. It is important to
note that the impact of each component on efficiency is not obvious (except
adaptive thresholding), so we only discuss the ablation results on
performance.

#### Dilated Convolution

We set all dilation rates to 1. The dilated convolution can capture more local
context information with less parameters and more speed. Without dilated
convolutions, the ability to capture long-distance dependencies decreases
significantly, resulting in a decline in the overall performance.

#### Gated Unit

The gated unit part ($\otimes{\rm{sigmoid}}(\mathbf{Y}_{b})$) in Equation (4)
is removed. The purpose of using gated unit is controlling information flow
and selecting the most important information dynamically. Removing the gated
units has a greater performance impact than removing dilated convolution and
residual connection.

#### Residual Connection

We remove the residual connection part ($+\mathbf{X}$) in Equation (4).
Generally, the residual connection can prevent gradient vanishing, but it
still has a positive impact on performance. The residual connection transmits
information in multiple channels in multi-layer neural networks, resulting in
simpler learning process and better generalization.

#### Mapping Mechanism

Without the mapping mechanism, namely, we set all entity types to [ANY], and
[ANY] can map to all the relations, that causes a small drop in performance.
However, one drawback of the mapping mechanism is that it can only deal with
the case where the mapping from relation to entity type is uniquely
determined. More general methods are worth exploring in the future.

#### Adaptive Thresholding

We perform sigmoid operation on all the positions and tuning a global
threshold. We find that using the global threshold improves the F1 scores on
validation sets. However, the F1 scores on test sets with the global threshold
are nearly 1% lower than the results under the adaptive thresholding, which
indicates that using adaptive thresholding makes the model have better
generalization. We also observe that the convergence speed using adaptive
thresholding loss is twice that using cross entropy loss. Sigmoid operation is
simple but suffers from imbalanced classes issue as it produces plenty of 0
positions which affects the convergence speed. Rank-based adaptive
thresholding loss balances the contribution of positive and negative classes
in loss, making the training easier and faster.

###  3.7 Discussion

#### Other baselines and benchmarks

Some latest work such as REBEL Huguet Cabot and Navigli (2021) and ReRe Xie et
al. (2021) are not addressed in baseline experiments. Both models have
comparable performance but lower efficiency compared to FastRE. REBEL (BART-
large as backbone) cannot run in our experimental settings as it requires more
computing resources. The parameters in REBEL are 400 $\times$ as FastRE and
REBEL mainly focuses on performance rather than efficiency, which are
different from the goal of our work. FastRE achieves the comparable
performance as ReRe on NYT10 (73.8 vs 73.8) and NYT11 (56.3 vs 56.2), while
the inference speed is 20 $\times$ faster than ReRe. The reason is that ReRe
adopts pipeline extraction and uses two separate BERT models in relation
classification and entity extraction. Furthermore, we have conducted
systematic experiments on other benchmarks such as ADE and WebNLG, and the
results consistently support the findings of our work. Due to space
limitation, we cannot report all results.

#### Trade off between performance and efficiency

The goal of FastRE is to accelerate RE efficiency while maintaining good
performance. It employs a simple architecture to achieve a balance between
performance and efficiency, because increasing the model complexity will
decrease the efficiency. We examined the NYT24 and found that 90.49% triplets
overlapping in test set and training set, while it is only 72.64% and 41.62%
in NYT10 and NYT11. This would leads BERT based models and Seq2seq based
models to gain high scores because of rote memorization and exposure bias.
Therefore, the results on NYT24 is not suitable to test the model
generalization. Finally, we directly replaced the BERT encoder in CasRel with
our convolutional encoder and the results show that the performance drops
1.8%, 0.7%, 3.3% on three benchmarks, which shows that the representation
ability of convolutional encoder is not as good as BERT on RE task. To sum up,
FastRE has somewhat limited representation ability as its simplicity, but it
achieves better generalization and is more suitable and applicable for
unbiased datasets like NYT11 rather than NYT24.

#### Explainability

FastRE improves training and inference speed while keeps promising
performance, and it could be somewhat explained by the gradient techniques.
Firstly, we visualized the distribution of attention scores on two separate
self-attention layers, and found that the score of entity boundary is
significantly higher than that of other words in both layers, which highlights
the boundary features for subsequent entity prediction. Secondly, we collected
the gradient of the loss w.r.t the input and computed the norm of the gradient
for each word vector in the input. After normalization on each word gradient
norm, we found that the important predicates in sentences (which indicate the
relationships between entities) always receive higher (relative) gradient
norms, while trivial conjunctions and prepositions have gradient norms close
to 0.

##  4 Conclusion

This paper proposes a simple and fast relation extraction model, FastRE, to
significantly reduce training and inference time but retain competitive
performance. We propose a novel convolutional encoder combined with dilated
convolution, gated unit and residual connection for RE. We also improve the
cascade binary tagging framework with type-relation mapping mechanism and
position-dependent adaptive thresholding. Experimental results show that
FastRE achieves faster training and inference speed compared to the state-of-
the-art models, while the performance is competitive.

## Acknowledgements

We thank the anonymous reviewers for their insightful comments. The work was
supported in part by the 13th Five-Year All-Army Common Information System
Equipment Pre-Research Project (Grant Nos. 31514020501, 31514020503). All
opinions are of the authors and do not reflect the view of sponsors.

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that a continuous set of solutions for the gap equation emerges at $M_{cr}+0$.

We first briefly describe the extension procedure. The two coupled Eliashberg
equations are for the pairing vertex $\Phi(\omega_{m})$ and the self-energy
$\Sigma(\omega_{m})$. For $\gamma=2$, the equations are

$\displaystyle\Phi(\omega_{m})={\bar{g}}^{2}\pi T\sum_{m^{\prime}\neq
m}\frac{\Phi(\omega_{m^{\prime}})}{\sqrt{{\tilde{\Sigma}}^{2}(\omega_{m^{\prime}})+\Phi^{2}(\omega_{m^{\prime}})}}~{}\frac{1}{|\omega_{m}-\omega_{m^{\prime}}|^{2}},$
$\displaystyle{\tilde{\Sigma}}(\omega_{m})=\omega_{m}+{\bar{g}}^{2}\pi
T\sum_{m^{\prime}\neq
m}\frac{{\tilde{\Sigma}}(\omega_{m^{\prime}})}{\sqrt{{\tilde{\Sigma}}^{2}(\omega_{m^{\prime}})+\Phi^{2}(\omega_{m^{\prime}})}}~{}\frac{1}{|\omega_{m}-\omega_{m^{\prime}}|^{2}}$
(115)

where ${\tilde{\Sigma}}(\omega_{m})=\omega_{m}+\Sigma(\omega_{m})$. At $T=0$,
the r.h.s. of each of the two equations contains a divergent integral $\int
dx/x^{2}$. To regularize the divergencies, we keep the temperature small but
finite and set $T=0$ at the end of calculations. At a finite $T$, the sum over
$m^{\prime}$ is non-singular as singular self-action term with $m^{\prime}=m$
cancels out by the same reason as the contributions from non-magnetic
impurities.

We then introduce

$\displaystyle{\bar{\Phi}}(\omega_{m})$ $\displaystyle=$
$\displaystyle\Phi(\omega_{m})\left(1-{\bar{g}}^{2}\frac{\zeta(2)}{(2\pi
T)}\frac{1}{\sqrt{\tilde{\Sigma}^{2}(\omega_{m})+\Phi^{2}(\omega_{m})}}\right)$
$\displaystyle{\bar{\tilde{\Sigma}}}(\omega_{m})$ $\displaystyle=$
$\displaystyle{\tilde{\Sigma}}(\omega_{m})\left(1-{\bar{g}}^{2}\frac{\zeta(2)}{(2\pi
T)}\frac{1}{\sqrt{\tilde{\Sigma}^{2}(\omega_{m})+\Phi^{2}(\omega_{m})}}\right)$
(116)

where $\zeta(2)=\pi^{2}/6=\sum_{n=1}^{\infty}1/n^{2}$. Because
$\Phi(\omega_{m})/{\tilde{\Sigma}}(\omega_{m})={\bar{\Phi}}(\omega_{m})/{\bar{\tilde{\Sigma}}}(\omega_{m})$,
Eqs. (C) can be re-expressed solely in terms of ${\bar{\Phi}}(\omega_{m})$ and
${\bar{\tilde{\Sigma}}}(\omega_{m})$:

$\displaystyle{\bar{\Phi}}(\omega_{m})={\bar{g}}^{2}\pi T\sum_{m^{\prime}\neq
m}\left(\frac{{\bar{\Phi}}(\omega_{m^{\prime}})}{\sqrt{{\bar{\tilde{\Sigma}}}^{2}(\omega_{m^{\prime}})+{\bar{\Phi}}^{2}(\omega_{m^{\prime}})}}-\frac{{\bar{\Phi}}(\omega_{m})}{\sqrt{{\bar{\tilde{\Sigma}}}^{2}(\omega_{m})+{\bar{\Phi}}^{2}(\omega_{m})}}\right)~{}\frac{1}{|\omega_{m}-\omega_{m^{\prime}}|^{2}},$
$\displaystyle{\bar{\tilde{\Sigma}}}(\omega_{m})=\omega_{m}$
$\displaystyle+{\bar{g}}^{2}\pi T\sum_{m^{\prime}\neq
m}\left(\frac{{\bar{\tilde{\Sigma}}}(\omega_{m^{\prime}})}{\sqrt{{\bar{\tilde{\Sigma}}}^{2}(\omega_{m^{\prime}})+{\bar{\Phi}}^{2}(\omega_{m^{\prime}})}}-\frac{{\bar{\tilde{\Sigma}}}(\omega_{m})}{\sqrt{{\bar{\tilde{\Sigma}}}^{2}(\omega_{m})+{\bar{\Phi}}^{2}(\omega_{m})}}\right)~{}\frac{1}{|\omega_{m}-\omega_{m^{\prime}}|^{2}}$
(117)

These equations are now free from singularities at $T=0$, when the summation
over Matsubara numbers is replaced by the integration over $\omega_{m}$.

We now extend the modified Eliashberg equations (C) by multiplying the
interaction in the particle-particle channel by a factor $1/M$:

${\bar{\Phi}}(\omega_{m})=\frac{{\bar{g}}^{2}}{2M}\int
d\omega^{\prime}_{m}\left(\frac{{\bar{\Phi}}(\omega^{\prime}_{m})}{\sqrt{{\bar{\tilde{\Sigma}}}^{2}(\omega^{\prime}_{m})+{\bar{\Phi}}^{2}(\omega^{\prime}_{m})}}-\frac{{\bar{\Phi}}(\omega_{m})}{\sqrt{{\bar{\tilde{\Sigma}}}^{2}(\omega_{m})+{\bar{\Phi}}^{2}(\omega_{m})}}\right)~{}\frac{1}{|\omega_{m}-\omega^{\prime}_{m}|^{2}}$
(118)

The gap function $\Delta(\omega_{m})$ is expressed via
${\bar{\Phi}}(\omega_{m})$ and ${\bar{\tilde{\Sigma}}}(\omega_{m})$ in the
same way as via the original $\Phi(\omega_{m})$ and
${\tilde{\Sigma}}(\omega_{m})$:
$\Delta(\omega_{m})=\omega_{m}{\bar{\Phi}}(\omega_{m})/{\bar{\tilde{\Sigma}}}(\omega_{m})$.
The equation on $\Delta(\omega_{m})$ is

$\Delta(\omega_{m})=\frac{{\bar{g}}^{2}}{2M}\int~{}\frac{d\omega^{\prime}_{m}}{|\omega_{m}-\omega^{\prime}_{m}|^{2}}\left(\frac{\Delta(\omega^{\prime}_{m})-M\frac{\Delta(\omega_{m})}{\omega_{m}}\omega^{\prime}_{m}}{\sqrt{\Delta^{2}(\omega^{\prime}_{m})+(\omega^{\prime}_{m})^{2}}}-\frac{\Delta(\omega_{m})(1-M)}{\sqrt{\Delta^{2}(\omega_{m})+\omega^{2}_{m}}}\right)$
(119)

Absorbing $1/M$ into ${\bar{g}}^{2}_{M}={\bar{g}}^{2}/M$, introducing a
dimensionless $\bar{\omega}_{m}=\omega_{m}/{\bar{g}}_{M}$,
$D(\bar{\omega}_{m})=\Delta(\bar{\omega}_{m})/\bar{\omega}_{m}$ and re-
arranging, we obtain from (119)

$\displaystyle
D(\bar{\omega}_{m})\left(\bar{\omega}+\frac{1-M}{2}\int~{}\frac{d\bar{\omega}^{\prime}_{m}}{|\bar{\omega}_{m}-\bar{\omega}^{\prime}_{m}|^{2}}\left(\frac{\operatorname{sgn}\bar{\omega}_{m}}{\sqrt{1+D^{2}(\bar{\omega}_{m})}}-\frac{\operatorname{sgn}\bar{\omega}^{\prime}_{m}}{\sqrt{1+D^{2}(\bar{\omega}^{\prime}_{m})}}\right)\right)=$
$\displaystyle\frac{1}{2}\int~{}\frac{d\bar{\omega}^{\prime}_{m}}{|\bar{\omega}_{m}-\bar{\omega}^{\prime}_{m}|^{2}}\frac{D(\bar{\omega}^{\prime}_{m})-D(\bar{\omega}_{m})}{\sqrt{1+D^{2}(\bar{\omega}^{\prime}_{m})}}\operatorname{sgn}\bar{\omega}^{\prime}_{m}$
(120)

Both integrals in (120) are free from singularities and infra-red convergent.

For infinitesimally small $D(\bar{\omega}_{m})$, Eq. (120) becomes

$D(\bar{\omega}_{m})\left(\bar{\omega}_{m}+\frac{1-M}{\bar{\omega}_{m}}\right)=\frac{1}{2}\int
d\bar{\omega}^{\prime}_{m}\frac{D(\bar{\omega}^{\prime}_{m})-D(\bar{\omega}_{m})}{|\bar{\omega}_{m}-\bar{\omega}^{\prime}_{m}|^{2}}\operatorname{sgn}\omega^{\prime}_{m},$
(121)

At small $\bar{\omega}_{m}$, the solution of the gap equation is

$D(\bar{\omega}_{m})=2\epsilon\cos{\left(\beta_{M}\log{\bar{\omega}^{2}_{m}}+\phi\right)}\operatorname{sgn}\omega_{m},$
(122)

It has the same form as Eq. (13), but now $\beta^{2}_{M}=M/\pi^{2}$. This form
implies that $M_{cr}=0$.

We now assume that $M$ is small and solve the non-linear gap equation. Our key
intension is to check whether we still have a continuum of solutions. For this
purpose, it is sufficient to focus on small $\bar{\omega}_{m}$, when we can
neglect bare $\bar{\omega}_{m}$ in the l.h.s. of (120).

As in Sec. III.3, we search for the solution of (120) in the series in
$\epsilon^{2}$ for both $D(\bar{\omega}_{m})$ and $\beta$. To leading order in
$M$, we obtain

$D(\bar{\omega}_{m})=\frac{2\epsilon}{\pi}M^{1/2}\log{\bar{\omega}^{2}_{m}}\left(1-3\epsilon^{2}+....\right)^{1/2}$
(123)

where dots stand for $\epsilon^{4}$ and higher order terms. The $M^{1/2}$
dependence (same as $(M-M_{cr})^{1/2}$ as $M_{cr}=0$) is an expected one. The
logarithmic dependence on frequency is consistent with the result in Paper I,
where we obtained $\log{\bar{\omega}^{\gamma}_{m}}$ dependence at $N=N_{cr}$.
However, there such dependence exists only for $N=N_{cr}$, while here we have
an infinite set of solutions with the same frequency dependence, but different
amplitudes, parametrized by $\epsilon$. All solutions appear simultaneously at
$M=0+$

A complimentary piece of evidence for multiple solutions comes about if we
simplify the l.h.s. of (120) by dropping $D^{2}$ terms in the denominator of
the $(1-M)$ term. The gap equation then reduces to

$D(\bar{\omega}_{m})\left(\bar{\omega}+\frac{1-M}{2\bar{\omega}}\right)=\frac{1}{2}\int~{}\frac{d\bar{\omega}^{\prime}_{m}}{|\bar{\omega}_{m}-\bar{\omega}^{\prime}_{m}|^{2}}\frac{D(\bar{\omega}^{\prime}_{m})-D(\bar{\omega}_{m})}{\sqrt{1+D^{2}(\bar{\omega}^{\prime}_{m})}}\mbox{\text{sign}}\bar{\omega}^{\prime}_{m}$
(124)

The full gap equation for the original $\gamma$ model is reproduced if we set
$M=1$, so Eq. (124) can be viewed as another extension of the original model.
The solution of the linearized gap equation is the same as before, with
$\beta^{2}_{M}=M/\pi$, hence $M_{cr}=0$. At $M\to 0$, the expansion in
$\epsilon^{2}$ now yields, to leading order in $M\equiv M-M_{cr}$:

$D(\bar{\omega}_{m})=2\epsilon\left(\cos{f_{M}(\bar{\omega}_{m})}-\frac{\epsilon^{2}}{16M}\cos{3f_{M}(\bar{\omega}_{m})}+...\right)$
(125)

where

$f_{M}(\bar{\omega}_{m})=\beta_{M,\epsilon}\log{\bar{\omega}^{2}_{m}}+\phi$
(126)

and

$\beta^{2}_{M,\epsilon}=\beta^{2}_{M}\left(1-\frac{3\epsilon^{2}}{M}+...\right)$
(127)

We see that the expansion holds in powers of $\epsilon^{2}/M$ and is valid up
to $\epsilon\sim(M)^{1/2}$, at which $\beta^{2}_{M,\epsilon}$ vanishes. At
larger $\epsilon$, $\beta^{2}_{M,\epsilon}$ becomes negative, and the solution
disappears (there is no normalized solution of the linearized gap equation).
We see that there is again an infinite set of solutions, specified by
$\epsilon$, which runs between $0$ and $\epsilon_{cr}=O(\sqrt{M})$.

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