# Robust Lottery Tickets for Pre-trained Language Models

Rui Zheng★∗, Rong Bao★ , Yuhao Zhou★, Di Liang♠, Sirui Wang♠,
Wei Wu♠, Tao Gui◆, Qi Zhang★,♣, Xuanjing Huang★
★ School of Computer Science, Fudan University, Shanghai, China
◆ Institute of Modern Languages and Linguistics, Fudan University, Shanghai,
China
♣ Shanghai Collaborative Innovation Center of Intelligent Visual Computing,
Fudan University
♠ Meituan Inc., Beijing, China
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
∗ Equal contribution. Corresponding authors.

###### Abstract

Recent works on _Lottery Ticket Hypothesis_ have shown that pre-trained
language models (PLMs) contain smaller matching subnetworks (winning tickets)
which are capable of reaching accuracy comparable to the original models.
However, these tickets are proved to be not robust to adversarial examples,
and even worse than their PLM counterparts. To address this problem, we
propose a novel method based on learning binary weight masks to identify
robust tickets hidden in the original PLMs. Since the loss is not
differentiable for the binary mask, we assign the hard concrete distribution
to the masks and encourage their sparsity using a smoothing approximation of
$L_{0}$ regularization. Furthermore, we design an adversarial loss objective
to guide the search for robust tickets and ensure that the tickets perform
well both in accuracy and robustness. Experimental results show the
significant improvement of the proposed method over previous work on
adversarial robustness evaluation.

##  1 Introduction

Large-scale pre-trained language models (PLMs), such as BERT Devlin et al.
(2019), Roberta Liu et al. (2019) and T5 Raffel et al. (2019) have achieved
great success in the field of natural language processing. As more transformer
layers are stacked with larger self-attention blocks, the complexity of PLMs
increases rapidly. Due to the over-parametrization of PLMs, some Transformer
heads and even layers can be pruned without significant losses in performance
Michel et al. (2019); Kovaleva et al. (2019); Rogers et al. (2020).

The Lottery Ticket Hypothesis suggests an over-parameterized network contains
certain subnetworks (i.e., winning tickets) that can match the performance of
the original model when trained in isolation Frankle and Carbin (2019). Chen
et al. (2020); Prasanna et al. (2020) also find these winning tickets exist in
PLMs. Chen et al. (2020) prune BERT in an unstructured fashion and obtain
winning tickets at sparsity from 40% to 90%. Prasanna et al. (2020) aim at
finding structurally sparse tickets for BERT by pruning entire attention heads
and MLP. Previous works mainly focused on using winning tickets to reduce
model size and speed up training time Chen et al. (2021), while little work
has been done to explore more benefits, such as better adversarial robustness
than the original model.

As we all know, PLMs are vulnerable to adversarial examples that are
legitimately crafted by imposing imperceptible perturbations on normal
examples Jin et al. (2020); Garg and Ramakrishnan (2020); Wang et al. (2021).
Recent studies have shown that pruned subnetworks of PLMs are even less robust
than their PLM counterparts Xu et al. (2021); Du et al. (2021). Xu et al.
(2021) observe that when fine-tuning the pruned model again, the model yields
a lower robustness. Du et al. (2021) clarify the above phenomenon further: the
compressed models overfit on shortcut samples and thus perform consistently
less robust than the uncompressed large model on adversarial test sets.

In this work, our goal is to find robust PLM tickets that, when fine-tuned on
downstream tasks, achieve matching test performance but are more robust than
the original PLMs. In order to make the topology structure of tickets
learnable, we assign binary masks to pre-trained weights to determine which
connections need to be removed. To solve discrete optimization problem of
binary masks, we assume the masks follow a hard concrete distribution (a soft
version of the Bernoulli distribution), which can be solved using Gumbel-
Softmax trick Louizos et al. (2018). We then use an adversarial loss objective
to guide the search for robust tickets and an approximate $L_{O}$
regularization is used to encourage the sparsity of robust tickets. Robust
tickets can be used as a robust substitute of original PLMs to fine-tune
downstream tasks. Experimental results show that robust tickets achieve a
significant improvement in adversarial robustness on various tasks and
maintain a matching accuracy. Our codes are publicly available at
Github111https://github.com/ruizheng20/robust_ticket.

The main contributions of our work are summarized as follows:

  * •

We demonstrate that PLMs contain robust tickets with matching accuracy but
better robustness than the original network.

  * •

We propose a novel and effective technique to find the robust tickets based on
learnable binary masks rather than the traditional iterative magnitude-based
pruning.

  * •

We provide a new perspective to explain the vulnerability of PLMs on
adversarial examples: some weights of PLMs do not contribute to the accuracy
but may harm the robustness.

##  2 Related Work

###  2.1 Textual Adversarial Attack and Defense

Textual attacks typically generate explicit adversarial examples by replacing
the components of sentences with their counterparts and maintaining a high
similarity in semantics Ren et al. (2019) or embedding space Li et al. (2020).
These adversarial attackers can be divided into character-level Gao et al.
(2018), word-level Ren et al. (2019); Zang et al. (2020); Jin et al. (2020);
Li et al. (2020) and multi-level Li et al. (2018). In response to adversarial
attackers, various adversarial defense methods are proposed to improve model
robustness. Adversarial training solves a min-max robust optimization and is
generally considered as one of the strongest defense methods Madry et al.
(2018); Zhu et al. (2020); Li and Qiu (2020). Adversarial data augmentation
(ADA) has been widely adopted to improve robustness by adding textual
adversarial examples during training Jin et al. (2020); Si et al. (2021).
However, ADA is not sufficient to cover the entire perturbed search space,
which grows exponentially with the length of the input text. Some
regularization methods, such as smoothness-inducing regularization Jiang et
al. (2020) and information bottleneck regularization Wang et al. (2020), are
also beneficial for robustness. Different from the above methods, we dig
robust tickets from original BERT, and the subnetworks we find have better
robustness after fine-tuning.

###  2.2 Lottery Ticket Hypothesis

Lottery Ticket Hypothesis (LTH) suggests the existence of certain sparse
subnetworks (i.e., winning tickets) at initialization that can achieve almost
the same test performance compared to the original model Frankle and Carbin
(2019). In the field of NLP, previous works find that the winning tickets also
exist in Transformers and LSTM Yu et al. (2020); Renda et al. (2020). Evci et
al. (2020) propose a method to optimize the topology of the sparse network
during training without sacrificing accuracy relative to existing dense-to-
sparse training methods. Chen et al. (2020) find that PLMs such as BERT
contain winning tickets with a sparsity of 40% to 90%, and the winning tickets
found in the mask language modeling task can universally be transfered to
other downstream tasks. Prasanna et al. (2020) find structurally sparse
winning tickets for BERT, and they notice that all subnetworks (winning
tickets and randomly pruned subnetworks) have comparable performance when
fine-tuned on downstream tasks. Chen et al. (2021) propose an efficient BERT
training method using Early-bird lottery tickets to reduce the training time
and inference time. Some recent studies have tried to dig out more features of
winning tickets. Zhang et al. (2021) demonstrate that even in biased models
(which focus on spurious correlations) there still exist unbiased winning
tickets. Liang et al. (2021) observe that at a certain sparsity, the
generalization performance of the winning tickets can not only match but also
exceed that of the full model. Du et al. (2021); Xu et al. (2021) show that
the winning tickets that only consider accuracy are over-fitting on easy
samples and generalize poorly on adversarial examples. Our work makes the
first attempt to find the robust winning tickets for PLMs.

###  2.3 Robustness in Model Pruning

Learning to identify a subnetwork with high adversarial robustness is widely
discussed in the field of computer vision. Post-train pruning approaches
require a pre-trained model with adversarial robustness before pruning Sehwag
et al. (2019); Gui et al. (2019). In-train pruning methods integrate the
pruning process into the robust learning process, which jointly optimize the
model parameters and pruning connections Vemparala et al. (2021); Ye et al.
(2019). Sehwag et al. (2020) integrate the robust training objective into the
pruning process and remove the connections based on importance scores. In our
work, we focus on finding robust tickets hidden in original PLMs rather than
pruning subnetworks from a robust model.

##  3 The Robust Ticket Framework

In this section, we propose a novel pruning method to extract robust tickets
of PLMs by learning binary weights masks with an adversarial loss objective.
Furthermore, we articulate the Robust Lottery Ticket Hypothesis: the full PLM
contains subnetworks (robust tickets) that can achieve better adversarial
robustness and comparable accuracy.

###  3.1 Revisiting Lottery Ticket Hypothesis

Denote $f(\theta)$ as a PLM with parameters $\theta$ that has been fine-tuned
on a downstream task. A subnetwork of $f(\theta)$ can be denoted as
$f(m\odot\theta)$, where $m$ are binary masks with the same dimension as
$\theta$ and $\odot$ is the Hadamard product operator. LTH suggests that, for
a network initialized with $\theta_{0}$, the Iterative Magnitude Pruning (IMP)
can identify a mask $m$, such that the subnetwork $f(x;m\odot\theta_{0})$ can
be trained to almost the same performance to the full model $f(\theta_{0})$ in
a comparable number of iterations. Such a subnetwork $f(x;m\odot\theta_{0})$
is called as _winning tickets_ , including both the structure mask $m$ and
initialization $\theta_{0}$. IMP iteratively removes the weights with the
smallest magnitudes from $m\odot\theta$ until a certain sparsity is reached.
However, the magnitude-based pruning is not suitable for robustness-aware
techniques Vemparala et al. (2021); Sehwag et al. (2020).

###  3.2 Discovering Robust Tickets

Our goal is to learn the sparse subnetwork, however, the training loss is not
differentiable for the binary masks. A simple choice is to adopt a straight-
through estimator to approximate the derivative Bengio et al. (2013).
Unfortunately, this approach ignores the Heaviside function in the likelihood
and results in biased gradients. Thus, we resort to a practical method to
learn sparse neural networks Louizos et al. (2018).

In our method, we assume each mask $m_{i}$ to be a independent random variable
that follows a hard concrete distribution
$\mathrm{HardConcrete}(\log\alpha_{i},\beta_{i})$ with temperature $\beta_{i}$
and location $\alpha_{i}$ Louizos et al. (2018):

$\displaystyle\mu_{i}\sim\mathcal{U}\left(0,1\right),$ (1) $\displaystyle
s_{i}=\sigma\left(\frac{1}{\beta_{i}}\left(\log\frac{\mu_{i}}{1-\mu_{i}}+\log\alpha_{i}\right)\right),$
(2) $\displaystyle
m_{i}=\min\left(1,\max\left(0,s_{i}\left(\zeta-\gamma\right)+\gamma\right)\right),$
(3)

where $\sigma$ denotes the sigmoid function, $\gamma=-0.1$, $\zeta=1.1$ are
constants, and $u_{i}$ is the sample drawn from uniform distribution
$\mathcal{U}(0,1)$. The random variable $s_{i}$ follows a binary concrete (or
Gumbel-Softmax) distribution, which is a smoothing approximation of the
discrete Bernoulli distribution Maddison et al. (2017); Jang et al. (2017).
Samples from the binary concrete distribution are identical to samples from a
Bernoulli distribution with probability $\alpha_{i}$ as $\beta_{i}\rightarrow
0$. The location $\alpha_{i}$ in (2) allows for gradient-based optimization
through reparametrization tricks. Using (3), the $s_{i}$ larger than
$\frac{1-\gamma}{\zeta-\gamma}$ is rounded to $1$, whereas the value smaller
than $\frac{-\gamma}{\zeta-\gamma}$ is rounded to $0$. To encourage the
sparsity, we penalize the $L_{0}$ complexity of masks based on the probability
which are non-zero:

$\mathcal{R}(m)=\frac{1}{\left|m\right|}\sum_{i=1}^{\left|m\right|}\sigma\left(\log\alpha_{i}-\beta_{i}\log\frac{-\gamma}{\zeta}\right).$
(4)

During the inference stage, the mask $\hat{m}_{i}$ can be estimated through a
hard concrete gate:

$\min\left(1,\max\left(0,\sigma\left(\log\alpha_{i}\right)\left(\zeta-\gamma\right)+\gamma\right)\right).$
(5)

####  3.2.1 Adversarial Loss Objective

To find the connections responsible for adversarial robustness, we incorporate
the adversarial loss into the mask learning objective:

$\min_{m}\underbrace{\mathbb{E}_{(x,y)\sim\mathcal{D}}\max_{\lVert\delta\rVert\leq\epsilon}\mathcal{L}\left(f(x+\delta;m\odot\theta),y\right)}_{\mathcal{L}_{adv}(m)},$
(6)

where $(x,y)$ is a data point from dataset $\mathcal{D}$, $\delta$ is the
perturbation that constrained within the $\epsilon$ ball. The inner
maximization problem in (6) is to find the worst-case adversarial examples to
maximize the classification loss, while the outer minimization problem in (6)
aims at optimizing the masks to minimize the loss of adversarial examples,
i.e., $\mathcal{L}_{adv}(m)$.

Adversarial attack method, typically with PGD, can be used to solve the inner
maximization problem. PGD applies the $K$-step stochastic gradient descent to
search for the perturbation $\delta$ Madry et al. (2018):

$\delta_{k+1}=\prod_{\lVert\delta\rVert\leq\epsilon}\left(\delta_{k}+\eta\frac{g\left(\delta_{k}\right)}{\rVert
g\left(\delta_{k}\right)\rVert}\right),$ (7)

where
$g\left(\delta_{k}\right)=\nabla_{x}\mathcal{L}\left(f(x+\delta_{k};m\odot\theta),y\right)$,
$\delta_{k}$ is the perturbation in $k$-th step and
$\prod_{\lVert\delta\rVert\leq\epsilon}(\cdot)$ projects the perturbation back
onto the Frobenius normalization ball. Then robust training optimizes the
network on adversarially perturbed input $x+\delta_{K}$. Through the above
process, we can conveniently obtain a large number of adversarial examples for
training.

By integrating the $L_{0}$ complexity regularizer into the training process of
masks, our adversarial loss objective becomes:

$\min_{m}\mathcal{L}_{adv}(m)+\lambda\mathcal{R}(m),$ (8)

where $\lambda$ denotes regularization strength.

####  3.2.2 Effect of Regularization Strength

The selection of the regularization strength $\lambda$ decides the quality of
robust tickets. Results carried on SST-2 in Fig.1 show that eventually more
than $90\%$ of the masks will be very close to $0$ or $1$, and the $L_{0}$
complexity regularizer $\mathcal{R}(m)$ will converge to a fixed value. As
$\lambda$ increases, $\mathcal{R}(m)$ decreases (the sparsity of the
subnetwork increases). The training of the adversarial loss objective in (8)
is insensitive to the $\lambda$, and in all experiments, $\lambda$ is chosen
in the range $[0.1,1]$. In the Appendix A, we show more details about mask
learning process.

###  3.3 Drawing and Retraining Winning Tickets

After training the masks $m$, we use the location parameters $\log\alpha$ of
masks to extract robust tickets. For the Gumbel-Softmax distribution in (2),
$\alpha_{i}$ is the expectation (confidence) of random variable $s_{i}$, i.e,
$\mathbb{E}\\{s_{i}\\}=\alpha_{i}$. Thus, we prune the weights whose masks
have the smallest expectation. We prune all attention heads and intermediate
neurons in an unstructured manner, which empirically has better performance
than structured pruning. Unlike the Lottery Ticket Hypothesis that requires
iterative magnitude pruning, the proposed method is a one-shot pruning method
that can obtain subnetworks of any sparsity. Then we retrain (i.e., fine-tune)
the robust tickets $f(m\odot\theta_{0})$ on downstream tasks.

###  3.4 Robust Lottery Tickets Hypothesis

In the context of adversarial robustness, we seek winning tickets that balance
accuracy and robustness, and then we state and demonstrate Robust Lottery
Tickets Hypothesis.

Figure 1: Effect of regularization strength $\lambda$ on regularizer
$\mathcal{R}(m)$, and the percentage of masks that exact $0$ and $1$.

##### Robust Lottery Tickets Hypothesis:

A pre-trained language model, such as BERT, contains some subnetworks (robust
tickets) initialized by pre-trained weights, and when these subnetworks are
trained in isolation, they can achieve better adversarial robustness and
comparable accuracy. In addition, robust tickets retain an important
characteristic of traditional lottery tickets —the ability to speed up the
training process.

The practical merits of Robust Lottery Ticket Hypothesis: 1) It provides an
effective pruning method that can reduce memory constraints during inference
time by identifying well-performing smaller networks which can fit in memory.
2) Our proposed robust ticket is more robust than the existing defense
methods, so it can be used as a defense method.

##  4 Experiments

We conduct several experiments to demonstrate the effectiveness of our method.
We first compare the proposed method with baseline methods in terms of clean
accuracy and robust evaluation. Then, we perform an ablation study to
illustrate the role of sparse mask learning and adversarial loss objective in
our method. In addition, we try to further flesh out our method with several
additional analysis experiments. Following the official BERT implementation
Devlin et al. (2019); Wolf et al. (2020), we use BERTBASE as our backbone
model for all experiments.

###  4.1 Datasets

We evaluate our method mainly on three text classification datasets: Internet
Movie Database (IMDB, Maas et al., 2011) , AG News corpus (AGNEWS, Zhang et
al., 2015) and Stanford Sentiment Treebank of binary classification (SST-2,
Socher et al., 2013). We also test our method on other types of tasks in GLUE,
such as MNLI, QNLI, QQP. The labels of GLUE test sets are not available, so
GLUE test sets cannot be used for adversarial attacks. The results of GLUE
tasks are tested on the official development set, and we divide $10\%$
training data as the development set.

###  4.2 Baselines

We compare our RobustT (Robust Tickets) with recently proposed adversarial
defense methods and the standard lottery ticket.

Fine-tune Devlin et al. (2019): The official BERT implementation on downstream
tasks. FreeLB Zhu et al. (2020): An enhanced gradient-based adversarial
training method which is not targeted at specific attack methods. InfoBERT
Wang et al. (2020): A learning framework for robust model fine-tuning from an
information-theoretic perspective. This method claims that it has obtained a
better representation of data features. LTH Chen et al. (2020): For a range of
downstream tasks, BERT contains winning lottery tickets at 40% to 90%
sparsity. Random: Subnetworks with the same layer-wise sparsity of the above
RobustT, but their structures are randomly pruned from the original BERT.

###  4.3 Robust Evaluation

Three widely accepted attack methods are used to verify the ability of our
proposed method against baselines Li et al. (2021). BERT-Attack Li et al.
(2020) is a method using BERT to generate adversarial text, and thus the
generated adversarial examples are fluent and semantically preserved.
TextFooler Jin et al. (2020) first identify the important words in the
sentences, and then replace them with synonyms that are semantically similar
and grammatically correct until the prediction changes. TextBugger Li et al.
(2018) is an adversarial attack method that generates misspelled words by
using character-level and word-level perturbations.

The evaluation metrics adopted in our experimental analyses are listed as
follows: Clean accuracy (Clean$\%$) denotes the accuracy on the clean test
dataset. Accuracy under attack (Aua$\%$) refers to the model’s prediction
accuracy facing specific adversarial attacks. Attack success rate (Suc$\%$) is
the ratio of the number of texts successfully perturbed by an attack method to
the total number of texts to be attempted. Number of Queries (#Query) is the
average number of times the attacker queries the model, which means the more
the average query number is, the harder the defense model is to be
compromised. For a robust method, higher clean accuracy, accuracy under
attack, and query times are expected, as well as lower attack success rate.

###  4.4 Implementation Details

We fine-tune the original BERT using the default settings on downstream tasks.
We train 20 epochs to discover the robust tickets from the fine-tuned BERT,
and then we retrain the robust tickets using default settings of BERT-base.
The $K$-step PGD requires $K$ forward-backward passes through the network,
which is time consuming. Thus, we turn to FreeLB, which accumulates gradients
in multiple forward passes and then passing gradients backward once. For our
approach, we prune robust tickets in the range of 10% and 90% sparsity and
report the best one in terms of robustness in our main experiments. For a fair
comparison, the sparsity of LTH is the same as that of robust tickets. All
experimental results are the average of $5$ trials with different seeds. More
implementation details and hyperparameters are provided in the Appendix B. We
implement all models in MindSpore.

###  4.5 Main Results on Robustness Evaluation

Dataset | Method | Clean$\%$ | BERT-Attack | TextFooler | TextBugger
---|---|---|---|---|---
Aua$\%$ | Suc$\%$ | #Query | Aua$\%$ | Suc$\%$ | #Query | Aua$\%$ | Suc$\%$ | #Query
IMDB | Fine-tune | $94.1$ | $7.8$ | $91.7$ | $1572.2$ | $12.2$ | $87.0$ | $1209.8$ | $25.8$ | $72.5$ | $783.2$
$\textrm{LTH}_{20\%}$ | $94.0$ | $3.6$ | $96.2$ | $1074.44$ | $7.2$ | $92.3$ | $894.1$ | $16.0$ | $83.0$ | $574.0$
FreeLB | $94.8$ | $22.6$ | $76.2$ | $1954.7$ | $27.2$ | $71.3$ | $1479.1$ | $36.0$ | $62.0$ | $907.3$
InfoBERT | $\mathbf{95.2}$ | $26.0$ | $72.7$ | $2326.0$ | $32.4$ | $66.0$ | $1572.2$ | $43.6$ | $54.2$ | $969.8$
$\textrm{Rand}_{20\%}$ | $93.1$ | $6.8$ | $92.8$ | $731.5$ | $7.4$ | $92.1$ | $598.7$ | $8.4$ | $91.9$ | $464.3$
$\textbf{RobustT}_{20\%}$ | $93.8$ | $\mathbf{55.2}$ | $\mathbf{41.2}$ | $\mathbf{3128.0}$ | $\mathbf{55.6}$ | $\mathbf{40.7}$ | $\mathbf{1988.4}$ | $\mathbf{57.6}$ | $\mathbf{38.6}$ | $\mathbf{1149.1}$
AGNEWS | Fine-tune | $94.7$ | $3.8$ | $96.0$ | $436.7$ | $14.9$ | $84.2$ | $333.2$ | $41.5$ | $56.1$ | $178.3$
$\textrm{LTH}_{40\%}$ | $93.7$ | $2.5$ | $97.3$ | $394.4$ | $11.0$ | $88.3$ | $295.2$ | $36.8$ | $60.7$ | $179.7$
FreeLB | $\mathbf{95.2}$ | $10.8$ | $88.6$ | $563.9$ | $24.3$ | $74.4$ | $394.6$ | $51.7$ | $45.5$ | $190.4$
InfoBERT | $94.4$ | $11.1$ | $88.3$ | $517.0$ | $25.1$ | $73.4$ | $374.7$ | $47.9$ | $49.3$ | $193.1$
$\textrm{Rand}_{40\%}$ | $94.0$ | $1.3$ | $98.6$ | $357.2$ | $6.3$ | $93.2$ | $275.1$ | $27.5$ | $70.1$ | $148.7$
$\textbf{RobustT}_{40\%}$ | $94.9$ | $\mathbf{12.1}$ | $\mathbf{87.2}$ | $\mathbf{607.7}$ | $\mathbf{28.5}$ | $\mathbf{70.0}$ | $\mathbf{442.1}$ | $\mathbf{53.4}$ | $\mathbf{43.7}$ | $\mathbf{207.8}$
SST-2 | Fine-tune | $92.0$ | $2.9$ | $96.8$ | $114.2$ | $5.0$ | $94.6$ | $98.4$ | $29.4$ | $68.3$ | $49.7$
$\textrm{LTH}_{30\%}$ | $92.1$ | $2.2$ | $97.6$ | $98.9$ | $4.1$ | $95.5$ | $90.5$ | $29.1$ | $68.4$ | $49.6$
FreeLB | $91.6$ | $10.2$ | $88.9$ | $154.6$ | $14.4$ | $84.2$ | $123.8$ | $\mathbf{42.4}$ | $\mathbf{53.7}$ | $\mathbf{54.9}$
InfoBERT | $\mathbf{92.1}$ | $14.4$ | $84.4$ | $162.3$ | $18.3$ | $80.1$ | $121.4$ | $40.3$ | $56.3$ | $51.2$
$\textrm{Rand}_{30\%}$ | $83.2$ | $2.1$ | $97.5$ | $89.4$ | $2.4$ | $97.1$ | $75.6$ | $16.5$ | $80.2$ | $44.2$
$\textbf{RobustT}_{30\%}$ | $90.9$ | $\mathbf{17.9}$ | $\mathbf{80.3}$ | $\mathbf{164.9}$ | $\mathbf{26.7}$ | $\mathbf{70.6}$ | $\mathbf{149.8}$ | $42.1$ | $53.7$ | $53.9$
Table 1: Main results on adversarial robustness evaluation. Fine-tuning RobustT for downstream tasks achieves a significant improvement of robustness. The percentage on the subscript denotes the sparsity of the subnetworks. The best performance is marked in bold. Suc$\%$ lower is better. Dataset | Method | Clean$\%$ | Aua$\%$
---|---|---|---
TextFooler | TextBugger
QNLI | Fine-tune | $\mathbf{91.6}$ | $4.7$ | $10.5$
FreeLB | $90.5$ | $12.8$ | $12.0$
InfoBERT | $91.5$ | $16.4$ | $20.9$
$\textbf{RobustT}_{30\%}$ | $91.5$ | $\mathbf{17.0}$ | $\mathbf{25.9}$
MNLI | Fine-tune | $\mathbf{84.4}$ | $7.7$ | $4.3$
FreeLB | $82.9$ | $11.0$ | $8.4$
InfoBERT | $84.1$ | $10.8$ | $8.4$
$\textbf{RobustT}_{30\%}$ | $84.0$ | $\mathbf{18.4}$ | $\mathbf{22.6}$
QQP | Fine-tune | $91.3$ | $24.8$ | $27.8$
FreeLB | $91.2$ | $27.4$ | $28.1$
InfoBERT | $\mathbf{91.9}$ | $34.4$ | $35.9$
$\textbf{RobustT}_{30\%}$ | $91.5$ | $\mathbf{47.2}$ | $\mathbf{46.0}$
Table 2: Adversarial robustness evaluation of RobustT on QNLI, MNLI and QQP datasets. Compare with the original BERT, fine-tuning on robust tickets improves the adversarial robustness. Dataset | Method | Clean$\%$ | Aua$\%$
---|---|---|---
IMDB | $\textbf{RobustT}_{20\%}$ | $93.8$ | $\mathbf{55.6}$
w/o Mask Leaning | $\mathbf{94.0}$ | $15.1$
w/o Adv | $93.4$ | $5.4$
AGNEWS | $\textbf{RobustT}_{40\%}$ | $\mathbf{94.9}$ | $\mathbf{28.5}$
w/o Mask Learning | $94.2$ | $16.1$
w/o Adv | $94.5$ | $8.8$
SST-2 | $\textbf{RobustT}_{30\%}$ | $90.9$ | $\mathbf{26.7}$
w/o Mask Learning | $\mathbf{92.2}$ | $6.2$
w/o Adv | $91.2$ | $3.5$
Table 3: Ablation study on text classification datasets. Aua$\%$ is obtained
after using TextFooler attack.

Table 1 shows the results of robust tickets and other baselines under
adversarial attack. We can observe that: 1) Original BERT and BERT-tickets
fail to perform well on adversarial robustness evaluation, and the BERT-
tickets even show lower robustness than BERT, indicating that it is difficult
for the pruned subnetworks to fight against adversarial attacks when only test
accuracy is considered. This result is consistent with the results in Du et
al. (2021); Xu et al. (2021). 2) The proposed robust ticket achieves a
significant improvement of robustness over the original BERT and other
adversarial defense methods. Robust tickets use a better robust structure to
resist adversarial attacks, which is different from the previous methods aimed
at solving robust optimization problems. 3) In both AGNEWS and IMDB, the
randomly pruned subnetwork loses only about $1$ performance point in test
accuracy, but performs poorly in adversarial robustness. This suggests that
robust tickets are more difficult to discovered than traditional lottery
tickets. 4) Robust tickets sacrifice accuracy performance in SST-2 and IMDB.
We speculate that this may be due to the trade-off between accuracy and
robustness Tsipras et al. (2019).

We also evaluate the performance of our proposed method on more tasks. From
Table 2, we can see that our proposed method yields significant improvements
of robustness over the original BERT on QNLI, MNLI and QQP datasets. There is
a significant improvement even compared with InfoBERT and FreeLB.

###  4.6 Ablation Study

(a) IMDB

(b) AGNEWS

(c) SST-2

Figure 2: Fine-tuning evaluation results of the robust ticket, the traditional
lottery ticket, FreeLB and the original BERT fine-tuning under various
sparsity levels. The adversarial robustness improves as the compression ratio
grows until a certain threshold, then the robustness deteriorates. Aua$\%$ is
obtained after using TextFooler attack.

To better illustrate the contribution of each component of our method, we
perform the ablation study by removing the following components: sparse mask
learning (but with IMP instead) and adversarial loss objective (Adv). The test
results are shown in Table 3. We can observe that: 1) Mask learning is
important for performance and IMP does not identify robust subnetworks well
Vemparala et al. (2021). 2) Without adversarial loss objective, the proposed
method identifies subnetworks that perform well in terms of clean accuracy,
but does not provide any improvement in terms of robustness.

##  5 Discussion

In this section, we study how the implementation of robust tickets affects the
model’s robustness.

###  5.1 Impact of Sparsity on Robust Tickets

(a) IMDB

(b) SST-2

(c) AGNEWS

(d) MNLI

(e) QNLI

(f) QQP

Figure 3: Heatmaps of sparsity patterns found on different tasks, each cell
gives the percentage of surviving weights in self-attention heads and MLPs.
The sparsity patterns on IMDB and SST-2 are similar, which may be due to the
fact that they are both text classification datasets based on movie reviews.

The proposed method can prune out a subnetwork with arbitrary sparsity based
on the confidence of masks. In Fig.2, we compare the robust tickets and
traditional lottery tickets across all sparsities. When the sparsity increases
to a certain level, the robustness decreases faster than the accuracy, which
indicates that the robustness is more likely to be affected by the model
structure than the accuracy. Therefore, it is more difficult to find a robust
ticket from BERT. The accuracy of the subnetwork is slowly decreasing with
increasing sparsity, but the robustness shows a different trend. The change in
robustness can be roughly divided into three phases: The robustness improves
as the sparsity grows until a certain threshold; beyond this threshold, the
robustness deteriorates but is still better than that of the lottery tickets.
In the end, when being highly compressed, the robust network collapses into a
lottery network. A similar phenomenon is also be observed Liang et al. (2021).
The robustness performance curve is not as smooth as the accuracy, this may be
due to the gap between the adversarial loss objective and the real textual
attacks.

###  5.2 Sparsity Pattern

Fig.3 shows the sparsity patterns of robust tickets on all six datasets. We
can clearly find that the pruning rate increases from bottom to top on the
text classification tasks (IMDB, SST2, AGNEWS), while it is more uniform in
the natural language inference tasks (MNLI and QNLI) and Quora question pairs
(QQP). Recent works show that BERT encodes a rich hierarchy of linguistic
information. Taking the advantage of the probing task, Jawahar et al. (2019)
indicate that the surface information features are encoded at the bottom,
syntactic information features are in the middle network, and semantic
information features in the top. Therefore, we speculate that the sparsity
pattern of robust tickets is task-dependent.

(a) IMDB

(b) AGNEWS

(c) SST-2

Figure 4: Clean accuracy and accuracy under attack as training proceeds.
Robust tickets accelerate both accuracy and robustness. Aua$\%$ is obtained
after using TextFooler attack.

###  5.3 Speedup Training Process

An important property of winning tickets is to accelerate the convergence of
the training process Chen et al. (2021); You et al. (2020). The training curve
in Fig.4 shows that the convergence speed of robust tickets is much faster
compared with the default fine-tuning and FreeLB. Moreover, the convergence
rate of both accuracy and robustness is accelerating. The traditional lottery
tickets converge faster than our method, which may be due to the fact that
robust tickets require maintaining a trade-off between robustness and
accuracy.

###  5.4 The Importance of Robust Tickets Initialization and Structure

To better understand which factor, initialization or structure, has a greater
impact on the robust ticket, we conduct corresponding analysis studies. We
avoid the effect of initializations by re-initializing the weights of robust
tickets. To avoid the effect of structures and preserve the effect of
initializations, we use the full BERT and re-initialize the weights that are
not contained in the robust tickets. Aua$\%$ is obtained after using
TextFooler attack. The results are shown in Table 4.

Dataset | Method | Clean$\%$ | Aua$\%$
---|---|---|---
IMDB | $\textbf{RobustT}_{20\%}$ | $\mathbf{93.7}$ | $\mathbf{55.6}$
w/o Initialization | $87.9$ | $0.2$
w/o Structure | $93.7$ | $13.4$
w/o Structure+Longer | $93.6$ | $18.6$
AGNEWS | $\textbf{RobustT}_{40\%}$ | $\mathbf{94.9}$ | $\mathbf{28.5}$
w/o Initialization | $92.4$ | $0.4$
w/o Structure | $94.9$ | $21.8$
w/o Structure+Longer | $94.8$ | $24.6$
SST-2 | $\textbf{RobustT}_{30\%}$ | $90.9$ | $26.7$
w/o Initialization | $83.1$ | $2.1$
w/o Structure | $\mathbf{92.0}$ | $15.7$
w/o Structure+Longer | $91.9$ | $\mathbf{27.5}$
Table 4: Importance of robust ticket initialization and structure. Our results
show that the initialization of robust tickets seems to be more important than
the structure, although both of them play a role.

####  5.4.1 Importance of initialization

LTH suggests that the winning tickets can not be learned effectively without
its original initialization. For our robust BERT tickets, their
initializations are pre-trained weights. Table 4 shows the failure of robust
tickets when the random re-initialization is performed.

####  5.4.2 Importance of structure

Frankle and Carbin (2019) hypothesize that the structure of winning tickets
encodes an inductive bias customized for the learning task at hand. Although
removing this inductive bias reduces performance compared to the robust
tickets, it still outperforms the original BERT, and its performance improves
further with longer training time (3 epochs $\rightarrow$ 10 epochs). It can
be seen that the initializations of some pre-training weights may lead to a
decrease in the robustness of the model.

##  6 Conclusion

In this paper, we articulate and demonstrate the Robust Lottery Ticket
Hypothesis for PLMs: the full PLM contains subnetworks (robust tickets) that
can achieve a better robustness performance. We propose an effective method to
solve the ticket selection problem by encouraging weights that are not
responsible for robustness to become exactly zero. Experiments on various
tasks corroborate the effectiveness of our method. We also find that pre-
trained weights may be a key factor affecting the robustness on downstream
tasks.

## Acknowledgements

The authors wish to thank the anonymous reviewers for their helpful comments.
This work was partially funded by National Natural Science Foundation of China
(No. 62076069, 61976056). This research was supported by Meituan, Beijing
Academy of Artificial Intelligence(BAAI), and CAAI-Huawei MindSpore Open Fund.

## References

  * Bengio et al. (2013) Yoshua Bengio, Nicholas Léonard, and Aaron C. Courville. 2013.  Estimating or propagating gradients through stochastic neurons for conditional computation.  _ArXiv preprint_ , abs/1308.3432.
  * Chen et al. (2020) Tianlong Chen, Jonathan Frankle, Shiyu Chang, Sijia Liu, Yang Zhang, Zhangyang Wang, and Michael Carbin. 2020.  The lottery ticket hypothesis for pre-trained BERT networks.  In _Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurIPS 2020, December 6-12, 2020, virtual_.
  * Chen et al. (2021) Xiaohan Chen, Yu Cheng, Shuohang Wang, Zhe Gan, Zhangyang Wang, and Jingjing Liu. 2021.  EarlyBERT: Efficient BERT training via early-bird lottery tickets.  In _Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers)_ , pages 2195–2207, Online. Association for Computational Linguistics.
  * Devlin et al. (2019) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2019.  BERT: Pre-training of deep bidirectional transformers for language understanding.  In _Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers)_ , pages 4171–4186, Minneapolis, Minnesota. Association for Computational Linguistics.
  * Du et al. (2021) Mengnan Du, Subhabrata Mukherjee, Yu Cheng, Milad Shokouhi, Xia Hu, and Ahmed Hassan Awadallah. 2021.  What do compressed large language models forget? robustness challenges in model compression.  _ArXiv preprint_ , abs/2110.08419.
  * Evci et al. (2020) Utku Evci, Trevor Gale, Jacob Menick, Pablo Samuel Castro, and Erich Elsen. 2020\.  Rigging the lottery: Making all tickets winners.  In _Proceedings of the 37th International Conference on Machine Learning, ICML 2020, 13-18 July 2020, Virtual Event_ , volume 119 of _Proceedings of Machine Learning Research_ , pages 2943–2952. PMLR.
  * Frankle and Carbin (2019) Jonathan Frankle and Michael Carbin. 2019.  The lottery ticket hypothesis: Finding sparse, trainable neural networks.  In _7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019_. OpenReview.net.
  * Gao et al. (2018) Ji Gao, Jack Lanchantin, Mary Lou Soffa, and Yanjun Qi. 2018.  Black-box generation of adversarial text sequences to evade deep learning classifiers.  _2018 IEEE Security and Privacy Workshops (SPW)_ , pages 50–56.
  * Garg and Ramakrishnan (2020) Siddhant Garg and Goutham Ramakrishnan. 2020.  BAE: BERT-based adversarial examples for text classification.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , pages 6174–6181, Online. Association for Computational Linguistics.
  * Gui et al. (2019) Shupeng Gui, Haotao Wang, Haichuan Yang, Chen Yu, Zhangyang Wang, and Ji Liu. 2019\.  Model compression with adversarial robustness: A unified optimization framework.  In _Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information Processing Systems 2019, NeurIPS 2019, December 8-14, 2019, Vancouver, BC, Canada_ , pages 1283–1294.
  * Jang et al. (2017) Eric Jang, Shixiang Gu, and Ben Poole. 2017.  Categorical reparameterization with gumbel-softmax.  In _5th International Conference on Learning Representations, ICLR 2017, Toulon, France, April 24-26, 2017, Conference Track Proceedings_. OpenReview.net.
  * Jawahar et al. (2019) Ganesh Jawahar, Beno\̂mathbf{i}t Sagot, and Djamé Seddah. 2019.  What does BERT learn about the structure of language? In _Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics_ , pages 3651–3657, Florence, Italy. Association for Computational Linguistics.
  * Jiang et al. (2020) Haoming Jiang, Pengcheng He, Weizhu Chen, Xiaodong Liu, Jianfeng Gao, and Tuo Zhao. 2020.  SMART: Robust and efficient fine-tuning for pre-trained natural language models through principled regularized optimization.  In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pages 2177–2190, Online. Association for Computational Linguistics.
  * Jin et al. (2020) Di Jin, Zhijing Jin, Joey Tianyi Zhou, and Peter Szolovits. 2020.  Is BERT really robust? A strong baseline for natural language attack on text classification and entailment.  In _The Thirty-Fourth AAAI Conference on Artificial Intelligence, AAAI 2020, The Thirty-Second Innovative Applications of Artificial Intelligence Conference, IAAI 2020, The Tenth AAAI Symposium on Educational Advances in Artificial Intelligence, EAAI 2020, New York, NY, USA, February 7-12, 2020_ , pages 8018–8025. AAAI Press.
  * Kovaleva et al. (2019) Olga Kovaleva, Alexey Romanov, Anna Rogers, and Anna Rumshisky. 2019.  Revealing the dark secrets of BERT.  In _Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)_ , pages 4365–4374, Hong Kong, China. Association for Computational Linguistics.
  * Li et al. (2018) Jinfeng Li, Shouling Ji, Tianyu Du, Bo Li, and Ting Wang. 2018.  Textbugger: Generating adversarial text against real-world applications.  _ArXiv preprint_ , abs/1812.05271.
  * Li et al. (2020) Linyang Li, Ruotian Ma, Qipeng Guo, Xiangyang Xue, and Xipeng Qiu. 2020.  BERT-ATTACK: Adversarial attack against BERT using BERT.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , pages 6193–6202, Online. Association for Computational Linguistics.
  * Li and Qiu (2020) Linyang Li and Xipeng Qiu. 2020.  Tavat: Token-aware virtual adversarial training for language understanding.  _ArXiv preprint_ , abs/2004.14543.
  * Li et al. (2021) Zongyi Li, Jianhan Xu, Jiehang Zeng, Linyang Li, Xiaoqing Zheng, Qi Zhang, Kai-Wei Chang, and Cho-Jui Hsieh. 2021.  Searching for an effective defender: Benchmarking defense against adversarial word substitution.  In _Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing_ , pages 3137–3147, Online and Punta Cana, Dominican Republic. Association for Computational Linguistics.
  * Liang et al. (2021) Chen Liang, Simiao Zuo, Minshuo Chen, Haoming Jiang, Xiaodong Liu, Pengcheng He, Tuo Zhao, and Weizhu Chen. 2021.  Super tickets in pre-trained language models: From model compression to improving generalization.  In _Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers)_ , pages 6524–6538, Online. Association for Computational Linguistics.
  * Liu et al. (2019) Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov. 2019.  Roberta: A robustly optimized bert pretraining approach.  _ArXiv preprint_ , abs/1907.11692.
  * Loshchilov and Hutter (2019) Ilya Loshchilov and Frank Hutter. 2019.  Decoupled weight decay regularization.  In _7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019_. OpenReview.net.
  * Louizos et al. (2018) Christos Louizos, Max Welling, and Diederik P. Kingma. 2018.  Learning sparse neural networks through l_0 regularization.  In _6th International Conference on Learning Representations, ICLR 2018, Vancouver, BC, Canada, April 30 - May 3, 2018, Conference Track Proceedings_. OpenReview.net.
  * Maas et al. (2011) Andrew L. Maas, Raymond E. Daly, Peter T. Pham, Dan Huang, Andrew Y. Ng, and Christopher Potts. 2011.  Learning word vectors for sentiment analysis.  In _Proceedings of the 49th Annual Meeting of the Association for Computational Linguistics: Human Language Technologies_ , pages 142–150, Portland, Oregon, USA. Association for Computational Linguistics.
  * Maddison et al. (2017) Chris J. Maddison, Andriy Mnih, and Yee Whye Teh. 2017.  The concrete distribution: A continuous relaxation of discrete random variables.  In _5th International Conference on Learning Representations, ICLR 2017, Toulon, France, April 24-26, 2017, Conference Track Proceedings_. OpenReview.net.
  * Madry et al. (2018) Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and Adrian Vladu. 2018.  Towards deep learning models resistant to adversarial attacks.  In _6th International Conference on Learning Representations, ICLR 2018, Vancouver, BC, Canada, April 30 - May 3, 2018, Conference Track Proceedings_. OpenReview.net.
  * Michel et al. (2019) Paul Michel, Omer Levy, and Graham Neubig. 2019.  Are sixteen heads really better than one?  In _Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information Processing Systems 2019, NeurIPS 2019, December 8-14, 2019, Vancouver, BC, Canada_ , pages 14014–14024.
  * Morris et al. (2020) John Morris, Eli Lifland, Jin Yong Yoo, Jake Grigsby, Di Jin, and Yanjun Qi. 2020\.  TextAttack: A framework for adversarial attacks, data augmentation, and adversarial training in NLP.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing: System Demonstrations_ , pages 119–126, Online. Association for Computational Linguistics.
  * Prasanna et al. (2020) Sai Prasanna, Anna Rogers, and Anna Rumshisky. 2020.  When BERT Plays the Lottery, All Tickets Are Winning.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , pages 3208–3229, Online. Association for Computational Linguistics.
  * Raffel et al. (2019) Colin Raffel, Noam M. Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J. Liu. 2019.  Exploring the limits of transfer learning with a unified text-to-text transformer.  _ArXiv preprint_ , abs/1910.10683.
  * Ren et al. (2019) Shuhuai Ren, Yihe Deng, Kun He, and Wanxiang Che. 2019.  Generating natural language adversarial examples through probability weighted word saliency.  In _Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics_ , pages 1085–1097, Florence, Italy. Association for Computational Linguistics.
  * Renda et al. (2020) Alex Renda, Jonathan Frankle, and Michael Carbin. 2020.  Comparing rewinding and fine-tuning in neural network pruning.  In _8th International Conference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020_. OpenReview.net.
  * Rogers et al. (2020) Anna Rogers, Olga Kovaleva, and Anna Rumshisky. 2020.  A primer in BERTology: What we know about how BERT works.  _Transactions of the Association for Computational Linguistics_ , 8:842–866.
  * Sehwag et al. (2020) Vikash Sehwag, Shiqi Wang, Prateek Mittal, and Suman Jana. 2020.  Hydra: Pruning adversarially robust neural networks.  In _Advances in Neural Information Processing Systems_ , volume 33, pages 19655–19666. Curran Associates, Inc.
  * Sehwag et al. (2019) Vikash Sehwag, Shiqi Wang, Prateek Mittal, and Suman Sekhar Jana. 2019.  Towards compact and robust deep neural networks.  _ArXiv preprint_ , abs/1906.06110.
  * Si et al. (2021) Chenglei Si, Zhengyan Zhang, Fanchao Qi, Zhiyuan Liu, Yasheng Wang, Qun Liu, and Maosong Sun. 2021.  Better robustness by more coverage: Adversarial and mixup data augmentation for robust finetuning.  In _Findings of the Association for Computational Linguistics: ACL-IJCNLP 2021_ , pages 1569–1576, Online. Association for Computational Linguistics.
  * Socher et al. (2013) Richard Socher, Alex Perelygin, Jean Wu, Jason Chuang, Christopher D. Manning, Andrew Ng, and Christopher Potts. 2013.  Recursive deep models for semantic compositionality over a sentiment treebank.  In _Proceedings of the 2013 Conference on Empirical Methods in Natural Language Processing_ , pages 1631–1642, Seattle, Washington, USA. Association for Computational Linguistics.
  * Tsipras et al. (2019) Dimitris Tsipras, Shibani Santurkar, Logan Engstrom, Alexander Turner, and Aleksander Madry. 2019.  Robustness may be at odds with accuracy.  In _7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019_. OpenReview.net.
  * Vemparala et al. (2021) Manoj Rohit Vemparala, Nael Fasfous, Alexander Frickenstein, Sreetama Sarkar, Qi Zhao, Sabine Kuhn, Lukas Frickenstein, Anmol Singh, Christian Unger, Naveen Shankar Nagaraja, Christian Wressnegger, and Walter Stechele. 2021.  Adversarial robust model compression using in-train pruning.  _2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW)_ , pages 66–75.
  * Wang et al. (2020) Boxin Wang, Shuohang Wang, Yu Cheng, Zhe Gan, Ruoxi Jia, Bo Li, and Jingjing Liu. 2020.  Infobert: Improving robustness of language models from an information theoretic perspective.  In _International Conference on Learning Representations_.
  * Wang et al. (2021) Xiao Wang, Qin Liu, Tao Gui, Qi Zhang, Yicheng Zou, Xin Zhou, Jiacheng Ye, Yongxin Zhang, Rui Zheng, Zexiong Pang, Qinzhuo Wu, Zhengyan Li, Chong Zhang, Ruotian Ma, Zichu Fei, Ruijian Cai, Jun Zhao, Xingwu Hu, Zhiheng Yan, Yiding Tan, Yuan Hu, Qiyuan Bian, Zhihua Liu, Shan Qin, Bolin Zhu, Xiaoyu Xing, Jinlan Fu, Yue Zhang, Minlong Peng, Xiaoqing Zheng, Yaqian Zhou, Zhongyu Wei, Xipeng Qiu, and Xuanjing Huang. 2021.  TextFlint: Unified multilingual robustness evaluation toolkit for natural language processing.  In _Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing: System Demonstrations_ , pages 347–355, Online. Association for Computational Linguistics.
  * Wolf et al. (2020) Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, Rémi Louf, Morgan Funtowicz, Joe Davison, Sam Shleifer, Patrick von Platen, Clara Ma, Yacine Jernite, Julien Plu, Canwen Xu, Teven Le Scao, Sylvain Gugger, Mariama Drame, Quentin Lhoest, and Alexander M. Rush. 2020.  Huggingface’s transformers: State-of-the-art natural language processing.
  * Xu et al. (2021) Canwen Xu, Wangchunshu Zhou, Tao Ge, Ke Xu, Julian McAuley, and Furu Wei. 2021.  Beyond preserved accuracy: Evaluating loyalty and robustness of BERT compression.  In _Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing_ , pages 10653–10659, Online and Punta Cana, Dominican Republic. Association for Computational Linguistics.
  * Ye et al. (2019) Shaokai Ye, Xue Lin, Kaidi Xu, Sijia Liu, Hao Cheng, Jan-Henrik Lambrechts, Huan Zhang, Aojun Zhou, Kaisheng Ma, and Yanzhi Wang. 2019.  Adversarial robustness vs. model compression, or both?  In _2019 IEEE/CVF International Conference on Computer Vision, ICCV 2019, Seoul, Korea (South), October 27 - November 2, 2019_ , pages 111–120. IEEE.
  * You et al. (2020) Haoran You, Chaojian Li, Pengfei Xu, Yonggan Fu, Yue Wang, Xiaohan Chen, Yingyan Lin, Zhangyang Wang, and Richard G. Baraniuk. 2020.  Drawing early-bird tickets: Toward more efficient training of deep networks.  In _International Conference on Learning Representations_.
  * Yu et al. (2020) Haonan Yu, Sergey Edunov, Yuandong Tian, and Ari S. Morcos. 2020.  Playing the lottery with rewards and multiple languages: lottery tickets in RL and NLP.  In _8th International Conference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020_. OpenReview.net.
  * Zang et al. (2020) Yuan Zang, Fanchao Qi, Chenghao Yang, Zhiyuan Liu, Meng Zhang, Qun Liu, and Maosong Sun. 2020.  Word-level textual adversarial attacking as combinatorial optimization.  In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pages 6066–6080, Online. Association for Computational Linguistics.
  * Zhang et al. (2021) Dinghuai Zhang, Kartik Ahuja, Yilun Xu, Yisen Wang, and Aaron C. Courville. 2021\.  Can subnetwork structure be the key to out-of-distribution generalization?  In _ICML_.
  * Zhang et al. (2015) Xiang Zhang, Junbo Jake Zhao, and Yann LeCun. 2015.  Character-level convolutional networks for text classification.  In _Advances in Neural Information Processing Systems 28: Annual Conference on Neural Information Processing Systems 2015, December 7-12, 2015, Montreal, Quebec, Canada_ , pages 649–657.
  * Zhu et al. (2020) Chen Zhu, Yu Cheng, Zhe Gan, Siqi Sun, Tom Goldstein, and Jingjing Liu. 2020.  Freelb: Enhanced adversarial training for natural language understanding.  In _8th International Conference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020_. OpenReview.net.

##  Appendix A The Effect of Regularization Strength during Mask Learning

(a) IMDB

(b) AGNEWS

(c) QNLI

(d) QQP

Figure 5: Effect of regularization strength during mask learning.

In section 3.2.2, we show the mask learning curves for various regularization
strengths $\lambda$ in SST-2 dataset. The results on more datasets are shown
in the Fig.5, where we can observe that the mask learning process is
insensitive to the regularization strength, and the convergence of masks is
eventually achieved.

##  Appendix B Implementation Details

###  B.1 Details for Fine-tuning Models

We report the hyperparameters used for fine-tuning the BERT-base and
retraining the winning tickets in table 5.

Hypeparameters | Values
---|---
Optimizer | AdamwLoshchilov and Hutter (2019)
Learning rate | $2\times 10^{-5}$
Dropout | $0.1$
Weight decay | $1\times 10^{-2}$
Batch size |  $16$ or $32$
Gradient clip | $(-1,1)$
Epochs | $3$
Bias-correction | True
Table 5: Hyperparameters used for fine-tuning the BERT-base and retraining the
winning tickets.

###  B.2 Details for Adversarial Attack

We use textattack Morris et al. (2020) to implement the adversarial attack
methods. For all attack methods, we use the default parameters of third-party
libraries. Adversarial robustness evaluation metrics (e.g., Aua$\%$ and
#Query) are evaluated on the all $872$ test samples for SST-2, $500$ randomly
selected test samples for IMDB, and $1000$ randomly selected test samples for
other datasets.

###  B.3 Hyperparameters

Adversarial loss objective introduces four widely used hyperparameters: the
perturbation step size $\eta$, the initial magnitude of perturbations
$\epsilon_{0}$, the number of adversarial steps $s$, and we do not constrain
the bound of perturbations. In addition, we also report two important
hyperparameters during mask learning. They are mask learning rate $\gamma$ and
regularization penalty coefficient $\lambda$. The weight decay $wd$ in the
optimizer are also changed compared with default settings to make the mask
sparsity rate converge better. We list the hyperparameters used for each tasks
in Table 6.

Datasets | $\eta$ | $\gamma$ | $\lambda$ | $\epsilon_{0}$ | $s$ | $wd$
---|---|---|---|---|---|---
SST2 | $0.03$ | $0.1$ | $0.5$ | $0.05$ | $5$ | $1e-6$
AGNEWS | $0.03$ | $0.05$ | $0.5$ | $0.05$ | $5$ | $1e-6$
IMDB | $0.03$ | $0.1$ | $0.5$ | $0.05$ | $5$ | $1e-6$
QQP | $0.04$ | $0.05$ | $0.1$ | $0.05$ | $3$ | $1e-6$
QNLI | $0.04$ | $0.05$ | $0.1$ | $0.05$ | $3$ | $1e-6$
MNLI | $0.2$ | $0.1$ | $0.1$ | $0.05$ | $2$ | $1e-6$
Table 6: Hyperparameters used during mask learning.


# New holographic dark energy model inspired by the DGP braneworld

A. Sheykhi,1,2, M. H. Dehghani 1,2 and S. Ghaffari1
asheykhi<EMAIL_ADDRESS>1 Physics Department and Biruni
Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran
2 Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O.
Box 55134-441, Maragha, Iran

###### Abstract

The energy density of the holographic dark energy is based on the area law of
entropy, and thus any modification of the area law leads to a modified
holographic energy density. Inspired by the entropy expression associated with
the apparent horizon of a Friedmann-Robertson-Walker (FRW) Universe in DGP
braneworld, we propose a new model for the holographic dark energy in the
framework of DGP brane cosmology. We investigate the cosmological consequences
of this new model and calculate the equation of state parameter by choosing
the Hubble radius, $L=H^{-1}$, as the system’s IR cutoff. Our study show that,
due to the effects of the extra dimension (bulk), the identification of IR-
cutoff with Hubble radius, can reproduce the present acceleration of the
Universe expansion. This is in contrast to the ordinary holographic dark
energy in standard cosmology which leads to the zero equation of state
parameter in the case of choosing the Hubble radius as system’s IR cutoff in
the absence of interaction between dark matter and dark energy.

##  I Introduction

Nowadays there are many strong evidences, on the observational side, that our
Universe is experiencing a phase of accelerated expansion likely driven by
some unknown energy component usually dubbed “dark energy” (DE) Riess . Recent
astronomical observations indicate that more than $70$ percent of the Universe
consists of DE with negative pressure Riess . Disclosing the nature and origin
of such a DE has been one of the biggest challenges of the modern cosmology.
Many theoretical candidates have been proposed as DE. Among them, those which
originate from fundamental theory such as quantum gravity has arisen a lot of
enthusiasm, recently. For a comprehensive review on DE models see Li2011 .

One of the dramatic candidate for DE, that arose a lot of enthusiasm recently,
is the so-called “holographic dark energy” (HDE) proposal (see Hsu ; Li ;
Wang1 ; Wang2 ; cop ; Pav`on ; Zim ; shey0 ; shey1 ; shey2  and references
therein). This model is based on the holographic principle which states that
the number of degrees of freedom of a physical system should scale with its
bounding area rather than with its volume Suss1  and it should be constrained
by an infrared cutoff Cohen . It was shown that in quantum field theory, the
UV cutoff $\Lambda$ should be related to the IR cutoff $L$ due to the limit
set of forming a black hole Cohen . If $\rho_{D}=\Lambda^{4}$ is the vacuum
energy density caused by UV cutoff, the total energy of size $L$ should not
exceed the mass of the system-size black hole

$E_{D}\leq E_{BH}\rightarrow L^{3}\rho_{D}\leq M_{p}^{2}L.$ (1)

If the largest cutoff $L$ is taken for saturating this inequality, we get the
energy density of HDE as Li

$\rho_{D}=\frac{3c^{2}M_{p}^{2}}{L^{2}}=\frac{3c^{2}}{8\pi GL^{2}}.$ (2)

The HDE model has been investigated widely in the literature and has also been
tested and constrained by various astronomical observation Xin ; Feng . It is
fair to claim that simplicity and reasonability of HDE model provides more
reliable frame to investigate the problem of DE rather than other models
proposed in the literature. It is worth mentioning that in the derivation of
HDE density (2), the black hole entropy $S$ plays a crucial role Cohen .
Indeed, the definition and derivation of holographic energy density depend on
the entropy-area relation $S\sim A\sim L^{2}$ of black holes, where $A$
represents the area of the horizon Cohen . Any modification of the black holes
entropy due to the quantum correction Sau  or extra dimension such as in
braneworld scenarios Sheykhi1 ; Sheykhi2  will affect directly on the
definition of the energy density of the HDE and leads to new cosmological
consequences.

The profound connection between thermodynamics and gravity has now well
established through a numerous and complementary theoretical investigations
Padm . It has been shown that the gravitational field equations in a wide
range of theories can be recast as the first law of thermodynamics on the
boundary of spacetime Jac ; CaiKim ; Par ; Pad ; Cai0 ; Cai1 ; Cai2 ; Cai3 ;
ShHL ; CaiHL . The studies were also generalized to the brane cosmology, where
it was shown that the differential form of the Friedmann equation on the brane
can be transformed to the first law of thermodynamics on the apparent horizon
CaiCao ; Nozari . This procedure leads to extract an expression for the
apparent horizon entropy in braneworld scenarios which is useful in studying
black holes physics on the brane Sheykhi1 ; Sheykhi2 . These results indicate
the holographic properties of the gravitational field equations in a wide
range of gravity theories. The deep connection between the gravitational
equation describing the gravity in the bulk and the first law of
thermodynamics on the apparent horizon reflects some deep ideas of holography.

According to the braneworld scenario, all matter fields in standard model of
particle physics are confined to a brane embedded in a higher-dimensional
bulk, while the gravitational field, in contrast, is usually considered to
live in the bulk spacetime. A simple and well studied version of braneword
model was proposed by Dvali-Gabadadze-Porrati (DGP) DGP ; Def1 ; Def2 . In
this model our four-dimensional Universe is a FRW brane embedded in a five-
dimensional Minkowski bulk. It is important to note that the self-accelerating
DGP solution has ghost instabilities and it is impossible to realize phantom
divide crossing in this branch of solutions without adding extra component(s).
To realize phantom divide crossing in the self-accelerating branch, it is
necessary to add at least a component as DE on the brane. On the other hand,
the normal DGP branch cannot explain late time cosmic speed-up, but it has the
potential to realize a phantom-like phase by dynamical screening of the brane
cosmological constant. Adding a DE component to the normal branch solution
brings new facilities to explain late time acceleration and also better
matching with observations. These are the motivations to add DE to this
braneworld setup Nozari . In this paper we would like to propose a new
modified HDE model in the context of the DGP braneworld.

The structure of this paper is as follows. In the next section, we first show
that the corresponding Friedmann equation of a flat FRW Universe in DGP
braneworld, can be rewritten as the first law of thermodynamics on the
apparent horizon. This procedure allows us to derive the entropy expression
associated with the apparent horizon in brane cosmology. Then, having the
entropy expression at hand, we propose a new modified HDE model in DGP
braneworld. In section III, we consider the Hubble radius as IR cutoff and
study cosmological implications of the proposed model. In section IV, we
extend the study to the case where there is an interaction between the two
dark components of the Universe. The last section is devoted to conclusions
and discussions.

##  II The model

We consider a homogeneous and isotropic FRW Universe on the brane which is
described by the line element

${\rm d}s^{2}=-{\rm d}t^{2}+a^{2}(t)\left(\frac{{\rm
d}r^{2}}{1-kr^{2}}+r^{2}{\rm d}\Omega^{2}\right),$ (3)

where $k=0,1,-1$ represent a flat, closed and open maximally symmetric space
on the brane, respectively. Using the spherical symmetry, we can rewrite the
metric as

$ds^{2}=h_{ab}dx^{a}dx^{b}+\tilde{r}^{2}d\Omega^{2},$ (4)

where $\tilde{r}=a(t)r$, $x^{0}=t$, $x^{1}=r$ and $h_{ab}$= diag
$\left(-1,\,\frac{a^{2}}{1-kr^{2}}\right)$ is the two dimensional sub-
manifold. The dynamical apparent horizon is determined by equation
$h^{ab}\partial_{a}\tilde{r}\partial_{b}\tilde{r}=0$, which implies that the
vector $\nabla\tilde{r}$ is null on the apparent horizon surface Bak ; Hayward
. The explicit evaluation of the apparent horizon for the FRW Universe gives
the apparent horizon radius

$\tilde{r}_{A}={1}/\sqrt{H^{2}+{k}/{a^{2}}}.$ (5)

where $H=\dot{a}/{a}$ is the Hubble parameter. The associated temperature
$T=|\kappa|/2\pi$ with the apparent horizon is defined through the surface
gravity

$\kappa=\frac{1}{2\sqrt{-h}}\partial_{a}\Big{(}\sqrt{-h}h^{ab}\partial_{b}\tilde{r}\Big{)},$
(6)

The explicit evolution of the surface gravity at the apparent horizon of FRW
Universe reads

$\kappa=-\frac{1}{\tilde{r}_{A}}\bigg{(}1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\bigg{)}.$
(7)

Thus the temperature associated with the apparent horizon is given by

$T=\frac{1}{2\pi\tilde{r}_{A}}\bigg{(}1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\bigg{)}.$
(8)

where we assume $\dot{\tilde{r}}_{A}<2H\tilde{r}_{A}$. It was shown that there
is indeed a Hawking radiation associated with the apparent horizon cao  which
gives more solid physical implication of the temperature associated with the
apparent horizon.

We are going to rewrite the Friedmann equation in the form of the first law of
thermodynamics at apparent horizon. The Friedmann equation in DGP braneworld
has the following form Def1

$H^{2}+\frac{k}{a^{2}}=\left(\sqrt{\frac{\rho}{3M_{\rm
p}^{2}}+\frac{1}{4r^{2}_{c}}}+\frac{\epsilon}{2r_{c}}\right)^{2},$ (9)

where $\epsilon=\pm 1$ corresponds to the two branches of solutions Def1 , and
$\rho=\rho_{m}+\rho_{D}$ is the total energy density of the fluid on the brane
where $\rho_{D}$ and $\rho_{m}$ are energy density of dark matter (DM) and DE
on the brane, respectively. In the above equation $r_{c}$ is the crossover
length scale between the small and large distances given by

$r_{c}=\frac{M_{\rm p}^{2}}{2M_{5}^{3}}=\frac{G_{5}}{2G_{4}}.$ (10)

The $\epsilon=+1$ is the self-accelerating solution in which the Universe may
enter an accelerating phase in late time without presence of additional DE
component. The $\epsilon=-1$ branch has been named as the normal branch, where
acceleration only appears if the DE component is included on the brane. For
$r_{c}\gg 1$, the Friedmann equation in standard cosmology is recovered,

$H^{2}+\frac{k}{a^{2}}=\frac{\rho}{3M_{\rm p}^{2}}.$ (11)

Recent observations indicate that our Universe is spatially flat. For a flat
FRW universe on the brane, Eq. (9) reduces to

$H^{2}-\frac{\epsilon}{r_{c}}H=\frac{\rho}{3M_{\rm p}^{2}}.$ (12)

As we mentioned for $\epsilon=+1$ and in the absence of any kind of energy or
matter field on the brane ($\rho=0$), there is a de Sitter solution for Eq.
(12) with constant Hubble parameter

$H=\frac{1}{r_{c}}\Rightarrow a(t)=a_{0}e^{\frac{t}{r_{c}}},$ (13)

which leads to an accelerating Universe with constant equation of state
parameter, $w_{D}=-1$, exactly like cosmological constant. However, this
solution suffers some unsatisfactory problems. First of all, it suffers the
well-known cosmological constant problems namely, the fine-tuning and the
coincidence problems. Besides, it leads to a constant $w_{D}$, while many
cosmological evidences, especially the analysis of the type Ia supernova data
indicates that the time varying DE gives a better fit than a cosmological
constant Planck1 ; Planck2 ; Alam . In addition, to arrive at Eq. (13) one
ignores all parts of energy on the brane including DE, DM and byronic matter,
which is not a reasonable assumption in a real universe even at the late time.

We also assume that there is no energy exchange between the brane and the bulk
and so the energy conservation equation holds on the brane

$\dot{\rho}+3H(1+w)\rho=0,$ (14)

while in general there is an interaction between DM and DE on the brane. Thus
both components do not obey the conservation equation, and they obey instead

$\displaystyle\dot{\rho}_{m}+3H\rho_{m}$ $\displaystyle=$ $\displaystyle Q,$
(15) $\displaystyle\dot{\rho}_{D}+3H\rho_{D}(1+\omega_{D})$ $\displaystyle=$
$\displaystyle-Q,$ (16)

where $\omega_{D}={p_{D}}/{\rho_{D}}$ is the equation of state parameter of DE
and $Q=\Gamma\rho_{D}$ shows the interaction between the DE and DM on the
brane. Using the apparent horizon radius (5) we can write the Friedmann
equation (9) as

$\rho_{m}+\rho_{D}=3M_{\rm
pl}^{2}\Big{[}\Big{(}\frac{1}{\tilde{r}_{A}}-\frac{\epsilon}{2r_{c}}\Big{)}^{2}-\frac{1}{4{r_{c}^{2}}}\Big{]}.$
(17)

To obtain a differential form of the Friedmann equation in favor of the first
law of thermodynamics, we take the differential of equation (17) and then by
using Eqs. (15) and (16) we find

$H\rho_{D}(1+u+\omega_{D})dt=2M_{\rm
p}^{2}\Big{(}\frac{1}{\tilde{r}_{A}}-\frac{\epsilon}{2r_{c}}\Big{)}\frac{d\tilde{r}_{A}}{\tilde{r}_{A}^{2}},$
(18)

where $u={\rho_{D}}/{\rho_{m}}$ is the ratio of energy densities of the two
dark components. Multiplying both sides of equation (18) by a factor of
$4\pi\tilde{r}_{A}^{3}(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}})$, and
using expression (7) for surface gravity, we arrive at

$-8\pi M_{\rm
p}^{2}\kappa\left(\frac{1}{r_{A}}-\frac{\epsilon}{2r_{c}}\right)\tilde{r}_{A}^{2}d\tilde{r}_{A}=4\pi
H\tilde{r}_{A}^{3}\rho_{D}(1+u+\omega_{D})dt-2\pi\tilde{r}_{A}^{2}\rho_{D}(1+u+\omega_{D})d\tilde{r}_{A}.$
(19)

Next, we assume $E=(\rho_{m}+\rho_{D})\frac{4\pi}{3}\tilde{r}_{A}^{3}$ is the
total energy inside the $3$-sphere on the brane, where
$V=\frac{4\pi}{3}\tilde{r}_{A}^{3}$ is the volume enveloped by $3$-dimensional
sphere. Taking differential form of total energy, $E$, we find

$dE=4\pi\tilde{r}_{A}^{2}(\rho_{m}+\rho_{D})d\tilde{r}_{A}+\frac{4\pi}{3}\tilde{r}_{A}^{3}(\dot{\rho}_{m}+\dot{\rho}_{D})dt.$
(20)

Using Eqs. (15) and (16), we get

$dE=4\pi\tilde{r}_{A}^{2}\rho_{D}(1+u)d\tilde{r}_{A}-4\pi\tilde{r}_{A}^{3}H\rho_{D}(1+u+\omega_{D})dt.$
(21)

Combining Eq. (21) with (19), we obtain

$dE-WdV=16\pi^{2}M_{\rm
p}^{2}T\left(\frac{1}{\tilde{r}_{A}}-\frac{\epsilon}{2r_{c}}\right)\tilde{r}_{A}^{2}d\tilde{r}_{A},$
(22)

where we have defined the work density as Hayward

$W=\frac{1}{2}(\rho-p)=\frac{1}{2}\rho_{D}(1+u-\omega_{D}).$ (23)

The work term $WdV$ is defined as the work done by the change of the apparent
horizon surface. Expression (22) is just the first law of thermodynamics at
the apparent horizon on the brane, namely $dE=TdS+WdV$. We can define the
entropy associated with the apparent horizon on the brane as

$S=\int^{\tilde{r}_{A}}_{0}dS=16\pi^{2}M_{\rm
p}^{2}\int^{\tilde{r}_{A}}_{0}{\left(\frac{1}{\tilde{r}_{A}}-\frac{\epsilon}{2r_{c}}\right)\tilde{r}_{A}^{2}d\tilde{r}_{A}}=\frac{A}{4G_{4}}\left(1-\frac{\epsilon\tilde{r}_{A}}{3r_{c}}\right),$
(24)

where we have used $M_{\rm pl}^{2}=(8\pi G_{4})^{-1}$ and
$A=4\pi\tilde{r}_{A}^{2}$ is the area of the apparent horizon. Let us note
that for $\epsilon=-1$, Eq. (24) is similar to the result obtained in Sheykhi1
. In the limiting case where $\tilde{r}_{A}\ll 3r_{c}$, one recovers the area
law for the apparent horizon entropy. Physically, this means that the apparent
horizon is not extended in the bulk and located totally on the brane. As a
result, the effect of the extra dimension does not appear in the entropy
expression.

It is well-known that in each gravity theory, the entropy expression
associated with the apparent horizon in cosmology is the same as the entropy
associated with the black hole horizon. The only change which is needed to
replace the horizon radius $r_{+}$ of black hole with the apparent horizon
$\tilde{r}_{A}$ Cai2 ; SMR . Thus we propose the entropy of the black hole
horizon in the DGP braneworld scenario to be given by

$S=\frac{A}{4G_{4}}\left(1-\frac{\epsilon r_{+}}{3r_{c}}\right),$ (25)

where $A=4\pi r_{+}^{2}$ is the area of the black hole horizon. It is
important to note that the definition and derivation of holographic energy
density ($\rho_{D}=3c^{2}M^{2}_{p}/L^{2}$) depends on the entropy-area
relationship $S\sim A\sim L^{2}$ of black holes, where $A$ represents the area
of the horizon Cohen . Inspired by the entropy relation (25) and following the
derivation of HDE Gub  and entropy-corrected HDE Wei ; SJ , we can easily
obtain the corresponding energy density of the HDE in the DGP braneworld as

$\rho_{D}=\frac{3c^{2}M_{\rm p}^{2}}{L^{2}}\left(1-\frac{\epsilon
L}{3r_{c}}\right).$ (26)

When $L\ll 3r_{c}$, the above equation yields the well-known holographic
energy density, as expected. The significant of the corrected term in various
regions depends on the fraction $L/r_{c}$. We emphasize here that for studying
the HDE in the framework of DGP braneworld, it is more reasonable to take the
energy density of HDE in the form of (26) instead of (2). This is due to the
fact that the well-known area law for the black holes entropy no longer holds
on the brane and the entropy associated with the horizon on the brane should
be modified as relation (25). This is an important point which was not taken
into account in the previous studies on HDE in the DGP braneworld HDEDGP . In
the remaining part of this paper, we shall investigate the cosmological
implications of the HDE density (26). Since the simple and natural choice for
the systems’s IR cutoff is the Hubble radius $L=H^{-1}$, thus in this paper we
consider this choice in two different cases, namely in the absence of the
interaction and then we consider the interacting case. In both cases we find
the equation of state and deceleration parameters of a FRW Universe on the
brane. We also plot the related figures to show the evolution of them in each
case.

##  III Non interacting new HDE model

Let us consider, for simplicity, the flat FRW universe. The modified Friedmann
equation is given by Eq. (12) where $\rho=\rho_{m}+\rho_{D}$. In the absence
of interaction between DE and DM, the continuity equations read

$\displaystyle\dot{\rho}_{m}+3H\rho_{m}=0,$ (27)
$\displaystyle\dot{\rho}_{D}+3H(1+\omega_{D})\rho_{D}=0.$ (28)

Note that since the matter component is mainly contributed by the cold DM we
ignore the baryon matter component here for simplicity. From (12), we can
write

$\Omega_{m}+\Omega_{D}=1-2\epsilon\sqrt{\Omega_{r_{c}}},$ (29)

where we have used the following definitions

$\Omega_{m}=\frac{\rho_{m}}{3M_{p}^{2}H^{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}\Omega_{D}=\frac{\rho_{D}}{3M_{p}^{2}H^{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Omega_{r_{c}}=\frac{1}{4r_{c}^{2}H^{2}}.$
(30)

Considering the Hubble radius $L=H^{-1}$ as the system’s IR cutoff, we can
write (26) as

$\rho_{D}=3c^{2}M_{p}^{2}H^{2}\left(1-\frac{2\epsilon\sqrt{\Omega_{r_{c}}}}{3}\right).$
(31)

Using (30), we have

$\Omega_{D}=c^{2}\left(1-\frac{2\epsilon\sqrt{\Omega_{r_{c}}}}{3}\right).$
(32)

This equation implies that for HDE in standard cosmology ($\Omega_{r_{c}}=0$),
with Hubble radius as the IR cutoff, $\Omega_{D}=c^{2}$ and thus DE has no
evolution during the history of the Universe. As we know this is not a
reasonable result. Thus our model may resolve this problem, since in our case
$\Omega_{D}$ given in (32) is no longer a constant.

Taking the time derivative of Eq. (12) and using Eqs. (27), (28), (29) and
(30) we can easily find

$\frac{\dot{H}}{H^{2}}=-\frac{3}{2}\left[\frac{1-\Omega_{D}-2\epsilon\sqrt{\Omega_{r_{c}}}}{1-\Omega_{D}-\epsilon\sqrt{\Omega_{r_{c}}}-\frac{\epsilon
c^{2}}{3}\sqrt{\Omega_{r_{c}}}}\right].$ (33)

Taking the time derivative of Eq. (31), we find

$\dot{\Omega}_{D}=\frac{2\epsilon
c^{2}\sqrt{\Omega_{r_{c}}}}{3}\frac{\dot{H}}{H}.$ (34)

Using the fact that $\dot{\Omega_{D}}=H{\Omega}^{\prime}_{D}$, after
substituting Eq. (33) in (34) the evolution of dimensionless ECHDE density as

${\Omega}^{\prime}_{D}=-\epsilon
c^{2}\sqrt{\Omega_{r_{c}}}\left[\frac{1-\Omega_{D}-2\epsilon\sqrt{\Omega_{r_{c}}}}{1-\Omega_{D}-\epsilon\sqrt{\Omega_{r_{c}}}-\frac{\epsilon
c^{2}}{3}\sqrt{\Omega_{r_{c}}}}\right],$ (35)

where the prime denotes derivative with respect to $x=\ln a$.

Figure 1: The evolution of the EoS parameter $\Omega_{D}$ versus redshift
parameter $z$.

Using (32) we can omit $\Omega_{r_{c}}$ and rewrite the above relation in the
form

${\Omega}^{\prime}_{D}=-3(c^{2}-\Omega_{D})\left[\frac{2c^{2}+\Omega_{D}c^{2}-3\Omega_{D}}{c^{2}+\Omega_{D}c^{2}+c^{4}-3\Omega_{D}}\right].$
(36)

The evolution of the dimensionless HDE density parameter $\Omega_{D}$ as a
function of $1+z=a^{-1}$ is shown in Fig.(1). From this figure we see that at
the early Universe where $z\rightarrow\infty$ we have $\Omega_{D}\rightarrow
0$, while at the late time, where $z\rightarrow-1$, the DE dominated, namely
$\Omega_{D}\rightarrow 1$.

Combining Eqs. (28) and (31) yields

$\omega_{D}=-1-\frac{2}{3}\left(1+\frac{\epsilon
c^{2}\sqrt{\Omega_{r_{c}}}}{3\Omega_{D}}\right)\frac{\dot{H}}{H^{2}},$ (37)

Figure 2: The evolution of the EoS and deceleration versus redshift parameter
$z$ for $c=0.9$.

and thus the equation of state parameter is obtained as

$\omega_{D}=-1+\left(1+\frac{\epsilon
c^{2}\sqrt{\Omega_{r_{c}}}}{3\Omega_{D}}\right)\left[\frac{1-\Omega_{D}-2\epsilon\sqrt{\Omega_{r_{c}}}}{1-\Omega_{D}-\epsilon\sqrt{\Omega_{r_{c}}}-\frac{\epsilon
c^{2}}{3}\sqrt{\Omega_{r_{c}}}}\right].$ (38)

In the limiting case where $\Omega_{r_{c}}\rightarrow 0$, we obtain
$\omega_{D}=0$ which is a wrong equation of state for DE and cannot derive the
acceleration of the Universe expansion Hsu . This is an expected result, since
in this regime the effects of the extra dimension disappear and the standard
cosmology is recovered. It was already shown that in the absence of
interaction, choosing $L=H^{-1}$ cannot produce the cosmic acceleration Hsu .
However, as one can see from Eq. (38) in the framework of DGP braneworld with
modified HDE density (26), the identification of IR-cutoff with Hubble radius,
$L=H^{-1}$, can lead to accelerated expansion. If we substitute
$\Omega_{r_{c}}$ from Eq. (32), we can further rewrite,

$\omega_{D}=\frac{2c^{2}(c^{2}-\Omega_{D})}{\Omega_{D}(c^{2}+\Omega_{D}c^{2}+c^{4}-3\Omega_{D})}.$
(39)

From Eq. (32), we see that the limiting case $\Omega_{r_{c}}=0$ can be
translated to $\Omega_{D}=c^{2}$, and thus from (39) we have $\omega_{D}=0$.
Therefore, the result of standard cosmology is recovered; no acceleration with
Hubble radius as IR cutoff.

The deceleration parameter is given by

$q=-1-\frac{\dot{H}}{H^{2}}.$ (40)

Substituting Eq. (33) into (40) one can obtain the deceleration parameter as

$q=-1+\frac{3}{2}\left[\frac{1-\Omega_{D}-2\epsilon\sqrt{\Omega_{r_{c}}}}{1-\Omega_{D}-\epsilon\sqrt{\Omega_{r_{c}}}-\frac{\epsilon
c^{2}}{3}\sqrt{\Omega_{r_{c}}}}\right].$ (41)

Using relation (32) one can rewrite Eq. (41) in terms of $c^{2}$ and
$\Omega_{D}$,

$q=-\frac{c^{4}+6\Omega_{D}-2c^{2}\Omega_{D}-5c^{2}}{c^{2}+\Omega_{D}c^{2}+c^{4}-3\Omega_{D}}.$
(42)

Again for $\Omega_{D}=c^{2}$ we have $q=\frac{1}{2}$ which implies a
decelerated phase for the Universe, corresponding to HDE with the Hubble
radius as IR cut-off in standard cosmology Hsu . The evolution of equation of
state and deceleration parameters are shown in figure 2 for different value of
the parameter $c^{2}$. From these figures we see that the Universe has a
transition from deceleration to the acceleration phase
$\omega_{D}<-\frac{1}{3}$. In this model the EoS parameter cannot cross the
phantom line $\omega_{D}=-1$ in future. In addition the transition from
deceleration to the acceleration phase occurs around $z\simeq 0.6$ compatible
with the recent observations Daly ; Kom1 ; Kom2 .

It is worth noting that although the equation of state and the deceleration
parameters do not depend explicitly on the crossover length scale $r_{c}$
which is the characterization of the DGP braneworld, they depend on $r_{c}$
via the relation between the dimensionless density parameters in Eq. (32). For
$r_{c}\gg 1(\Omega_{r_{c}}\rightarrow 0)$, the effects of the extra dimension
vanish and the results of the standard cosmology are restored Hsu .

##  IV Interacting new HDE model

In the presence of the interaction between DE and DM, the conservation
equation do no hold separately, they instead obey (15) and (16). Recent
observational evidences provided by the galaxy cluster Abell A586 supports the
interaction between DE and DM Bertolami2 . The dynamics of interacting DE
models with different $Q$-classes have been studied in ample detail in
Amendola . Here we choose $Q=3b^{2}H(\rho_{D}+\rho_{m})$ as the interaction
term, where $b^{2}$ is a coupling constant.
Taking the time derivative of Eq. (12) and using Eqs. (29),(30), (15) and
(16), we find

$\frac{\dot{H}}{H^{2}}=\frac{-3(1-\Omega_{D}-2\epsilon\sqrt{\Omega_{r_{c}}})+3b^{2}(1-2\epsilon\sqrt{\Omega_{r_{c}}})}{2(1-\Omega_{D}-\epsilon\sqrt{\Omega_{r_{c}}}-\frac{c^{2}\epsilon}{3}\sqrt{\Omega_{r_{c}}})}.$
(43)

Using Eqs. (34) and (43) we can obtain the evolution of the dimensionless
density $\Omega_{D}$ as

${\Omega}^{\prime}_{D}=3(c^{2}-\Omega_{D})\frac{(-2c^{2}-c^{2}\Omega_{D}+3\Omega_{D})+b^{2}(-2c^{2}+3\Omega_{D})}{\Big{(}c^{2}-3\Omega_{D}+c^{2}\Omega_{D}+c^{4}\Big{)}},$
(44)

where we have used Eq. (32) for omitting $\Omega_{r_{c}}$. The evolution of
the dimensionless HDE density parameter $\Omega_{D}$ as a function of
$1+z=a^{-1}$ is shown in Fig. (3). Again we see that at the early time
$\Omega_{D}\rightarrow 0$, while at the late time, $\Omega_{D}\rightarrow 1$.

Combining Eqs. (16), (29) and (30) we arrive at

$\omega_{D}=-1-\frac{2}{3H}\left(\frac{\dot{H}}{H^{2}}\right)-\frac{2c^{2}\epsilon\sqrt{\Omega_{r_{c}}}}{9\Omega_{D}}\frac{\dot{H}}{H^{2}}-\frac{b^{2}(1-2c^{2}\epsilon\sqrt{\Omega_{r_{c}}})}{\Omega_{D}}.$
(45)

Figure 3: The evolution of $\Omega_{D}$ versus redshift parameter $z$ for
interacting new HDE in DGP braneworld.

Substituting Eq. (43) into Eq. (45), we find the equation of state parameter
of interacting new HDE as

$\omega_{D}=\frac{\sqrt{\Omega_{r_{c}}}(-\epsilon\Omega_{D}+\frac{\epsilon
c^{2}}{3}-\frac{2\epsilon^{2}c^{2}}{3}\sqrt{\Omega_{r_{c}}})-{b^{2}(1-2\epsilon\sqrt{\Omega_{r_{c}}})(1-\epsilon\sqrt{\Omega_{r_{c}}})}}{\Omega_{D}(1-\epsilon\sqrt{\Omega_{r_{c}}}-\Omega_{D}-\frac{c^{2}\epsilon}{3}\sqrt{\Omega_{r_{c}}})}.$
(46)

Substituting Eq. (43) into (40) we obtain the deceleration parameter as

$q=\frac{1-4\epsilon\sqrt{\Omega_{r_{c}}}-\Omega_{D}+\frac{2c^{2}\epsilon}{3}\sqrt{\Omega_{r_{c}}}+3b^{2}(1-2\epsilon\sqrt{\Omega_{r_{c}}})}{2(1-\epsilon\sqrt{\Omega_{r_{c}}}-\Omega_{D}-\frac{c^{2}\epsilon}{3}\sqrt{\Omega_{r_{c}}})}.$
(47)

Figure 4: The evolution of the EoS parameter $\omega_{D}$ versus redshift
parameter $z$ for interacting new HDE in DGP braneworld.

Figure 5: The evolution of the deceleration parameter $q$ versus redshift
parameter $z$ for interacting new HDE in DGP braneworld.

Using relation (32), one can rewrite Eqs. (46) and (47) in the following form

$\displaystyle\omega_{D}$ $\displaystyle=$
$\displaystyle\frac{2c^{2}(c^{2}-\Omega_{D})+\frac{b^{2}}{c^{2}}(3\Omega_{D}-2c^{2})(3\Omega_{D}-c^{2})}{\Omega_{D}\Big{(}c^{2}+\Omega_{D}c^{2}+c^{4}-3\Omega_{D}\Big{)}},$
(48) $\displaystyle q$ $\displaystyle=$
$\displaystyle-\frac{c^{4}+6\Omega_{D}-2c^{2}\Omega_{D}-5c^{2}+3b^{2}(2c^{2}-3\Omega_{D})}{\Big{(}c^{2}+\Omega_{D}c^{2}+c^{4}-3\Omega_{D}\Big{)}}.$
(49)

In the absence of the interaction, i.e., $b^{2}=0$, Eqs. (48) and (49) reduce
to (39) and (42), respectively. Again from relation (32) we see that the limit
of standard cosmology where $\Omega_{r_{c}}$ corresponds to
$\Omega_{D}=c^{2}$, and in this limit the results of standard cosmology for
interacting HDE with Hubble radius as systems’s IR cutoff is restored Pav`on ,

$\displaystyle\omega_{D}$ $\displaystyle=$
$\displaystyle-\frac{b^{2}}{c^{2}(1-c^{2})},$ (50) $\displaystyle q$
$\displaystyle=$ $\displaystyle\frac{1}{2}-\frac{3b^{2}}{2(1-c^{2})}.$ (51)

Again, we see that for the choice of $L=H^{-1}$, in standard cosmology, an
interaction is the only way to have an EoS different from that for dust Pav`on
; Zim . However, for the model presented in this paper, and in the context of
DGP braneworld, even in the absence of interaction, the the natural and simple
choice of $L=H^{-1}$, can leads to the accelerated Universe. This is one of
the main result we have addressed in the present paper.

The evolution of the $\omega_{D}$ and $q$ are shown in figures 4 and 5 for
different model parameters. In one figure we fix $b^{2}$ and set different
values for $c$, while in another figure we fix $c$ and set different values
for $b^{2}$. From these figures we clearly see that the transition from the
deceleration into acceleration phase can occur around $z\sim 0.6$, which is
consist with recent observations.

##  V Conclusions and discussions

The energy density expression of the HDE is based on the area law of the black
hole entropy, and thus any modification of the area law leads to a modified
holographic energy density. Since in the DGP braneworld the area law of the
entropy can be modified as given in (25), thus it is more reasonable to
propose a new energy density for the HDE which is based on this modified
entropy expression.

In this paper, we first showed that the Friedmann equation describing the
evolution of the FRW Universe in DGP can be recast as the first law of
thermodynamics on the apparent horizon. This procedure leads to extract an
entropy expression associated with the apparent horizon. We expect the entropy
of the black hole horizon to have the same expression but replacing the
apparent horizon radius $\tilde{r}_{A}$ with the black hole horizon radius
$r_{+}$. Then, inspired by the entropy expression associated with the apparent
horizon of FRW universe in DGP braneworld, we proposed a new model for the HDE
in the framework of the DGP brane cosmology. We believe that if one is
interested in studying the HDE in the framework of DGP braneworld, one should
take the energy density in the form of (26) instead of the usual form
$\rho_{D}={3c^{2}M_{\rm p}^{2}}/{L^{2}}$. Compared to the energy density of
the HDE in standard cosmology, the new HDE describing by expression (26)
consists an additional term which incorporates the effects of un-compact extra
dimension onto the brane. Clearly in the limiting case where the effects of
the extra dimension vanishes,($L\ll 3r_{c}$), the energy density of usual HDE
is recovered.

Then, we studied the cosmological implications of this new model for flat FRW
Universe on the brane. For this purpose, we chose the Hubble radius
$L=H^{-1}$, as system IR cutoff. The Hubble radius is not only the most
natural and obvious but also the simplest choice for IR cutoff in flat
universe. It was already shown that the Hubble radius in flat Universe can
result an accelerated Universe provided the interaction between the two dark
components of the universe is taken into account Pav`on . In other words, in
the absence of interaction it leads to the EoS of dust, $w\omega_{D}=0$ Hsu .
Interestingly enough, we found that, for the new HDE in DGP braneworld, the
identification of IR cutoff with Hubble horizon, $L=H^{-1}$, can lead to an
acceleration of the universe expansion, even in the absence of the interaction
between two dark components. This is contrast to the HDE in standard
cosmology, where $\omega_{D}=0$ if one choose $L=H^{-1}$ in the absence of
interaction between the two dark components. We also examined our model by
taking into account the interaction term and derived the cosmological
parameters. In order to see the behavior of the EoS and deceleration
parameter, we plotted these parameters versus redshift in both cases. Our
studies show that in both cases, our Universe has a transition from
deceleration to the acceleration phase around $z\approx 0.6$, which is
compatible with observational evidences Daly ; Kom1 ; Kom2 .

###### Acknowledgements.

We thank from the Research Council of Shiraz University. This work has been
supported financially by Research Institute for Astronomy & Astrophysics of
Maragha (RIAAM), Iran.

## References

  * (1) A.G. Riess, et al., Astron. J. 116, 1009 (1998);
S. Perlmutter, et al., Astrophys. J. 517, 565 (1999);
S. Perlmutter, et al., Astrophys. J. 598, 102 (2003);
A.G. Riess, et al., Astrophys. J. 607, 665 (2004).

  * (2) M. Li, X. D. Li, S. Wang, Y. Wang, Commun. Theor. Phys. 56, 525 (2011).
  * (3) S. D. H. Hsu, Phys. Lett. B 594, 13 (2004).
  * (4) M. Li, Phys. Lett. B 603, 1 (2004);
Q. G. Huang, M. Li, JCAP 0408, 013 (2004).

  * (5) B. Wang, Y. Gong and E. Abdalla, Phys. Lett. B 624, 141 (2005);
B. Wang, C. Y. Lin and E. Abdalla, Phys. Lett. B 637, 357 (2005);
M. R. Setare, Phys. Lett. B 642,1 (2006).

  * (6) B. Wang, E. Abdalla, R. K. Su, Phys. Lett. B 611 (2005) 21;
J. Y. Shen, B. Wang, E. Abdalla, R. K. Su, Phys. Lett. B 609, 200 (2005);
C. Feng, B. Wang, Y. Gong, R. K. Su, JCAP 0709, 005 (2007) ;
B. Wang, C. Y. Lin. D. Pavon and E. Abdalla, Phys. Lett. B 662, 1 (2008).

  * (7) E. J. Copeland, M. Sami and S. Tsujikawa, Int. J. Mod. Phys. D 15 (2006) 1753.
  * (8) D.Pav on and W. Zimdahl, Phys. Lett. B 628 206 (2005).
  * (9) W. Zimdahl and D. Pavon, Classical Quantum Gravity 24, 5461 (2007).
  * (10) A. Sheykhi, Phys. Lett. B 681, 205 (2009);
A. Sheykhi, Phys. Rev. D 84, 107302 (2011).

  * (11) A. Sheykhi, Class. Quantum Grav. 27, 025007 (2010);
A. Sheykhi, Mubasher Jamil, Phys. Lett. B 694, 284 (2011).

  * (12) A. Sheykhi, M. Jamil, Gen Relativ Gravit 43, 2661 (2011).
  * (13) G. t Hooft, gr-qc/9310026;
L. Susskind, J. Math. Phys. 36 (1995) 6377\.

  * (14) A. Cohen, D. Kaplan, A. Nelson, Phys. Rev. Lett. 82 (1999) 4971\.
  * (15) X. Zhang, F. Q. Wu, Phys. Rev. D 72, 043524 (2005);
X. Zhang, F. Q. Wu, Phys. Rev. D 76, 023502 (2007);
Q. G. Huang, Y. G. Gong, JCAP 0408, 006 (2004);
K. Enqvist, S. Hannestad, M. S. Sloth, JCAP 0502, 004 (2005);
J.y. Shen, B. Wang, E. Abdalla, R.K. Su, Phys. Lett. B 609, 200 (2005);
Y. Wang, L. Xu, Phys. Rev. D 81, 083523 (2010);
S. M. R. Micheletti, JCAP 1005, 009(2010);
M. Li, JCAP 0906, 036 (2009);
X. Zhang, Phys. Rev. D 79, 103509 (2009.

  * (16) B. Feng, X. Wang, X. Zhang, Phys. Lett. B 607, 35 (2005) ;
H.C. Kao, W.L. Lee, F.L. Lin, Phys. Rev. D 71 123518, (2005).

  * (17) S. Das, S. Shankaranarayanan and S. Sur, Phys. Rev. D 77 (2008) 064013.
  * (18) A. Sheykhi, B. Wang, R.G. Cai, Nucl. Phys. B 779, 1 (2007).
  * (19) A. Sheykhi, B. Wang and R. G. Cai, Phys. Rev. D 76, 023515 (2007);
A. Sheykhi, B. Wang, Phys. Lett. B 678, 434 (2009);
A. Sheykhi, J. Cosm. Astro. Phys. 05, 019 (2009) 019.

  * (20) T. Padmanabhan, Rep. Prog. Phys. 73 (2010) 046901\.
  * (21) T. Jacobson, Phys. Rev. Lett. 75, 1260 (1995).
  * (22) R. G. Cai and S. P. Kim, J. High Energy Phys 0502, 050 (2005).
  * (23) A. Paranjape, S. Sarkar and T. Padmanabhan, Phys. Rev. D 74, 104015 (2006).
  * (24) T. Padmanabhan, Mod. Phys. Lett. A 25 (2010) 1129;
T. Padmanabhan, Phys. Rev. D 81, 124040 (2010).

  * (25) M. Akbar and R. G. Cai, Phys. Lett. B 648, 243 (2007).
  * (26) M. Akbar and R. G. Cai, Phys. Lett. B 635, 7 (2006).
  * (27) M. Akbar and R. G. Cai, Phys. Rev. D 75, 084003 (2007).
  * (28) R. G. Cai and L. M. Cao, Phys.Rev. D 75, 064008 (2007).
  * (29) A. Sheykhi, Eur. Phys. J. C 69, 265 (2010);
A. Sheykhi, Phys. Rev. D 87, 024022 (2013).

  * (30) R. G. Cai and N. Ohta, Phys. Rev. D 81, (2010) 084061\.
  * (31) R. G. Cai and L. M. Cao, Nucl. Phys. B 785, 135 (2007).
  * (32) K. Nozari, N. Behrouz, A. Sheykhi, Int. J. Theor. Phys. 52, 2351 (2013).
  * (33) G. Dvali, G. Gabadadze, M. Porrati, Phys. Lett. B 485, 208 (2000).
  * (34) C. Deffayet, Phys. Lett. B 502, 199 (2001).
  * (35) C. Deffayet and G. Dvali, Phys. Rev. D 65, 044023 (2002).
  * (36) D. Bak, S.J. Rey, Class. Quantum Gravity 17, L83 (2000)
  * (37) S.A. Hayward, S. Mukohyana, M.C. Ashworth, Phys. Lett. A 256, 347 (1999);
S.A. Hayward, Class. Quantum Gravity 15, 3147 (1998).

  * (38) R.G. Cai, L.M. Cao, Y.P. Hu, Class. Quant. Grav. 26, 155018 (2009).
  * (39) J.Q. Xia, H. Li, X. Zhang, Phys. Rev. D 88, 063501 (2013).
  * (40) W. Yang, L. Xu, Phys. Rev. D 89, 083517 (2014) ;
B. Novosyadlyj, O. Sergijenko, R. Durrer, V. Pelykh, arXiv:1312.6579

  * (41) U. Alam, V. Sahni, A.A. Starobinsky, JCAP 0406, 008 (2004);
D. Huterer, A. Cooray, Phys. Rev. D 71, 023506 (2005);
Y.G. Gong, Int. J. Mod. Phys. D 14, 599 (2005);
Y.G. Gong, Class. Quantum Grav. 22, 2121 (2005).

  * (42) A. Sheykhi, H. Moradpour, and N. Riazi, Gen. Relativ. Gravit. 45 (2013) 1033\.
  * (43) B. Guberina, R. Horvat and H. Nikolic, JCAP 01 (2007) 012.
  * (44) H. Wei, Commun. Theor. Phys. 52 (2009) 743.
  * (45) A. Sheykhi and M. Jamil, Gen. Relativ. Gravit., 43 (2011) 2661.
  * (46) J. Dutta, S. Chakraborty, M. Ansari, Mod. Phys. Lett. A 25, 3069 (2010);
D. J. Liu, H. Wang, B. Yang, Phys. Lett. B 694, 6 (2010);
N.Cruz, S. Lepe, F. Pena, arXiv:1109.2090.

  * (47) R.A. Daly et al., Astrophys. J. 677, 1 (2008).
  * (48) E. Komatsu et al. [WMAP Collaboration], Astrophys. J. Suppl. 192, 18 (2011).
  * (49) V. Salvatelli, A. Marchini, L. L. Honorez and O. Mena, Phys. Rev. D 88, 023531 (2013).
  * (50) O. Bertolami, F. Gil Pedro, M. Le Delliou, Phys. Lett. B 654 (2007) 165; O. Bertolami, F. Gil Pedro, M. Le Delliou, Gen. Rel. Grav. 41 (2009) 2839; E. Abdalla, L.R. Abramo, L. Sodre, B. Wang, Phys. Lett. B 673 (2009) 107.
  * (51) L. Amendola, Phys. Rev. D 60 043501 (1999);
L. Amendola, Phys. Rev. D 62 043511 (2000);
B. Wang, Y. Gong and E. Abdalla, Phys. Lett. B 624 141 (2005);
D.Pav on and W. Zimdahl, Phys. Lett. B 628 206 (2005);
M. Szydlowski, 2006 Phys. Lett. B 632 1 (2006);
S. Tsujikawa, Phys. Rev. D 73 103504 (2006);
Z.K. Guo, N. Ohta and S. Tsujikawa, Phys. Rev. D 76 023508 (2007);
G. Caldera-Cabral, R. Maartens and L.A. Urena-Lopez, Phys. Rev. D 79 063518
(2009).