# On the Importance of Effectively Adapting Pretrained Language Models for
Active Learning

Katerina Margatina♠ Loïc Barrault♣ Nikolaos Aletras♠
♠University of Sheffield, ♣University of Le Mans
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

Recent Active Learning (AL) approaches in Natural Language Processing (NLP)
proposed using off-the-shelf pretrained language models (LMs). In this paper,
we argue that these LMs are not adapted effectively to the downstream task
during AL and we explore ways to address this issue. We suggest to first adapt
the pretrained LM to the target task by continuing training with all the
available unlabeled data and then use it for AL. We also propose a simple yet
effective fine-tuning method to ensure that the adapted LM is properly trained
in both low and high resource scenarios during AL. Our experiments demonstrate
that our approach provides substantial data efficiency improvements compared
to the standard fine-tuning approach, suggesting that a poor training strategy
can be catastrophic for AL.111For all experiments in this paper, we have used
the code provided by Margatina et al. (2021):
https://github.com/mourga/contrastive-active-learning

##  1 Introduction

Active Learning (AL) is a method for training supervised models in a data-
efficient way Cohn et al. (1996); Settles (2009). It is especially useful in
scenarios where a large pool of unlabeled data is available but only a limited
annotation budget can be afforded; or where expert annotation is prohibitively
expensive and time consuming. AL methods iteratively alternate between (i)
model training with the labeled data available; and (ii) data selection for
annotation using a stopping criterion, e.g. until exhausting a fixed
annotation budget or reaching a pre-defined performance on a held-out dataset.

Data selection is performed by an acquisition function that ranks unlabeled
data points by some informativeness metric aiming to improve over random
selection, using either uncertainty Lewis and Gale (1994); Cohn et al. (1996);
Gal et al. (2017); Kirsch et al. (2019); Zhang and Plank (2021), diversity
Brinker (2003); Bodó et al. (2011); Sener and Savarese (2018), or both Ducoffe
and Precioso (2018); Ash et al. (2020); Yuan et al. (2020); Margatina et al.
(2021).

Previous AL approaches in NLP use task-specific neural models that are trained
from scratch at each iteration Shen et al. (2017); Siddhant and Lipton (2018);
Prabhu et al. (2019); Ikhwantri et al. (2018); Kasai et al. (2019). However,
these models are usually outperformed by pretrained language models (LMs)
adapted to end-tasks Howard and Ruder (2018), making them suboptimal for AL.
Only recently, pretrained LMs such as Bert Devlin et al. (2019) have been
introduced in AL settings Yuan et al. (2020); Ein-Dor et al. (2020); Shelmanov
et al. (2021); Karamcheti et al. (2021); Margatina et al. (2021). Still, they
are trained at each AL iteration with a standard fine-tuning approach that
mainly includes a pre-defined number of training epochs, which has been
demonstrated to be unstable, especially in small datasets Zhang et al. (2020);
Dodge et al. (2020); Mosbach et al. (2021). Since AL includes both low and
high data resource settings, the AL model training scheme should be robust in
both scenarios.222During the first few AL iterations the available labeled
data is limited (low-resource), while it could become very large towards the
last iterations (high-resource).

To address these limitations, we introduce a suite of effective training
strategies for AL (§2). Contrary to previous work Yuan et al. (2020); Ein-Dor
et al. (2020); Margatina et al. (2021) that also use Bert Devlin et al.
(2019), our proposed method accounts for various data availability settings
and the instability of fine-tuning. First, we continue pretraining the LM with
the available unlabeled data to adapt it to the task-specific domain. This
way, we leverage not only the available labeled data at each AL iteration, but
the entire unlabeled pool. Second, we further propose a simple yet effective
fine-tuning method that is robust in both low and high resource data settings
for AL.

We explore the effectiveness of our approach on five standard natural language
understandings tasks with various acquisition functions, showing that it
outperforms all baselines (§3). We also conduct an analysis to demonstrate the
importance of effective adaptation of pretrained models for AL (§4). Our
findings highlight that the LM adaptation strategy can be more critical than
the actual data acquisition strategy.

##  2 Adapting & Fine-tuning Pretrained Models for Active Learning

Given a downstream classification task with $C$ classes, a typical AL setup
consists of a pool of unlabeled data $\mathcal{D}_{\textbf{pool}}$, a model
$\mathcal{M}$, an annotation budget $b$ of data points and an acquisition
function $a(.)$ for selecting $k$ unlabeled data points for annotation (i.e.
acquisition size) until $b$ runs out. The AL performance is assessed by
training a model on the actively acquired dataset and evaluating on a held-out
test set $\mathcal{D}_{\textbf{test}}$.

#### Adaptation (tapt)

Inspired by recent work on transfer learning that shows improvements in
downstream classification performance by continuing the pretraining of the LM
with the task data Howard and Ruder (2018) we add an extra step to the AL
process by continuing pretraining the LM (i.e. Task-Adaptive Pretraining
tapt), as in Gururangan et al. (2020). Formally, we use an LM, such as Bert
Devlin et al. (2019), $\mathcal{P}(x;W_{0})$ with weights $W_{0}$, that has
been already pretrained on a large corpus. We fine-tune $\mathcal{P}(x;W_{0})$
with the available unlabeled data of the downstream task
$\mathcal{D}_{\textbf{pool}}$, resulting in the task-adapted LM
$\mathcal{P}_{\text{TAPT}}(x;W_{0}^{\prime})$ with new weights
$W_{0}^{\prime}$ (cf. line 2 of algorithm 1).

#### Fine-tuning (ft+)

We now use the adapted LM $\mathcal{P}_{\text{TAPT}}(x;W_{0}^{\prime})$ for
AL. At each iteration $i$, we initialize our model $\mathcal{M}_{i}$ with the
pretrained weights $W_{0}^{\prime}$ and we add a task-specific feedforward
layer for classification with weights $W_{c}$ on top of the [CLS] token
representation of Bert-based $\mathcal{P}_{\text{TAPT}}$. We fine-tune the
classification model $\mathcal{M}_{i}(x;[W_{0}^{\prime},W_{c}])$ with all
$x\in\mathcal{D}_{\textbf{lab}}$. (cf. line 6 to 8 of algorithm 1).

Recent work in AL Ein-Dor et al. (2020); Yuan et al. (2020) uses the standard
fine-tuning method proposed in Devlin et al. (2019) which includes a fixed
number of $3$ training epochs, learning rate warmup over the first $10\%$ of
the steps and AdamW optimizer Loshchilov and Hutter (2019) without bias
correction, among other hyperparameters.

We follow a different approach by taking into account insights from few-shot
fine-tuning literature Mosbach et al. (2021); Zhang et al. (2020); Dodge et
al. (2020) that proposes longer fine-tuning and more evaluation steps during
training. 333In this paper we use few-shot to describe the setting where there
are few labeled data available and therefore few-shot fine-tuning corresponds
to fine-tuning a model on limited labeled training data. This is different
than the few-shot setting presented in recent literature Brown et al. (2020),
where no model weights are updated.  We combine these guidelines to our fine-
tuning approach by using early stopping with $20$ epochs based on the
validation loss, learning rate $2e-5$, bias correction and $5$ evaluation
steps per epoch. However, increasing the number of epochs from $3$ to $20$,
also increases the warmup steps ($10\%$ of total steps444Some guidelines
propose an even smaller number of warmup steps, such as $6\%$ in RoBERTa Liu
et al. (2020).) almost $7$ times. This may be problematic in scenarios where
the dataset is large but the optimal number of epochs may be small (e.g. $2$
or $3$). To account for this limitation in our AL setting where the size of
training set changes at each iteration, we propose to select the warmup steps
as $min(10\%\text{ of total steps},100)$. We denote standard fine-tuning as
sft and our approach as ft+.

1

2

3

Input: unlabeled data $\mathcal{D}_{\textbf{pool}}$, pretrained LM
$\mathcal{P}(x;W_{0})$, acquisition size $k$, AL iterations $T$, acquisition
function $a$

4

5$\mathcal{D}_{\textbf{lab}}\leftarrow\emptyset$

6

7$\mathcal{P}_{\text{TAPT}}(x;W_{0}^{\prime})\leftarrow$ Train
$\mathcal{P}(x;W_{0})$ on $\mathcal{D}_{\textbf{pool}}$

8

9$\mathcal{Q}_{0}\leftarrow$ RANDOM$(.),|\mathcal{Q}_{0}|=k$

10

11$\mathcal{D}_{\textbf{lab}}=\mathcal{D}_{\textbf{lab}}\cup\mathcal{Q}_{0}$

12
$\mathcal{D}_{\textbf{pool}}=\mathcal{D}_{\textbf{pool}}\setminus\mathcal{Q}_{0}$

13 for  _ $i\leftarrow 1$ to $T$_ do

14 $\mathcal{M}_{i}(x;[W_{0}^{\prime},W_{c}])\leftarrow$ Initialize from
$\mathcal{P}_{\text{TAPT}}(x;W_{0}^{\prime})$

15 $\mathcal{M}_{i}(x;W_{i})\leftarrow$ Train model on
$\mathcal{D}_{\textbf{lab}}$

16 $\mathcal{Q}_{i}\leftarrow
a(\mathcal{M}_{i},\mathcal{D}_{\textbf{pool}},k)$

17 $\mathcal{D}_{\textbf{lab}}=\mathcal{D}_{\textbf{lab}}\cup\mathcal{Q}_{i}$

18
$\mathcal{D}_{\textbf{pool}}=\mathcal{D}_{\textbf{pool}}\setminus\mathcal{Q}_{i}$

19

20 end for

Output: $\mathcal{D}_{\textbf{lab}}$

Algorithm 1 AL with Pretrained LMs

datasets | train | val | test | $k$ | $C$
---|---|---|---|---|---
trec-6 |  $4.9$K | $546$ | $500$ | $1\%$ | $6$
dbpedia |  $20$K |  $2$K |  $70$K |  $1$% | $14$
imdb |  $22.5$K |  $2.5$K |  $25$K |  $1$% | $2$
sst-2 |  $60.6$K |  $6.7$K | $871$ |  $1$% | $2$
agnews |  $114$K |  $6$K |  $7.6$K | $0.5\%$ | $4$
|  |  |  |  |

Table 1: Datasets statistics for $\mathcal{D}_{\textbf{pool}}$,
$\mathcal{D}_{\textbf{val}}$ and $\mathcal{D}_{\textbf{test}}$ respectively.
$k$ stands for the acquisition size (% of $\mathcal{D}_{\textbf{pool}}$) and
$C$ the number of classes.

Figure 1: Test accuracy during AL iterations. We plot the median and standard
deviation across five runs.

##  3 Experiments & Results

#### Data

We experiment with five diverse natural language understanding tasks: question
classification (trec-6; Voorhees and Tice (2000)), sentiment analysis (imdb;
Maas et al. (2011), sst-2 Socher et al. (2013)) and topic classification
(dbpedia, agnews; Zhang et al. (2015)), including binary and multi-class
labels and varying dataset sizes (Table 1). More details can be found in
Appendix A.1.

#### Experimental Setup

We perform all AL experiments using BERT-base Devlin et al. (2019) and
Entropy, BertKM, Alps Yuan et al. (2020), Badge Ash et al. (2020) and Random
(baseline) as the acquisition functions. We pair our proposed training
approach tapt-ft+ with Entropy acquisition. We refer the reader to Appendix A
for an extended description of our experimental setup, including the datasets
used (§A.1), the training and AL details (§A.2), the model hyperparameters
(§A.3) and the baselines (§A.4).

#### Results

Figure 1 shows the test accuracy during AL iterations. We first observe that
our proposed approach (tapt-ft+) achieves large data efficiency reaching the
full-dataset performance within the $15\%$ budget for all datasets, in
contrast to the standard AL approach (Bert-sft). The effectiveness of our
approach is mostly notable in the smaller datasets. In trec-6, it achieves the
goal accuracy with almost $10\%$ annotated data, while in dbpedia only in the
first iteration with $2\%$ of the data. After the first AL iteration in imdb,
tapt-ft+, it achieves only $2.5$ points of accuracy lower than the performance
when using $100\%$ of the data. In the larger sst-2 and agnews datasets, it is
closer to the baselines but still outperforms them, achieving the full-dataset
performance with $8\%$ and $12\%$ of the data respectively. We also observe
that in all five datasets, the addition of our proposed pretraining step
(tapt) and fine-tuning technique (ft+) leads to large performance gains,
especially in the first AL iterations. This is particularly evident in trec-6,
dbpedia and imdb datasets, where after the first AL iteration (i.e. equivalent
to $2\%$ of training data) tapt+ft+ with Entropy is $45$, $30$ and $12$ points
in accuracy higher than the Entropy baseline with Bert and sft.

#### Training vs. Acquisition Strategy

We finally observe that the performance curves of the various acquisition
functions considered (i.e. dotted lines) are generally close to each other,
suggesting that the choice of the acquisition strategy may not affect
substantially the AL performance in certain cases. In other words, we conclude
that the training strategy can be more important than the acquisition
strategy. We find that uncertainty sampling with Entropy is generally the best
performing acquisition function, followed by Badge.555We provide results with
additional acquisition functions in the Appendix B.2 and B.3. Still, finding a
universally well-performing acquisition function, independent of the training
strategy, is an open research question.

##  4 Analysis & Discussion

###  4.1 Task-Adaptive Pretraining

We first present details of our implementation of tapt (§2) and reflect on its
effectiveness in the AL pipeline. Following Gururangan et al. (2020), we
continue pretraining Bert for the MLM task using all the unlabeled data
$\mathcal{D}_{\textbf{pool}}$ for all datasets separately. We plot the
learning curves of Bert-tapt for all datasets in Figure 2. We first observe
that the masked LM loss is steadily decreasing for dbpedia, imdb and agnews
across optimization steps, which correlates with the high early AL performance
gains of tapt in these datasets (Fig. 1). We also observe that the LM overfits
in trec-6 and sst-2 datasets. We attribute this to the very small training
dataset of trec-6 and the informal textual style of sst-2. Despite the fact
that the sst-2 dataset includes approximately $67$K of training data, the
sentences are very short (i.e. average length of $9.4$ words per sentence). We
hypothesize the LM overfits because of the lack of long and more diverse
sentences. We provide more details on tapt at the Appendix B.1.

Figure 2: Validation MLM loss during tapt.

Figure 3: Few-shot standard Bert fine-tuning.

###  4.2 Few-shot Fine-tuning

In this set of experiments, we aim to highlight that it is crucial to consider
the few-shot learning problem in the early AL stages, which is often neglected
in literature. This is more important when using pretrained LMs, since they
are overparameterized models that require adapting their training scheme in
low data settings to ensure robustness.

To illustrate the potential ineffectiveness of standard fine-tuning (sft), we
randomly undersample the agnews and imdb datasets to form low, medium and high
resource data settings (i.e. $100$, $1,000$ and $10,000$ training samples),
and train Bert for a fixed number of $3$, $10$, and $20$ epochs. We repeat
this process with $10$ different random seeds to account for stochasticity in
sampling and we plot the test accuracy in Figure 3. Figure 3 shows that sft is
suboptimal for low data settings (e.g. $100$ samples), indicating that more
optimization steps (i.e. epochs) are needed for the model to adapt to the few
training samples Zhang et al. (2020); Mosbach et al. (2021). As the training
samples increase (e.g. $1,000$), fewer epochs are often better. It is thus
evident that there is not a clearly optimal way to choose a predefined number
of epochs to train the model given the number of training examples. This
motivates the need to find a fine-tuning policy for AL that effectively adapts
to the data resource setting of each iteration (independently of the number of
training examples or dataset), which is mainly tackled by our proposed fine-
tuning approach ft+ (§2).

###  4.3 Ablation Study

We finally conduct an ablation study to evaluate the contribution of our two
proposed steps to the AL pipeline; the pretraining step (tapt) and fine-tuning
method (ft+). We show that the addition of both methods provides large gains
compared to standard fine-tuning (sft) in terms of accuracy, data efficiency
and uncertainty calibration. We compare Bert with sft, Bert with ft+ and Bert-
tapt with ft+. Along with test accuracy, we also evaluate each model using
uncertainty estimation metrics Ovadia et al. (2019): Brier score, negative log
likelihood (NLL), expected calibration error (ECE) and entropy. A well-
calibrated model should have high accuracy and low uncertainty.

Figure 4 shows the results for the smallest and largest datasets, trec-6 and
agnews respectively. For trec-6, training Bert with our fine-tuning approach
ft+ provides large gains both in accuracy and uncertainty calibration, showing
the importance of fine-tuning the LM for a larger number of epochs in low
resource settings. For the larger dataset, agnews, we see that Bert with sft
performs equally to ft+ which is the ideal scenario. We see that our fine-
tuning approach does not deteriorate the performance of Bert given the large
increase in warmup steps, showing that our simple strategy provides robust
results in both high and low resource settings. After demonstrating that ft+
yields better results than sft, we next compare Bert-tapt-ft+ against Bert-
ft+. We observe that in both datasets Bert-tapt outperforms Bert, with this
being particularly evident in the early iterations. This confirms our
hypothesis that by implicitly using the entire pool of unlabeled data for
extra pretraining (tapt), we boost the performance of the AL model using less
data.

Figure 4: Ablation study for tapt and ft+.

##  5 Conclusion

We have presented a simple yet effective training scheme for AL with
pretrained LMs that accounts for varying data availability and instability of
fine-tuning. Specifically, we propose to first continue pretraining the LM
with the available unlabeled data to adapt it to the task-specific domain.
This way, we leverage not only the available labeled data at each AL
iteration, but the entire unlabeled pool. We further propose a method to fine-
tune the model during AL iterations so that training is robust in both low and
high resource data settings.

Our experiments show that our approach yields substantially better results
than standard fine-tuning in five standard NLP datasets. Furthermore, we find
that the training strategy can be more important than the acquisition
strategy. In other words, a poor training strategy can be a crucial impediment
to the effectiveness of a good acquisition function, and thus limit its
effectiveness (even over random sampling). Hence, our work highlights how
critical it is to properly adapt a pretrained LM to the low data resource AL
setting.

As state-of-the-art models in NLP advance rapidly, in the future we would be
interested in exploring the use of larger LMs, such as Gpt-3 Brown et al.
(2020) and Flan Wei et al. (2022). These models have achieved impressive
performance in very low data resource settings (e.g. zero-shot and few-shot),
so we would imagine they would be good candidates for the challenging setting
of active learning.

## Acknowledgments

We would like to thank Giorgos Vernikos, our colleagues at the Sheffield NLP
group for feedback on an earlier version of this paper, and all the anonymous
reviewers for their constructive comments. KM and NA are supported by Amazon
through the Alexa Fellowship scheme.

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##  Appendix A Appendix: Experimental Setup

###  A.1 Datasets

We experiment with five diverse natural language understanding tasks including
binary and multi-class labels and varying dataset sizes (Table 1). The first
task is question classification using the six-class version of the small
trec-6 dataset of open-domain, fact-based questions divided into broad
semantic categories Voorhees and Tice (2000). We also evaluate our approach on
sentiment analysis using the binary movie review imdb dataset Maas et al.
(2011) and the binary version of the sst-2 dataset Socher et al. (2013). We
finally use the large-scale agnews and dbpedia datasets from Zhang et al.
(2015) for topic classification. We undersample the latter and form a
$\mathcal{D}_{\textbf{pool}}$ of $20$K examples and
$\mathcal{D}_{\textbf{val}}$ $2$K as in Margatina et al. (2021). For trec-6,
imdb and sst-2 we randomly sample $10\%$ from the training set to serve as the
validation set, while for agnews we sample $5$%. For the dbpedia dataset we
undersample both training and validation datasets (from the standard splits)
to facilitate our AL simulation (i.e. the original dataset consists of $560$K
training and $28$K validation data examples). For all datasets we use the
standard test set, apart from the sst-2 dataset that is taken from the glue
benchmark Wang et al. (2019) we use the development set as the held-out test
set (and subsample a development set from the original training set).

###  A.2 Training & AL Details

We use BERT-base Devlin et al. (2019) and fine-tune it (tapt §2) for $100$K
steps, with learning rate $2e-05$ and the rest of hyperparameters as in
Gururangan et al. (2020) using the HuggingFace library Wolf et al. (2020). We
evaluate the model $5$ times per epoch on $\mathcal{D}_{val}$ and keep the one
with the lowest validation loss as in Dodge et al. (2020). We use the code
provided by Kirsch et al. (2019) for the uncertainty-based acquisition
functions and Yuan et al. (2020) for Alps, Badge and BertKM. We use the
standard splits provided for all datasets, if available, otherwise we randomly
sample a validation set. We test all models on a held-out test set. We repeat
all experiments with five different random seeds resulting into different
initializations of $\mathcal{D}_{\textbf{lab}}$ and the weights of the extra
task-specific output feedforward layer. For all datasets we use as budget the
$15\%$ of $\mathcal{D}_{\textbf{pool}}$. Each experiment is run on a single
Nvidia Tesla V100 GPU.

###  A.3 Hyperparameters

For all datasets we train BERT-base Devlin et al. (2019) from the HuggingFace
library Wolf et al. (2020) in Pytorch Paszke et al. (2019). We train all
models with batch size $16$, learning rate $2e-5$, no weight decay, AdamW
optimizer with epsilon $1e-8$. For all datasets we use maximum sequence length
of $128$, except for imdb and agnews that contain longer input texts, where we
use $256$. To ensure reproducibility and fair comparison between the various
methods under evaluation, we run all experiments with the same five seeds that
we randomly selected from the range $[1,9999]$.

###  A.4 Baselines

#### Acquisition functions

We compare Entropywith four baseline acquisition functions. The first is the
standard AL baseline, Random, which applies uniform sampling and selects $k$
data points from $\mathcal{D}_{\textbf{pool}}$ at each iteration. The second
is Badge Ash et al. (2020), an acquisition function that aims to combine
diversity and uncertainty sampling. The algorithm computes gradient embeddings
$g_{x}$ for every candidate data point $x$ in $\mathcal{D}_{\textbf{pool}}$
and then uses clustering to select a batch. Each $g_{x}$ is computed as the
gradient of the cross-entropy loss with respect to the parameters of the
model’s last layer. We also compare against a recently introduced cold-start
acquisition function called Alps Yuan et al. (2020). Alps acquisition uses the
masked language model (MLM) loss of Bert as a proxy for model uncertainty in
the downstream classification task. Specifically, aiming to leverage both
uncertainty and diversity, Alps forms a surprisal embedding $s_{x}$ for each
$x$, by passing the unmasked input $x$ through the Bert MLM head to compute
the cross-entropy loss for a random 15% subsample of tokens against the target
labels. Alps clusters these embeddings to sample $k$ sentences for each AL
iteration. Last, following Yuan et al. (2020), we use BertKM as a diversity
baseline, where the $l_{2}$ normalized Bert output embeddings are used for
clustering.

#### Models & Fine-tuning Methods

We evaluate two variants of the pretrained language model; the original Bert
model, used in Yuan et al. (2020) and Ein-Dor et al. (2020)666Ein-Dor et al.
(2020) evaluate various acquisition functions, including entropy with MC
dropout, and use Bert with the standard fine-tuning approach (sft)., and our
adapted model Bert-tapt (§2), and two fine-tuning methods; our proposed fine-
tuning approach ft+ (§2) and standard Bert fine-tuning sft.

Model | trec-6 | dbpedia | imdb | sst-2 | agnews
---|---|---|---|---|---
Validation Set
bert | 94.4 | 99.1 | 90.7 | 93.7 | 94.4
bert-tapt | 95.2 | 99.2 | 91.9 | 94.3 | 94.5
Test Set
bert | 80.6 | 99.2 | 91.0 | 90.6 | 94.0
bert-tapt | 77.2 | 99.2 | 91.9 | 90.8 | 94.2
|  |  |  |  |

Table 2: Accuracy with $100\%$ of data over five runs (different random
seeds).

##  Appendix B Appendix: Analysis

###  B.1 Task-Adaptive Pretraining (tapt) & Full-Dataset Performance

As discussed in §2 and §4, we continue training the BERT-base Devlin et al.
(2019) pretrained masked language model using the available data
$\mathcal{D}_{\textbf{pool}}$. We explored various learning rates between
$1e-4$ and $1e-5$ and found the latter to produce the lowest validation loss.
We trained each model (one for each dataset) for up to $100$K optimization
steps, we evaluated on $\mathcal{D}_{\textbf{val}}$ every $500$ steps and
saved the checkpoint with the lowest validation loss. We used the resulting
model in our (Bert-tapt) experiments. We plot the learning curves of masked
language modeling task (tapt) for three datasets and all considered learning
rates in Figure 5. We notice that a smaller learning rate facilitates the
training of the MLM.

In Table 2 we provide the validation and test accuracy of Bert and Bert-tapt
for all datasets. We present the mean across runs with three random seeds. For
fine-tuning the models, we used the proposed approach ft+ (§2).

Figure 5: Learning curves of tapt for various learning rates. | trec-6 | sst-2 | imdb | dbpedia | agnews
---|---|---|---|---|---
Random | 0/0 | 0/0 | 0/0 | 0/0 | 0/0
Alps | 0/57 | 0/478 | 0/206 | 0/134 | 0/634
Badge | 0/63 | 0/23110 | 0/1059 | 0/192 | -
BertKM | 0/47 | 0/2297 | 0/324 | 0/137 | 0/3651
Entropy | 81/0 | 989/0 | 557/0 | 264/0 | 2911/0
Least Confidence | 69/0 | 865/0 | 522/0 | 256/0 | 2607/0
BALD | 69/0 | 797/0 | 524/0 | 256/0 | 2589/0
BatchBALD | 69/21 | 841/1141 | 450/104 | 256/482 | 2844/5611
|  |  |  |  |
Table 3: Runtimes (in seconds) for all datasets. In each cell of the table we
present a tuple $i/s$ where $i$ is the inference time and $s$ the selection
time. Inference time is the time for the model to perform a forward pass for
all the unlabeled data in $\mathcal{D}_{\textbf{pool}}$ and selection time is
the time that each acquisition function requires to rank all candidate data
points and select $k$ for annotation (for a single iteration). Since we cannot
report the runtimes for every model in the AL pipeline (at each iteration the
size of $\mathcal{D}_{\textbf{pool}}$ changes), we provide the median.

Figure 6: Comparison of acquisition functions using tapt and ft+ in training
Bert.

###  B.2 Performance of Acquisition Functions

In our Bert-tapt-ft+ experiments so far, we showed results with Entropy. We
have also experimented with various uncertainty-based acquisition functions.
Specifically, four uncertainty-based acquisition functions are used in our
work: Least Confidence, Entropy, BALD and BatchBALD. Least Confidence Lewis
and Gale (1994) sorts $\mathcal{D}_{pool}$ by the probability of not
predicting the most confident class, in descending order, Entropy Shannon
(1948) selects samples that maximize the predictive entropy, and BALD Houlsby
et al. (2011), short for Bayesian Active Learning by Disagreement, chooses
data points that maximize the mutual information between predictions and
model’s posterior probabilities. BatchBALD Kirsch et al. (2019) is a recently
introduced extension of BALD that jointly scores points by estimating the
mutual information between multiple data points and the model parameters. This
iterative algorithm aims to find batches of informative data points, in
contrast to BALD that chooses points that are informative individually. Note
that Least Confidence, Entropy and BALD have been used in AL for NLP by
Siddhant and Lipton (2018). To the best of our knowledge, BatchBALD is
evaluated for the first time in the NLP domain.

Instead of using the output softmax probabilities for each class, we use a
probabilistic formulation of deep neural networks in order to acquire better
calibrated scores. Monte Carlo (MC) dropout Gal and Ghahramani (2016) is a
simple yet effective method for performing approximate variational inference,
based on dropout Srivastava et al. (2014). Gal and Ghahramani (2016) prove
that by simply performing dropout during the forward pass in making
predictions, the output is equivalent to the prediction when the parameters
are sampled from a variational distribution of the true posterior. Therefore,
dropout during inference results into obtaining predictions from different
parts of the network. Our Bert-based $\mathcal{M}_{i}$ model uses dropout
layers during training for regularization. We apply MC dropout by simply
activating them during test time and we perform multiple stochastic forward
passes. Formally, we do $N$ passes of every $x\in\mathcal{D}_{\textbf{pool}}$
through $\mathcal{M}_{i}(x;W_{i})$ to acquire $N$ different output probability
distributions for each $x$. MC dropout for AL has been previously used in the
literature Gal et al. (2017); Shen et al. (2017); Siddhant and Lipton (2018);
Lowell and Lipton (2019); Ein-Dor et al. (2020); Shelmanov et al. (2021).

Our findings show that all functions provide similar performance, except for
BALD that slightly underperforms. This makes our approach agnostic to the
selected uncertainty-based acquisition method. We also evaluate our proposed
methods with our baseline acquisition functions, i.e. Random, Alps, BertKM and
Badge, since our training strategy is orthogonal to the acquisition strategy.
We compare all acquisition functions with Bert-tapt-ft+ for agnews and imdb in
Figure 6. We observe that in general uncertainty-based acquisition performs
better compared to diversity, while all acquisition strategies have benefited
from our training strategy (tapt and ft+).

###  B.3 Efficiency of Acquisition Functions

In this section we discuss the efficiency of the eight acquisition functions
considered in this work; Random, Alps, Badge, BertKM, Entropy, Least
Confidence, BALD and BatchBALD.

In Table 3 we provide the runtimes for all acquisition functions and datasets.
Each AL experiments consists of multiple iterations and (therefore multiple
models), each with a different training dataset $\mathcal{D}_{\textbf{lab}}$
and pool of unlabeled data $\mathcal{D}_{\textbf{pool}}$. In order to evaluate
how computationally heavy is each method, we provide the median of all the
models in one AL experiment. We calculate the runtime of two types of
functionalities. The first is the inference time and stands for the forward
pass of each $x\in\mathcal{D}_{\textbf{pool}}$ to acquire confidence scores
for uncertainty sampling. Random, Alps, Badge and BertKM do not require this
step so it is only applied of uncertainty-based acquisition where acquiring
uncertainty estimates with MC dropout is needed. The second functionality is
selection time and measures how much time each acquisition function requires
to rank and select the $k$ data points from $\mathcal{D}_{\textbf{pool}}$ to
be labeled in the next step of the AL pipeline. Random, Entropy, Least
Confidence and BALD perform simple equations to rank the data points and
therefore so do not require selection time. On the other hand, Alps, Badge,
BertKM and BatchBALD perform iterative algorithms that increase selection
time. From all acquisition functions Alps and BertKM are faster because they
do not require the inference step of all the unlabeled data to the model.
Entropy, Least Confidence and BALD require the same time for selecting data,
which is equivalent for the time needed to perform one forward pass of the
entire $\mathcal{D}_{\textbf{pool}}$. Finally Badge and BatchBALD are the most
computationally heavy approaches, since both algorithms require multiple
computations for the selection time. Random has a total runtime of zero
seconds, as expected.


# Perturbative $\lambda$-Supersymmetry and Small $\kappa$-Phenomenology

Sibo Zheng  Department of Physics, Chongqing University, Chongqing 401331, P.
R. China

###### Abstract

For the minimal $\lambda$-supersymmetry, it stays perturbative to the GUT
scale for $\lambda\leq 0.7$. This upper bound is relaxed when one either takes
the criteria that all couplings close to $\sim 4\pi$ for non-perturbation or
allows new fields at the intermediate scale between the weak and GUT scale. We
show that a hidden $U(1)_{X}$ gauge sector with spontaneously broken scale
$\sim 10$ TeV improves this bound as $\lambda\leq 1.23$ instead. This may
induce significant effects on Higgs physics such as decreasing fine tuning
involving the Higgs scalar mass, as well as on the small
$\kappa$-phenomenology.

##  I I. Introduction

The standard model (SM)-like Higgs boson with mass around 126 GeV h  reported
at the LHC strongly favors extension beyond the minimal supersymmetric model
(MSSM). A simple idea is adding a SM singlet $S$ to the MSSM, with
superpotential

$\displaystyle{}W=\lambda~{}SH_{u}H_{d}+\frac{\kappa}{3}S^{3}.$ (1)

Due to the Yukawa coupling $\lambda$ in Eq.(1), the Higgs boson mass can be
naturally uplifted to observed value for moderate value $\lambda\geq 0.5$ at
the electroweak (EW) scale. This specific extension is referred as the next-
to-minimal supersymmetric model (NMSSM) 0910.1785  for $\lambda\leq 0.7$, or
$\lambda$-supersymmetry ($\lambda$-SUSY) 0607332  for $0.7\leq\lambda\leq 2$.
Either the NMSSM or $\lambda$-SUSY is very attractive from the viewpoint of
naturalness argument 1112.2703 ; 1312.0181 , as the stringent tension on stop
masses required by the Higgs mass in the MSSM is dramatically reduced.

The origin of upper bound on $\lambda$ above can be seen from the beta
function $\beta_{\lambda}$ for $\lambda$. Since $\beta_{\lambda}$ is dominated
by the top Yukawa coupling $y_{t}$, the sign of which is positive, it imposes
an upper bound on the EW value of $\lambda$ in order to be still in the
perturbative region at high energy scale. Given this scale near the grand
unification (GUT) scale $\simeq 1.0\times 10^{15}$ GeV, the critical value was
found to be $\sim 0.7$ 9801437  in terms of the one-loop beta function 111In
this paper, we follow the convention in 9709356 , and present our $\beta$
functions in the $\overline{DR}$ scheme.,

$\displaystyle{}\beta_{\lambda}=\frac{\lambda}{16\pi^{2}}\left[4\lambda^{2}+3y^{2}_{t}+2\kappa^{2}-3g^{2}_{2}-\frac{3}{5}g^{2}_{1}\right],$
(2)

where $t\equiv\ln(\mu/\mu_{0})$, $\mu$ being the running renormalization scale
and $\mu_{0}\equiv 1$ TeV. On the other hand, the observed Higgs mass favors
large value of $\lambda$ 1310.0459 .

The intuition for solving the tension between the observed Higgs mass and
perturbativity is obvious in the context of SUSY. The main observation is that
Yukawa couplings in superpotential receive only wave-function induced
renormalizations due to the protection of SUSY, in contrast to non-
supersymmetric case 222The phenomenon of Yukawa coupling being asymptotically
free has been exposed in $\lambda\phi^{4}$ scalar field theory with dimension
lower than four by Wilson and Fisher Wilson , and in well-known nonlinear
sigma model Polyakov  together with an attempt to obtain a version of four-
dimensional supersymmetry Ketov . Similar phenomenon was also addressed by
authors of 9406199  in four-dimensional scalar field theory with nonpolynomial
potentials.. By following the fact that the beta functions of Yukawa couplings
are related to the anomalous dimensions of chiral fields 9709356 , and the
anomalous dimensions are proportional to quadratic Yukawa couplings with
positive coefficients, the sign of contribution to $\beta$ function due to
Yukawa interactions is thus always positive. A way to alleviate this is
introducing hidden super-confining gauge dynamics 0405267 ; 0311349 , from
which Yukawa couplings are asymptotically free, and $S$ is a composite other
than fundamental state, with large $\lambda$ at low energy as a result of this
asymptotical freedom. In this context the Higgs doublets can be either
fundamental 0405267  or composite states 0311349 .

In this paper, we will explore $\lambda$-SUSY that stays perturbative up to
the GUT scale in alternative way. In this framework we will obtain a well
defined $\lambda$-SUSY on the realm of perturbative analysis, and operators
such as Higgs doublets and singlet $S$ are all fundamental other than
composite states. This is our main motivation for this study.

The study of well defined $\lambda$-SUSY was initially addressed in
Ref.9801437 , in which it was found that for $\tan\beta$ smaller than $\sim
10$, $\lambda$ is the first Yukawa coupling running into the non-perturbative
region at high energy scale. It was also understood that either introducing
new matter fields at the intermediate scale between the weak and GUT scale or
adopting smaller EW value $\kappa(\mu_{0})$ can decrease the evolution rate
for $\lambda$. Given an initial EW value $\kappa(\mu_{0})$, one can derive the
upper bound on $\lambda(\mu_{0})$ or vice versa, once new matters which appear
at the intermediate scale are identified explicitly. The minimal content of
new fields only includes the messenger sector which communicates the SUSY
breaking effect to the visible sector, namely the $\lambda$-SUSY.

Together with the messengers a spontaneously broken $U(1)_{X}$ gauge group
will be considered as the new fields. We consider a class of models different
from those studied in the literature 0404251 , in which SM fermions and
sfermions except the Higgs doublets and $S$ are all charged under the hidden
$U(1)_{X}$. The abelian gauge coupling $g_{X}$ and the $U(1)_{X}$-breaking
scale $M_{X}$ enter into the parameter space as two new free parameters. In
comparison with traditional NMSSM or $\lambda$-SUSY, in our model the beta
function for top Yukawa coupling $\beta_{t}$ receives new and negative
contribution due to the hidden $U(1)_{X}$ sector, the magnitude of which is
determined by the hidden gauge coupling $g_{X}$ and scale $M_{X}$. As long as
$g_{X}$ is large enough but still valid for perturbative analysis, these new
effects decrease the slope of $\lambda$ as function of renormalization scale
$\mu$ above scale $M_{X}$, and therefore lead to larger critical value
$\lambda(\mu_{0})$.

The paper is organized as follows. In section 2, in terms of one-loop
renormalization group equations (RGEs) for relevant coupling constants we
estimate the critical values of $\kappa(\mu_{0})$ and $\lambda(\mu_{0})$ due
to the hidden $U(1)_{X}$ effect. In section 3, we revise the phenomenology for
large $\lambda$ but small $\kappa$. Finally we conclude in section 4. In
appendix A, we show the details of the hidden sector and briefly review
collider constraints on the model parameters.

##  II II. Perturbative $\lambda$-SUSY

As mentioned in the introduction, the beta function for Yukawa couplings of
superpotential can be easily estimated by using the non-renormalization of
superpotential. Firstly, in the case without hidden $U(1)_{X}$ gauge group the
one-loop beta functions for Yukawa couplings in $\lambda$-SUSY are given by
333The value of $\tan\beta$ defined as the ratio
$\left<H^{0}_{u}\right>/\left<H^{0}_{d}\right>$ is required to be smaller than
$\sim 10$ in light of the 126 Higgs mass in the NMSSM or $\lambda$-SUSY. In
0810.0989  it has been shown that experimental constraints require
$\tan\beta\leq 10$ for $\lambda\geq 1.0$. Thus, it is a good approximation to
ignore the bottom Yukawa coupling in compared with top Yukawa coupling.,

$\displaystyle{}\beta_{\lambda}$ $\displaystyle=$
$\displaystyle\frac{\lambda}{16\pi^{2}}\left[4\lambda^{2}+2\kappa^{2}+3y^{2}_{t}-3g_{2}^{2}-\frac{3}{5}g^{2}_{1}\right],$
$\displaystyle\beta_{y_{t}}$ $\displaystyle=$
$\displaystyle\frac{y_{t}}{16\pi^{2}}\left[6y^{2}_{t}+\lambda^{2}-\frac{16}{3}g^{2}_{3}-3g^{2}_{2}-\frac{13}{15}g^{2}_{1}\right],$
$\displaystyle\beta_{\kappa}$ $\displaystyle=$
$\displaystyle\frac{\kappa}{16\pi^{2}}\left[6\lambda^{2}+6\kappa^{2}\right].$
(3)

The one-loop beta functions for the SM gauge couplings are the same as the
MSSM,

$\displaystyle{}\frac{\partial}{\partial t}\alpha^{-1}_{i}$ $\displaystyle=$
$\displaystyle-\frac{b_{i}}{2\pi}.$ (4)

where $(b_{1},b_{2},b_{3})=(33/5,1,-3)$.

In the case with a hidden $U(1)_{X}$ gauge group, we study the model in which
SM fermions and sfermions all carry a hidden $U(1)_{X}$ charge, with the
spontaneously broken scale $M_{X}\simeq 10$ TeV. The Higgs doublets, however,
are singlets of this $U(1)_{X}$ symmetry. Anomaly free conditions require
three hidden matters $X_{1,2,3}$ with the same $U(1)_{X}$ charge added to the
model. For details about the matter representations, see appendix A. The
modifications to RGEs in Eq.(II) due to the hidden $U(1)_{X}$ effect are given
by,

$\displaystyle{}\delta\beta_{\lambda}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\delta\beta_{y_{t}}$ $\displaystyle=$
$\displaystyle-\frac{y_{t}}{16\pi^{2}}\left(\frac{12}{5}a^{2}\cdot
g^{2}_{X}\right),$ $\displaystyle\delta\beta_{\kappa}$ $\displaystyle=$
$\displaystyle 0,$ (5)

together with

$\displaystyle{}\beta_{g_{X}}=\frac{g^{3}_{X}}{16\pi^{2}}\left[\frac{3}{5}a^{2}\cdot\left(\frac{39}{2}n_{g}+3\right)\right],$
(6)

where $n_{g}=3$ denotes the number of SM fermion generations. It is obvious
that large EW value $g_{X}(\mu_{0})$ favors small $\mid a\mid$.

The RG running for $g_{X}$ with different values at $\mu_{0}$ is plotted in
Fig.1. In this figure non-perturbative region lies below the red line. It
shows the critical value $g_{X}(\mu_{0})\leq\\{2.0,1.18,0.82\\}$ for
$a=\\{\frac{1}{6},\frac{1}{3},\frac{1}{2}\\}$, respectively. Given the upper
bound on $g_{X}(\mu_{0})$, the upper bound on Yukawa coupling $\lambda$ can be
estimated in terms of RGEs in Eq.(II) and Eq.(II). The input parameters
related to the critical value $\lambda(\mu_{0})$ are composed of

$\displaystyle{}\\{y_{t}(\mu_{0}),~{}\kappa(\mu_{0}),~{}g_{X}(\mu_{0}),~{}M,~{}n_{5\bar{5}}\\},$
(7)

where $M$ denotes the messenger mass scale, $n_{5\bar{5}}$ represents the
number of $\bar{5}+5$ representation of $SU(5)$ for messengers. We will use
the updated pole mass of top quark $m_{t}=174$ GeV top  instead of $m_{t}=180$
GeV in 9801437  for our analysis, and take the criteria that the theory enters
into the non-perturbative region when any of coupling constants in the theory
is bigger than $\sim 4\pi$.

Figure 1: Critical value $g_{X}(\mu_{0})$ for
$a=\\{\frac{1}{6},\frac{1}{3},\frac{1}{2}\\}$, respectively.

We want to emphasize that $(i)$, below scale $M_{X}$, the modifications to
RGEs arising from $U(1)_{X}$ can be ignored. $(ii)$, above scale $M$,
coefficients $b_{i}$ in beta functions for SM gauge couplings should change as
$b_{i}\rightarrow b_{i}+n_{5\bar{5}}$. Fig. 2 shows the modifications to the
upper bound on $\kappa(\mu_{0})$ due to the hidden $U(1)_{X}$ sector given
fixed $\lambda(\mu_{0})$. We choose the initial EW value
$\lambda(\mu_{0})=1.0$, $g_{X}(\mu_{0})=0.82$ for $a=1/2$, and messenger
paramaters $M=10^{7}$ GeV and $n_{5\bar{5}}=\\{1,4\\}$ for illustration. Given
the critical value $\kappa(\mu_{0})$, the solid line in gray (green)
represents the RG running for $\lambda^{-1}$ without (with) $U(1)_{X}$ effect.
The RG runnings of $\kappa^{-1}$ (dotted) and $y^{-1}_{t}$ (dotdashed) shown
in the figure imply that the upper bound $\kappa_{0}\leq 0.8$ is improved to
$\kappa_{0}\leq 0.85\sim 0.86$ when the hidden $U(1)_{X}$ effect is taken into
account. Alternatively, given the same $\kappa(\mu_{0})$, the upper bound on
$\lambda(\mu_{0})$ can be improved by the $U(1)_{X}$ effect. Combination of
the left and right panel shows that the deviation due to the change of
$n_{5\bar{5}}$ is small, in comparison with the $U(1)_{X}$ effect. Note that
plots in Fig.1 are actually critical lines, because the change from the
perturbative to the non-perturbative region is rather abrupt.

Figure 2: RG running of $\lambda^{-1}$ (solid line) for initial EW value $\lambda(\mu_{0})=1.0$ and $g_{X}(\mu_{0})=0.82$ for $a=1/2$, and messenger parameters $M=10^{7}$ GeV and $n_{5\bar{5}}=1$ (left) and $n_{5\bar{5}}=4$ (right), respectively. The gray and green color corresponds to $\lambda$-SUSY without and with $U(1)_{X}$, respectively. The RG runnings of $\kappa^{-1}$ (dotted) and $y^{-1}_{t}$ (dotdashed) shown in the figure indicate that the upper bound $\kappa_{0}\leq 0.8$ is improved to $\kappa_{0}\leq 0.85\sim 0.86$ when the hidden $U(1)_{X}$ effect is taken into account. Horizontal line refers to critical point between perturbative and non-perturbative region.  $\lambda$(1TeV) |  without $U(1)_{X}$ |  with $U(1)_{X}$
---|---|---
1.0 | $\kappa(\mu_{0})\leq 0.80$ | $\kappa(\mu_{0})\leq 0.85$
1.10 | $\kappa(\mu_{0})=0$ | $\kappa(\mu_{0})\leq 0.81$
1.23 | not well defined | $\kappa(\mu_{0})=0$
Table 1: Upper bound on $\lambda(\mu_{0})$ for $g_{X}(\mu_{0})=0.82$ for
$a=1/2$, messenger parameters $M=10^{7}$ GeV and $n_{5\bar{5}}=1$, and
different initial values of $\kappa(\mu_{0})$. In contrast to the result
$\lambda(\mu_{0})\leq 0.8$ for $\kappa(\mu_{0})\simeq 0$ in 9801437 , we have
$\lambda(\mu_{0})\leq 1.23$ instead for the case with the hidden $U(1)_{X}$.

In table I, we show the improvement on the upper bound on $\lambda(\mu_{0})$
due to the hidden $U(1)_{X}$. Without $U(1)_{X}$ sector, the model is not well
defined up to the GUT scale when $\lambda>0.80$. Instead, $\lambda\leq 1.23$
stays perturbative up to the GUT scale when the hidden $U(1)_{X}$ is added to
the model. The critical value on $\lambda$ may be modified due to different
choices on parameters $\\{n_{5\bar{5}},M,a\\}$. As shown in Fig.2, the
deviation to $\lambda(\mu_{0})$ due to messenger parameters $M$ and
$n_{5\bar{5}}$ is rather small. Similar conclusion holds when one tunes $a$.
Because $a$ larger than $1/2$ leads to larger contribution to
$\delta\beta_{y_{t}}$ but unfortunately smaller $g_{X}(\mu_{0})$, and vice
versa.

In summary, in terms of introducing a hidden $U(1)_{X}$ sector with gauge
symmetry broken scale $\simeq 10$ TeV well defined $\lambda$-SUSY up to the
GUT scale is allowed for $\lambda\leq 1.23$. In this class of models all
fields including singlet $S$ and the Higgs doublets are fundamental. The
implication for $\lambda\sim 1-2$ has been addressed, e.g., in 1310.0459 . In
the next section, we revise the phenomenological implication for such large
$\lambda$ together with small $\kappa$.

##  III III. PQ Symmetry and Small $\kappa$-Phenomenology

This section is devoted to study the phenomenology in $\lambda$-SUSY with
small $\kappa$. Although it is a consequence of taking large $\lambda$ for
well defined $\lambda$-SUSY, the study of small $\kappa$-phenomenology can be
considered as an independent subject from the viewpoint of phenomenology. The
smallness of $\kappa$ can be understood due to a broken $U(1)$ global
symmetry, i.e., Peccei-Quinn (PQ) symmetry. Without $\kappa$, the model is
invariant under the following $U(1)$ symmetry transformation,

$\displaystyle{}H_{u}\rightarrow H_{u}\exp({i\phi}),~{}H_{d}\rightarrow
H_{d}\exp({i\phi}),~{}S\rightarrow S\exp({-2i\phi}),$

Terms like $\delta V=m^{2}S^{2}+B_{\mu}H_{u}H_{d}$ explicitly breaks this
symmetry. For these breaking small, we have a pseudo-Goldstone boson with
small mass.

The soft terms in the scalar potential for small $\kappa$-phenomenology is
given by,

$\displaystyle{}V_{soft}$ $\displaystyle=$ $\displaystyle m^{2}_{S}\mid
S\mid^{2}+m^{2}_{H_{u}}H^{2}_{u}+m^{2}_{H_{d}}H^{2}_{d}$ (9) $\displaystyle+$
$\displaystyle(A_{\lambda}\lambda SH_{u}H_{d}+\text{H.c}),$

where $\kappa$ relevant terms are ignored. The EW symmetry breaking vacuum can
be determined from Eq.(9). For details on analysis of EW breaking vacuum, see,
e.g., 0910.1785 ; 0712.2903 . There are five free parameters

$\displaystyle\\{m^{2}_{H_{u}},~{}m^{2}_{H_{d}},m^{2}_{S},~{}A_{\lambda},~{}\lambda\\}$
(10)

which define the small $\kappa$-phenomenology. The first two can be traded for
$\upsilon=174$ GeV and $\tan\beta$. In what follows we explore the constraints
on these parameters, the Higgs scalar spectrum, and their couplings to SM
particles.

The $3\times 3$ squared mass matrix of CP-even neutral scalars reads 444Here
we follow the conventions and notation in Ref.0712.2903 . ,

$\displaystyle{}\mathcal{M}^{2}=\left(\begin{array}[]{ccccc}\frac{A_{\lambda}^{2}}{1+x}+(M^{2}_{Z}-\lambda^{2}\upsilon^{2})\sin^{2}2\beta&-\frac{1}{2}(M^{2}_{Z}-\lambda^{2}\upsilon^{2})\sin
4\beta&-A_{\lambda}\lambda\upsilon\cos 2\beta\\\
&M^{2}_{Z}\cos^{2}2\beta+\lambda^{2}\upsilon^{2}\sin^{2}2\beta&-A_{\lambda}\lambda\upsilon\sin
2\beta\frac{x}{1+x}\\\ &*&\lambda^{2}\upsilon^{2}(1+x)\\\ \end{array}\right)$
(14)

in the basis $(H,h,s)$, where $x\equiv m^{2}_{S}/(\lambda\upsilon)^{2}$,
$H=\cos\beta h_{2}-\sin\beta h_{1}$ and $h=\cos\beta h_{1}+\sin\beta h_{2}$
under the compositions $H_{u}^{0}=\frac{1}{\sqrt{2}}(h_{1}+i\pi_{1})$,
$H_{d}^{0}=\frac{1}{\sqrt{2}}(h_{2}+i\pi_{2})$ and
$S=\frac{1}{\sqrt{2}}(s+i\pi_{s})$. The state $h$ mixes with other two states
$H$ and $s$ via $\mathcal{M}^{2}_{12}$ and $\mathcal{M}^{2}_{23}$ ,
respectively. Due to the smallness of $\mathcal{M}^{2}_{12}$, state $h$ mixes
dominately with $s$. Without such mixing ( i.e.,
$\mathcal{M}^{2}_{23}\rightarrow 0$ ), $h$ couples to SM gauge bosons and
fermions exactly as the SM Higgs. This implies that in the parameter space of
small $m_{S}$, $h$ is the SM-like scalar discovered at the LHC with
$\mathcal{M}^{2}_{22}=(126\ GeV)^{2}$, and $s$ decouples from SM gauge bosons
with mass $\lambda\upsilon\sim 209$ GeV for $\lambda=1.2$ . Such scalar $s$
easily escapes searches performed by the LEP2 and Run-I of LHC.

The precision measurement of $h$ including its couplings to SM gauge bosons
and fermions powerfully constrains the magnitude of mixing effect. After
mixing, we define the mass eigenstates as $h_{1}$ and $h_{2}$, where
$h_{1}=\cos\theta h-\sin\theta s$, $h_{2}=\cos\theta s+\sin\theta h$.
Normalized to SM Higgs boson couplings, $h_{1}$ and $h_{2}$ couple to SM
particles as,

$\displaystyle{}\xi_{h_{1}VV}$ $\displaystyle=$
$\displaystyle\frac{g^{2}_{h_{1}VV}}{g^{2}_{hVV}}=\cos^{2}\theta,~{}~{}\xi_{h_{2}VV}=\frac{g^{2}_{h_{2}VV}}{g^{2}_{hVV}}=\sin^{2}\theta.$

where $V=\\{W,Z,t,b,\cdots\\}$. At present status, LHC data suggests that
$0.96\leq\cos^{2}\theta\leq 1$ fit . Alternatively we have $\sin^{2}\theta\leq
0.04$.

In Fig.3 we show the parameter space in the two-parameter plane of $m_{S}$ and
$A_{\lambda}$, for $\lambda=1.2$ and $\tan\beta=\\{4.5,5,5.5\\}$,
respectively. For each $\tan\beta$ region below the color contour is excluded
by the condition of stability of potential and mass bound on chargino mass
$m_{\chi}>103$ GeV chargino . Region in the right up corner is excluded by the
precision measurement of Higgs coupling presently. In particular, $m_{S}$
heavier than $\sim$90, 105, and 110 GeV is excluded for $\tan\beta=4.5$, 5 and
5.5, respectively. In comparison with the choice $\lambda=0.7$ discussed in
0712.2903 , the main difference is that for $\lambda=1.2$ it allows larger
$m_{S}$.

As for the Higgs mass constraint, the discrepancy between
$\mathcal{M}^{2}_{22}$ and 126 GeV is compensated by the stop induced loop
correction. The stop mass for the zero mixing effect (i.e., $A_{t}\sim 0$) is
$\sim 340$ GeV for $\tan\beta=4.5$, $\sim 550$ GeV for $\tan\beta=5$ and $\sim
800$ GeV for $\tan\beta=5.5$. Stop mass beneath 1 TeV is favored by
naturalness. Nevertheless, there has tension for such light stop with present
LHC data. This problem can be resolved in some situation, which we will not
discuss here.

We show in Fig.4 the ratio $\xi_{h_{2}VV}$ defined in Eq.(III), which
determines the production rate for scalar $h_{2}$. Its magnitude increases
slowly as $m_{S}$ becomes larger. $\xi_{h_{2}VV}$ reaches $\sim 0.03$ at most
when $m_{S}$ closes to its upper bound (suggested by Fig. 3). For such
strength of coupling and mass $\sim 200$ GeV, $h_{2}$ is easily out of reach
of Run-I at the LHC and earlier attempts at the LEP2. Small ratio of strength
of coupling similarly holds for heaviest CP-even neutral state $H$. In this
sense, it is probably more efficient to probe charged Higgs scalar $H^{\pm}$,
CP-odd scalar $A$, or light pseudo-boson $G$. Studies along this line can be
found in, e.g, 0005308 .

Figure 3: Parameter space in the two-parameter plane of $m_{S}$ and
$A_{\lambda}$, for $\lambda=1.2$ and $\tan\beta=4.5$ (black), 5 (blue), 5.5
(red), respectively. For each $\tan\beta$ region below the color contour is
excluded by the condition of stability of potential and mass bound on chargino
mass $m_{\chi}>103$ GeV. Region in the right up corner is excluded by the fit
to Higgs couplings presently. Figure 4: Ratio of coupling strength for $h_{2}$
shown in the plane of $m_{S}-A_{\lambda}$ for $\lambda=1.2$ and
$\tan\beta=4.5$.

As for other scalar masses we show them in table II for three sets of
$A_{\lambda}$ and $m_{S}$, which are chosen in the parameter space shown in
Fig.3. It is shown that for each case the mass of charged Higgs boson exceeds
its experimental bound $\sim 300$ GeV, and $A$ scalar is always the heaviest
with mass around 600 GeV. In comparison with scalar mass spectrum in fat Higgs
model 0311349 , the spectrum in table II is similar to it, although their high
energy completions are rather different. As a final note we want to mention
that the smallness of $m_{S}$ in compared with $A_{\lambda}$ can be achieved
in model building, e.g., in gauge mediation. Because singlet $S$ only couples
to messengers indirectly through Higgs doublets.

$(A_{\lambda},m_{S})$ (GeV) |  $H^{\pm}$ (GeV) |  $H$ (GeV) |  $A$(GeV)
---|---|---|---
(500, 40) | 463 | 482 | 535
(560, 50) | 527 | 530 | 585
(620, 50) | 590 | 586 | 640
Table 2: Mass spectrum for three sets of choices, which are subtracted from
Fig.3.

##  IV IV. Conclusions

In this paper, we have studied $\lambda$-SUSY which stays perturbative up to
the GUT scale. We find that the bound $\lambda\leq 0.7\sim 0.8$ in the minimal
model is relaxed to $\lambda\leq 1.23$ if a hidden $U(1)_{X}$ gauge theory is
introduced above scale $\sim 10$ TeV and small $\kappa(\mu_{0})$ is assumed at
the same time. This improvement gives rise to several interesting consequences
in phenomenology. For example, the fine tuning related to 126 Higgs mass can
be reduced, and light stop beneath 1 TeV can be allowed. In the light of such
single $U(1)_{X}$, one may introduce multiple $U(1)$s or other gauge sectors
at the intermediate scale, and further uplift the bound on $\lambda$.

In the second part of the paper, we have revised small $\kappa$-phenomenology
for large $\lambda$. In comparison with fat Higgs model 0311349 , the spectrum
in the small $\kappa$-phenomenology is similar, although their high energy
completions are different. The null result for signals of the other two CP-
even neutral scalars $h_{2}$ and $H$ is due to the perfect match between the
scalar discovered at the LHC (Here its is referred as $h_{1}$) and the SM
Higgs. Because the perfect fit dramatically reduces the mixing effect between
$h_{1}$ and the others, which results in tiny strength of coupling for $h_{2}$
and $H$ relative to the SM expectation. The studies on signals of charged
scalar $H^{\pm}$, CP-odd scalar $A$ and pseudo-boson $G$ will shed light on
this type of model.

$\mathbf{Acknowledgement}$
The author thanks J-J. Cao for correspondence and the referee for valuable
suggestions. This work is supported in part by Natural Science Foundation of
China under Grant No.11247031 and 11405015.

##  Appendix A A. The Hidden $U(1)_{X}$ sector

| $SU(3)\times SU(2)\times U(1)_{Y}$ | $U(1)_{X}$
---|---|---
$Q_{i}$ | $(\mathbf{3},\mathbf{2},\frac{1}{6})$ | $a$
$\bar{u}_{i}$ | $(\bar{\mathbf{3}},\mathbf{1},-\frac{2}{3})$ | $-a$
$\bar{d}_{i}$ | $(\bar{\mathbf{3}},\mathbf{1},\frac{1}{3})$ | $-a$
$L_{i}$ | $(\mathbf{1},\mathbf{2},-\frac{1}{2})$ | $-3a$
$\bar{e}_{i}$ | $(\mathbf{1},\mathbf{1},1)$ | $3a$
$H_{u}$ | $(\mathbf{1},\mathbf{2},\frac{1}{2})$ | $0$
$H_{d}$ | $(\mathbf{1},\mathbf{2},-\frac{1}{2})$ | $0$
$X_{1,2,3}$ | $(\mathbf{1},\mathbf{1},0)$ | $3a$
$S$ | $(\mathbf{1},\mathbf{1},0)$ | $0$
Table 3: Charge assignments for the matter superfileds in the visible and
hidden sectors. $Q_{i}$, $\bar{u}_{i}$, $\bar{d}_{i}$, $L_{i}$ and
$\bar{e}_{i}$ denote the three-generation matters of MSSM. $S$ is a singlet of
both SM gauge group and the hidden $U(1)_{X}$. $X_{1,2,3}$ are a set of hidden
matters charged under the hidden $U(1)_{X}$, which are responsible for the
$U(1)_{X}$ gauge symmetry breaking. $a$ is a real number.

$\mathbf{Realistic~{}Model}$. The model that we are going to study for the
case with hidden $U(1)_{X}$ gauge symmetry is presented in table III. The
$U(1)_{X}$ charges in this table must satisfy the gauge anomaly free
conditions and are consistent with the superpotential of visible sector,
$W\sim
y_{u}Q_{i}\bar{u}_{i}H_{u}+y_{d}Q_{i}\bar{d}_{i}H_{d}+y_{e}L_{i}\bar{e}_{i}H_{d}+\mu
H_{u}H_{d}$. We limit to the case in which the Higgs doublets $H_{u,d}$ are
singlets of $U(1)_{X}$. Without the three hidden matters $X_{1,2,3}$ with the
same $U(1)_{X}$ charge anomaly free conditions such as $U(1)_{X}-Graviton-
Graviton$ and $U(1)_{X}-U(1)_{X}-U(1)_{X}$ can not be satisfied 555 We thank
the referee for reminding us that the $U(1)_{X}$ charges in this model are
identical to $B-L$ quantum numbers of the SM fields, and the charges of the
”hidden” matter, $X_{1,2,3}$ are same as those of right-handed neutrino
fields..

$\mathbf{Symmetry~{}Breaking}$. There is no signal of $Z^{\prime}$ from broken
$U(1)_{X}$ gauge group yet, so it should be spontaneously broken above the
weak scale for $g_{X}$ of order SM gauge coupling. This can be achieved in
various ways. Here we simply take the gauge mediation for example. If the
hidden $U(1)_{X}$ gauge group communicates the D-type of SUSY-breaking effects
into $X$s, the sign of soft mass squared $m^{2}_{X}$ would be negative
0705.0865 . For earlier application of such property in gauge mediation, see,
e.g, 0812.4600 . Below SUSY-breaking scale the potential for $X_{i}$ we have

$\displaystyle{}V=-m^{2}_{X_{i}}X_{i}^{{\dagger}}X_{i}+\left(X_{i}^{{\dagger}}X_{i}\right)^{2}.$
(16)

which spontaneously breaks $U(1)_{X}$, with $U(1)_{X}$-breaking scale
$M_{X}\sim m_{X}$. Note that the magnitude of $m_{X}$ can be either larger or
smaller than the soft breaking masses in the visible sector, which depends on
the ratio of $D$ term relative to $F$ term and also the ratio of $g_{X}$
relative to SM gauge couplings. With a $D$ term which gives rise to an order
of magnitude larger than soft breaking masses and $g_{X}$ of the same order as
SM gauge coupling, one can obtain $M_{X}\simeq\mathcal{O}(10)$ TeV and
$m_{\tilde{f_{i}}}\sim\mathcal{O}(1)$ TeV in the visible sector.

$\mathbf{Limit~{}on~{}g_{X}(\mu_{0})}$. The experimental constraint on $M_{X}$
and gauge coupling $g_{X}$ is obtained from direct production of $Z^{\prime}$
at colliders through leptonic decays; and also indirect searches from flavor
violation and electroweak precision tests. For a review on the status of
hidden $U(1)_{X}$, see e.g., Zprime . The parameter $M_{X}$ is constrained to
be above $1\sim 2$ TeV for coupling $g_{X}\sim 1-2$. For $M_{X}\simeq 10$ TeV
adopted in this note, we have the experimental limit $g_{X}\leq M_{X}/(\mid
a\mid\cdot\text{a few TeV})\sim\mid a\mid^{-1}$ limit .

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