#  L-AutoDA: Leveraging Large Language Models for Automated Decision-based
Adversarial Attacks

Ping Guo, Fei Liu, Xi Lin, Qingchuan Zhao, Qingfu Zhang  Department of
Computer Science
City University of Hong Kong
Hong Kong
{pingguo5-c, fliu36-c, xi.lin, qizhao<EMAIL_ADDRESS>
###### Abstract

In the rapidly evolving field of machine learning, adversarial attacks present
a significant challenge to model robustness and security. Decision-based
attacks, which only require feedback on the decision of a model rather than
detailed probabilities or scores, are particularly insidious and difficult to
defend against. This work introduces L-AutoDA (Large Language Model-based
Automated Decision-based Adversarial Attacks), a novel approach leveraging the
generative capabilities of Large Language Models (LLMs) to automate the design
of these attacks. By iteratively interacting with LLMs in an evolutionary
framework, L-AutoDA automatically designs competitive attack algorithms
efficiently without much human effort. We demonstrate the efficacy of L-AutoDA
on CIFAR-10 dataset, showing significant improvements over baseline methods in
both success rate and computational efficiency. Our findings underscore the
potential of language models as tools for adversarial attack generation and
highlight new avenues for the development of robust AI systems.

###### Index Terms:

LLMs, Adversarial Attacks, Automated Algorithm Design

##  I Introduction

Deep neural network (DNN) models, despite their remarkable performance in a
broad spectrum of domains, remain prone to _adversarial attacks_ [1, 2], which
involve imperceptibly altering the input data to induce incorrect responses.
Such vulnerabilities undermine the integrity and reliability of machine
learning systems, particularly in safety-critical scenarios such as autonomous
vehicle driving [3] and medical diagnostics [4]. Attackers can conduct _white-
box attacks_ by exploiting full knowledge of the DNN, or resort to _black-box
attacks_ when the model’s details are concealed [5]. Among these, _decision-
based attacks_ , which require only the model’s final output, present a
significant threat in real-world environments [6, 7]. They are especially
dangerous for commercial platforms that typically only disclose the output
label to users, significantly jeopardizing their security [8].

The intensifying arms race between the advancement of attack methodologies and
the parallel development of defensive measures underscores a critical need for
the automation of adversarial attack algorithm generation [9]. The importance
of this automation is particularly pronounced in decision-based attacks, where
considerable manual effort is required for the development and refinement of
attack strategies. Presently, mainstream strategies for decision-based attacks
predominantly rely on manually crafted heuristics [10, 11, 12, 13, 14], which
imposes significant challenges for further advancements in their
effectiveness.

The automation of adversarial attack algorithm design, underpinned by
automatic program synthesis [15], involves generating programs that conform to
predefined attack specifications. This branch of program synthesis, known
within the machine learning community as AutoML [16], aims to produce
strategies with minimal manual intervention. AutoDA [17] represents a cutting-
edge effort in this domain, adopting a random search within a curated
collection of algebraic operations to develop adversarial attack algorithms.
However, this approach is inherently labor-intensive, as it demands the
development of domain-specific languages and the construction of automated
testing infrastructures. Despite the substantial efforts and resources poured
into this initiative, the autonomous evolution of innovative algorithms with
little reliance on human experts persists as a formidable challenge [17, 18].

Recent advancements highlight the potential of large language models (LLMs) in
algorithm design, as illustrated by Google’s FunSearch [19] and the
evolutionary algorithm community’s AEL [20]. These efforts have underscored
the feasibility of utilizing LLMs for autonomous algorithm design. The
advantages of employing LLMs are manifold: they process natural language
inputs, obviating the need for Domain Specific Language (DSL) encoding,
thereby enabling the construction of novel algorithms unfettered by
traditional encoding constraints; moreover, they can be readily integrated
into prevalent testing frameworks with negligible modifications to the test
scripts. A comparison of this methodology with manual algorithm design and
automatic program synthesis is detailed in TABLE I.

TABLE I: Comparison of strengths and weaknesses of different algorithm design
approaches.

Method | Time | Non-Expertise | Refinement
---|---|---|---
Domain | Extra
Manual | 1-2 Months | ✗ | ✓ | ✓
Automatic Synthesis | 1-2 Months | ✗ | ✗ | ✗
L-AutoDA (Ours) | 1-2 Days | ✓ | ✓ | ✓

In this study, we apply the AEL framework to introduce L-AutoDA, an innovative
automated framework designed for devising decision-based adversarial attacks.
To the best of our knowledge, this work constitutes the first attempt to
utilize LLMs in the development and autonomous evaluation of adversarial
attack algorithms. We have devised targeted prompts and employed a population-
based approach within the AEL framework [20] to derive these strategies.
Notably, the initial suite of algorithms was conceived solely by LLMs and did
not depend on established human-centric design principles. This signifies a
groundbreaking shift away from conventional approaches, featuring a new
paradigm in the autonomous generation of adversarial algorithms.

Our contributions are as follows:

  * •

We present L-AutoDA, an innovative automated framework that leverages LLMs for
developing decision-based adversarial attack algorithms. This constitutes the
first attempt to utilize LLMs in this domain, thereby opening new paradigms in
the field.

  * •

We have demonstrated the superiority of LLMs in crafting adversarial attack
algorithms compared to existing methods. These advantages include: _1)_ the
capacity to generate algorithms from interacting through interactions with
natural language prompts, circumventing reliance on human experts; _2)_ the
proficiency to generate more effective algorithms than those designed by
humans; and _3)_ the capability to produce algorithms that seamlessly
integrate with current testing codes.

  * •

Our experiments and analysis reveal the robust performance of the generated
algorithms, offering new insights into the design of decision-based
adversarial attacks and setting a precedent for future research in this area.

##  II Related Works

###  II-A Decision-based Adversarial Attacks

Decision-based adversarial attacks represent the most challenging scenarios
for attackers due to the minimal information, i.e., the output label,
available about the target model. Nevertheless, they constitute a significant
threat to machine learning applications. A pioneering study by Ilyas et al.
[6] utilized Natural Evolution Strategies (NES) to optimize a surrogate
function employing a constrained number of queries to the model. Subsequent
advancements have focused on refining gradient estimation techniques. For
example, the framework introduced by Cheng et al. in the OPT attack [14]
reformulates the primary optimization challenge. Enhanced techniques such as
Sign-OPT [13], prioritize gradient direction, i.e., sign gradient, over
magnitude, and HopSkipJump [12], which indicates successful gradient
estimation in the initial problem when paired with a binary search to maintain
the estimation proximate to the decision boundary. Additionally, the efficacy
of decision-based attacks has been evidenced by methods within the random walk
paradigm, including Boundary Attack [11] and Evolutionary Attack [10].

###  II-B Automatically Generating Adversarial Attacks.

The automation of adversarial attack algorithms has been a focal point in the
field of adversarial machine learning [21, 17]. The evolution of these
algorithms has seen a shift from elementary gradient-based techniques, such as
the Fast Gradient Sign Method (FGSM) which relies on actual gradient data [2],
to more intricate iterative and optimization-based methods like decision-based
attacks that only necessitate output label data [10, 11, 12, 13, 14].
Researchers have investigated employing genetic algorithms and other
evolutionary strategies for automatic creation of perturbation algorithms. The
inefficiency inherent in searching for an infinite function space prompted the
development of a DSL, constraining function lengths , leading to the state-of-
the-art AutoDA’s demonstrable superiority over traditional approaches [17].
Nonetheless, the design phase of these algorithms remains labor-intensive and
heavily dependent on human expertise, entailing the creation of a domain-
specific language, a code generator, and a testing framework.

###  II-C Large Language Models for Algorithm Design.

In recent years, the expansion of LLMs and the increased availability of
comprehensive training datasets have empowered them with extraordinary
capabilities [22, 23]. LLMs excel in a variety of research areas, especially
in conducting a broad spectrum of tasks in a zero-shot fashion [24, 25, 26,
27, 28, 29, 30, 31, 32, 33, 34, 35, 36]. These developments have opened up new
opportunities for LLMs to process and generate complex algorithmic content.

Expanding their utility further, LLMs have been employed as creative
instruments in algorithm design. For instance, numerous investigations have
capitalized on LLMs that function as black-box components for crafting
evolutionary algorithms, neural architectures, Monte Carlo Tree Search
algorithms, solutions for graph-based combinatorial optimization, genetic
programming, and open-ended challenges [22]. Relying solely on prompts for
interacting with LLMs, however, may lead to less than optimal results. In a
groundbreaking development, combining large language models with evolutionary
computation has led to the autonomous self-improvement of algorithms [20],
code [37], and mathematical functions [38]. This approach harnesses the
potential of LLMs to autonomously conceive and refine cutting-edge algorithms
and tactics within an evolutionary paradigm [39, 38].

Figure 1: Overview of the L-AutoDA Framework Methodology. This diagram
delineates the two core components of our L-AutoDA framework: the algorithm
generation and testing phases. In the algorithm generation phase, we adopt the
AEL framework, leveraging LLMs to guide an evolutionary search process. In the
testing phase, we employ existing decision-based attack testing code,
integrating these algorithms into the attack program to validate their
efficacy. This approach facilitates an efficient exploration of the
algorithmic space through natural language processing, obviating the need for
complex program synthesis frameworks and expediting the development cycle.

##  III Preliminaries

###  III-A Decision-based Adversarial Attacks

Consider an image classifier $\mathcal{M}:\mathcal{X}\to\mathcal{Y}$, hosted
on a cloud server to provide classification services. Here, $\mathcal{X}$
represents the input space containing images with $C$ channels of dimensions
$H\times W$ while $\mathcal{Y}$ represents the output space of $m$ potential
labels, expressed as probabilities. Specifically,
$\mathcal{X}\subseteq[0,1]^{C\times H\times W}$ encompasses all potential
input images, while $\mathcal{Y}\subseteq[0,1]^{m}$ defines the range of
output probabilities.

Attackers interact with this classifier by submitting an input query
$\bm{x}\in\mathcal{X}$ and receiving the output
$\mathcal{M}(\bm{x})\in\mathcal{Y}$ that reflects the classifier’s
predictions. To elucidate the decision-making process, we define the label of
input $\bm{x}$ as $C(\bm{x})=\arg\max_{i}\mathcal{M}_{i}(\bm{x})$,
representing the model’s most confident prediction. In decision-based attacks,
this label constitutes the sole piece of information the attacker obtains.

The decision-based attack introduces a small perturbation $\delta$ that
satisfies $\|\delta\|_{p}\leq\epsilon$ to the original input $\bm{x}$,
yielding a perturbed variant $\bm{x}+\delta$. The objective for the attacker
is to devise a $\delta$ so that the classifier falsely assigns a new label to
the perturbed input $\bm{x}+\delta$. This can be formulated as the
optimization problem below:

$\min\|\delta\|_{p}\quad\text{s.t.}\quad C(\bm{x}+\delta)\neq C(\bm{x}).$ (1)

A successful decision-based adversarial attack is indicated by
$\|\delta\|_{p}\leq\epsilon$. This paper mainly focuses on untargeted attacks
bounded by the $\ell_{2}$-norm ($p=2$), which is the most common setting in
the literature. Nevertheless, we can extend our method to targeted attacks by
simply modifying the constraint to $C(\bm{x}+\delta)=y$, where $y$ is a
specified target label.

###  III-B Algorithm Evolution using Large Language Model

In this study, we adopt the Algorithm Evolution with Large Language Models
(AEL) framework, as proposed by Liu et al. [20]. This framework integrates
evolutionary computing (EC) principles with the advanced capabilities of LLMs.
AEL is comprised of an iterative cycle that includes the fundamental EC
phases: initialization, evaluation, selection, crossover, mutation, and
population size control. The primary objective of AEL is to derive an optimal
algorithm for a specific problem through the repeated execution of these
phases.

Initialization. The initial population can be derived either from pre-existing
algorithms or by generating new ones using LLMs. This flexibility ensures a
thorough exploration of the algorithmic landscape. A carefully crafted prompt
is employed to generate this initial algorithmic population, further details
of which are elaborated in [20].

Evaluating Solutions. A critical aspect of AEL involves the evaluation of the
solution fitness. In our implementation, we utilize decision-based attack
testing to calculate an algorithm’s fitness, which is quantified by measuring
the $\ell_{2}$ distance between adversarial output and original input.

Generating New Solutions. This stage adheres to the established protocols of
EC.

  * •

Selection. Analogous to traditional EC practices, we select a predetermined
number of algorithms to be retained through each iteration.

  * •

Crossover. We facilitate the crossover operation by submitting a pair of
algorithmic candidates, along with guiding prompts, to the LLMs, which in
turn, synthesizes a potentially superior algorithm. This approach leverages
the LLMs’ ability to enhance the search process beyond the random search
capabilities offered by automated program synthesis.

  * •

Mutation. Introducing variation into the algorithmic pool is paramount for
fostering diversity. This is accomplished by instructing the LLMs to introduce
minor modifications to the current algorithms.

##  IV L-AutoDA: LLM-based Automated Decision-based Adversarial Attacks

In this section, we introduce our novel framework, L-AutoDA, which is designed
for automatically generating decision-based adversarial attacks. We begin by
delineating the problem formulation and examining the search space associated
with our framework (Section IV-A). Subsequently, we describe the comprehensive
structure of the L-AutoDA framework (Section IV-B) as well as elaborate on the
specifics of its implementation (Section IV-C). An illustrative overview of
the L-AutoDA architecture is depicted in Figure 1.

###  IV-A Search Space of Decision-based Attack Framework

The concept of designing and implementing a comprehensive algorithm capable of
crafting perturbations that successfully deceive a model constitutes a
significant challenge in automatic program synthesis [15]. The expansive
search space hinders the identification of an optimal algorithm solution,
although the inherent flexibility of this approach offers broad possibilities.

While this strategy represents a plausible path for refining our LLM-based
algorithmic framework and signifies a substantial advance beyond traditional
techniques, we delineate a particular search space for the LLM to explore.
Future work will include extensive experimentation with comprehensive
algorithms. We utilize a random-walk-based template to construct decision-
based attacks, as outlined in Algorithm 1. This approach integrates essential
features from predecessor methods, such as the Evolutionary Attack [10], the
Boundary Attack [11], and various other strategies.

The framework highlights two pivotal components for further improvement: the
generate function and the accompanying hyperparameters. The generate function
plays a vital role in the algorithm, processing the current adversarial
example $\bm{x}_{1}$, the original example $\bm{x}_{0}$, and yielding a new
adversarial instance $\bm{x}$. Hyperparameters are instrumental in regulating
the algorithm’s behavior, influencing elements like step size and iteration
count. To streamline the search process, we adopted the parameter tuning
strategy from Fu et al. [17], concentrating our efforts on refining the
generate function.

Algorithm 1 Random Walk Framework for Decision-Based Attacks under $\ell_{2}$
perturbation

1:Input: original example $\bm{x}_{0}$, adversarial starting point
$\bm{x}_{1}$

2:Output: An adversarial example $\bm{x}$.

3:Initialization: $\bm{x}\leftarrow\bm{x}_{1}$;
$d_{\min}\leftarrow\|\bm{x}-\bm{x}_{0}\|_{2}$

4:while query budget not reached do

5: $\bm{x}^{\prime}\leftarrow\texttt{generate}(\bm{x},\bm{x}_{0})$

6: if $\bm{x}^{\prime}$ is adversarial and
$\|\bm{x}^{\prime}-\bm{x}_{0}\|_{2}<d_{\min}$ then

7: $\bm{x}\leftarrow\bm{x}^{\prime}$;
$d_{\min}\leftarrow\|\bm{x}^{\prime}-\bm{x}_{0}\|_{2}$

8: end if

9: Update hyper-parameters.

10:end while

11:return $\bm{x}$

###  IV-B L-AutoDA

The L-AutoDA framework is an innovative system that integrates with the AEL
paradigm [20] to facilitate the generation of new adversarial attack
algorithms. At its core, L-AutoDA aims to search for a generate function
responsible for producing new adversarial solutions. The obtained generate
functions are then seamlessly integrated into standard decision-based attack
programs, as depicted in Figure 1. This integration enables the continuous
production and assessment of novel attack strategies.

In the L-AutoDA framework, algorithm generation is conducted using a
population-based approach, which involves creating a diverse set of candidate
algorithms. A dedicated test script, the specifics of which are elaborated in
Section. IV-C, is used to evaluate the performance of these candidates. The
performance is quantified by an objective value, defined as the mean distance
between the adversarial images produced by the algorithm and the original
images. This metric serves as a fitness function within the AEL framework,
guiding the evolutionary process. Utilizing this objective value, AEL conducts
evolutionary operations, such as selection, crossover, and mutation, to evolve
the population of algorithms. Throughout this process, the most promising
candidates—or “elite” algorithms—are identified and preserved. This
evolutionary cycle progresses iteratively, fostering the generation of
increasingly effective adversarial attack algorithms.

The generative process of L-AutoDA is designed to work in harmony with
existing decision-based attack programs. It assesses the quality of the
generated algorithms by examining the output the attack program produces when
provided with the generated generate function. This is a significant
advancement over traditional program synthesis methods, which often require
extensive validation to ensure the legitimacy and functional correctness of
the generated code. By focusing on the output and performance of algorithms
rather than their syntactic correctness, L-AutoDA streamlines the search
process and demonstrates its superiority.

###  IV-C Implementation

We delineate the detailed implementation of our L-AutoDA algorithm below.

Function Specification. The algorithm accepts four inputs: the original
example $\bm{x}_{0}$, the adversarial starting point $\bm{x}_{1}$, standard
random noise $\bm{r}$, and a dynamically adjusted hyperparameter $s$. It
produces the adversarial example $\bm{x}$. The algorithm aims to devise a
method that amalgamates these inputs, with the hyperparameter offering
guidance on the reference step size.

Hyper-parameter Tuning. Our approach to hyperparameter tuning adopts the
strategy presented in [17]. We introduce a piece-wise linear function $f(p)$
defined as:

$f(p)=\left\\{\begin{matrix}0.5+2p&0\leq p\leq 0.25\\\
\frac{5}{6}+\frac{2p}{3}&0.25<p\leq 1\\\ \end{matrix}\right.$ (2)

During each iteration, $p$ is updated in the following manner:

$p=0.95p+0.05k$ (3)

where $k$ represents the discovery of an improved adversarial point, taking on
the value of $1$ if a better point is found and $0$ otherwise. The
hyperparameter $s$ is then computed by:

$s=s\cdot[f(p)]^{0.1}$ (4)

This results in a negative feedback loop designed to stabilize $p$ near
$0.25$.

Testing Script. The AEL framework necessitates the use of a fitness function
value to steer its evolutionary process. Consequently, a testing script has
been developed to assess the performance of the evolved algorithms. Instead of
iterating over the whole test set, which is exceedingly time-consuming, we
selected a consistent subset (i.e., 8 images) from the dataset for our
attacks. These samples are used to compute the fitness value, utilizing
standardized attack settings. While this procedure may introduce bias,
empirical evidence has demonstrated its utility in expediting the search
process.

##  V Experiments

###  V-A Experimental Setup

L-AutoDA Generation. The experimental setup for the L-AutoDA algorithm
generation is divided into two distinct parts: 1) settings for the AEL running
process and 2) for the objective value evaluation. Note that the generation
process is conducted on CIFAR-10 dataset [40] and a ResNet-18 classification
model [41].

AEL Settings. In our setting, the AEL framework operates over 20 generations,
each comprising 10 algorithm candidates. Moreover, we set the crossover
probability at 1.0, ensuring that each pair of selected programs undergoes
recombination, and the mutation probability at 0.5 to introduce variability.
The default LLM for algorithm generation is GPT-3.5-turbo-1106, with plans to
expand testing to additional large language models in subsequent research.

Algorithm Evaluation. For the evaluation of the generated algorithms, our
testing script is configured to allow a maximum of 8,000 queries per
algorithm. We execute the algorithms on the first eight images of the CIFAR-10
test set to ensure a consistent and manageable testing environment. The
adversarial images produced are then used to calculate the $\ell_{2}$
distances relative to their original counterparts. The mean of these distances
is computed to serve as the fitness value, which is fed back into the AEL
framework, thereby informing the evolutionary search for more effective attack
algorithms.

TABLE II: The performance of L-AutoDA compared to three baseline algorithms
using the testing script. The mean distance of 1000 images are documented with
the standard variance to be the subscript. The best performance cell is marked
with light gray and the text within is bolded.

Boundary | HSJA | HSJA* | L-Auto-20
---|---|---|---
$0.3939$ | $1.3628$ | $1.2839$ | $\mathbf{0.2517}$

TABLE III: The full test performance of L-AutoDA-20 compared to three baseline
algorithms.

Attack Name | Distance ($\ell_{2}$-norm) | Attack Success Rate
---|---|---
# of Queries | $2500$ | $5000$ | $10000$ | $2500$ | $5000$ | $10000$
Boundary | $1.9107_{1.2665}$ | $1.0938_{0.7861}$ | $0.4495_{0.3340}$ | $\mathbf{14.7}$ | $\mathbf{26.2}$ | $65.5$
HSJA | $2.0512_{1.0876}$ | $1.2833_{0.7442}$ | $0.8978_{0.5360}$ | $9.2$ | $16.1$ | $24.6$
HSJA* | $2.6482_{1.5790}$ | $1.6532_{1.0347}$ | $1.1306_{0.6987}$ | $7.9$ | $13.9$ | $19.6$
L-AutoDA-20 | $\mathbf{1.5202}_{\mathbf{0.1337}}$ | $\mathbf{0.6171}_{\mathbf{0.1430}}$ | $\mathbf{0.3445}_{\mathbf{0.2386}}$ | $0.0$ | $0.5$ | $\mathbf{80.3}$

Attack Evaluation. The evaluation process for different attacks is a crucial
aspect of the experimental setup, providing a comprehensive assessment of the
generated adversarial algorithms’ performance.

Datasets. Our evaluation utilizes a subset of the CIFAR-10 dataset, comprising
100 randomly sampled images from each class, to ensure a diverse and
representative test bed. To facilitate a fair comparison across all attack
algorithms, we introduce a set of 10 images with incorrect labels as the
initial starting points for the attacks, ensuring that each algorithm begins
from a standardized baseline.

Comparative Algorithms. In our comparative analysis, we establish the Boundary
attack [11], which operates under the random walk framework, as the baseline
algorithm. Additionally, we include the widely acknowledged SOTA decision-
based attack algorithm, the HopSkipJumpAttack (HSJA) [12], which employs a
gradient-based approach. To further enrich our comparison, we introduce a
variant of HSJA that utilizes a grid search strategy instead of its default
geometric progression for step search, denoted as HSJA* in our paper. Our
future work anticipates the inclusion of more attack algorithms for a more
exhaustive comparison.

Detailed Parameters. Delving into the detailed parameter settings, for the
Boundary Attack, we configure the spherical step and the source step to be
0.01, with an adaptation rate for the step size set at 1.5. For the
HopSkipJump Attack, we set the value of $\gamma$ to 1.0, with the initial
gradient estimation steps at 100 and capped at a maximum of 10,000. Reflecting
the adaptive nature of the L-AutoDA-generated algorithms, a negative feedback
mechanism is employed to fine-tune the hyperparameter $s$, which is initially
set to 0.001.

###  V-B Algorithm Generation

The performance of the algorithms generated by L-AutoDA is encapsulated in
Figure 2, which demonstrates their compelling capabilities. Remarkably, the
initial iteration of L-AutoDA produced algorithms that outperformed HSJA.
Although this unexpected result may be partially attributed to the limited
subset of images used during testing, it nonetheless underscores the potential
of L-AutoDA in rapidly devising effective attack strategies. As the
evolutionary process progressed, L-AutoDA continued to refine its algorithms,
surpassing both HSJA and Boundary Attack by the 6th generation. This trend of
improvement was consistent, with each subsequent generation enhancing the
algorithms’ effectiveness.

Figure 2: Performance Trajectories of L-AutoDA. This graph illustrates the
comparative efficiency of our L-AutoDA framework against the human-best
gradient-based (HopSkipJump Attack) and gradient-free (Boundary Attack)
methods. L-AutoDA’s candidates demonstrate a breakthrough in the 13th
generation, surpassing the reference performance lines and continuing to
enhance efficiency in subsequent generations.

An intriguing aspect was the reduction in the variance of algorithm
performance within each generation. This convergence suggests a stabilization
of performance across the generated algorithms, indicating that L-AutoDA is
not only producing more effective algorithms over time but also more reliable
ones.

The results of the final round are documented in TABLE II. L-AutoDA’s best
algorithm within the 20th generation, denoted as L-Auto-20, achieved a mean
perturbation distance of 0.2517 across the test images. This represents a
significant improvement over the HSJA and Boundary Attack, which achieved mean
perturbation distances of 1.3628 and 0.3939, respectively.

###  V-C Attack Evaluation

To thoroughly evaluate the algorithms generated, we subjected them to tests on
an expanded subset as delineated in our experimental setup. The most effective
algorithm produced by the final iteration of L-AutoDA, referred to as
L-AutoDA-20, was selected for benchmark comparison.

Overall Results. We have documented the overall full test results in TABLE
III. The table reveals that L-AutoDA-20 is the most effective algorithm,
achieving the lowest mean distance across all query counts. This result is
particularly impressive given that L-AutoDA-20 was generated entirely from
scratch by the LLM, without any human intervention. As for the success rate,
L-AutoDA-20 achieved a 0% success rate at 2500 queries, which is expected
given the limited number of queries. The success rate then increased to 80.3%
at 10000 queries, surpassing all other algorithms. We delineate the
relationship between attack success rate and distance in the following
sections.

Attack Success Rate. Figure 3 illustrates the attack success rate with the
number of queries. A successful attack is defined by an $\ell_{2}$ norm less
than 0.5 between the adversarial example and the original image, consistent
with the widely accepted standard in the current benchmarks [42]. The figure
reveals that L-AutoDA-20’s performance is suboptimal at 2500 and 5000 queries.
However, there is a notable uptick in success rate when the query count
reaches 10000, surpassing all baseline algorithms. This pattern suggests that
L-AutoDA sacrifices initial search efficiency to enhance the quality of the
search at later stages, particularly after 8000 queries (testing script).

Figure 3: Attack Success Rate using different numbers of queries using
L-AutoDA-20 and other attack algorithms.

Distance. We present the comparative analysis of the perturbation distances in
Figure 4, where we plot the mean $\ell_{2}$ distance between the adversarial
and original images against the number of queries used. The shaded areas in
the figure represent a 0.25 multiplier of the standard deviation , providing
insight into the variability of each algorithm’s performance.

From Figure 4, it is evident that L-AutoDA-20 maintains the most consistent
performance across all tested query counts, as indicated by the smallest
standard deviation values. This consistency suggests that L-AutoDA-20 is less
sensitive to the variations in the input data, making it a robust choice for
generating adversarial examples. Although this robustness may come at the cost
of a reduced attack success rate in the initial phase, it becomes a
significant advantage in later stages, particularly beyond 8000 queries.

The stability of L-AutoDA-20 is particularly beneficial when the attack
requires subtlety, as it is capable of producing perturbations that are
minimally perceptible yet still effective. This characteristic is crucial for
scenarios where detectability is a concern and stealth is paramount.

Figure 4: Distance between adversarial examples and original images using
different numbers of queries using L-AutoDA-20 and other attack algorithms.
The lines denote the mean value of the test pairs and the shaded areas
represent a 0.25 multiplier of the standard deviation.

###  V-D Additional Results on Median Distance

To avoid the influence of variations with the images and better illustrate the
effectiveness of our framework, we have demonstrated the median distance of
the adversarial examples generated by different algorithms in TABLE IV. The
results are consistent with the previous analysis, with L-AutoDA-20 achieving
the lowest median distance across all query counts.

TABLE IV: Median distance of L-AutoDA-20 compared to three baseline
algorithms. The best performance cell is marked with light gray and the text
within is bolded.

| $2500$ | $5000$ | $10000$
---|---|---|---
Boundary | $1.7374$ | $0.9489$ | $0.3695$
HSJA | $2.0230$ | $1.2468$ | $0.8646$
HSJA* | $2.5150$ | $1.5580$ | $1.0618$
L-AutoDA-20 | $\mathbf{1.5301}$ | $\mathbf{0.5896}$ | $\mathbf{0.2862}$

###  V-E Interpretation of the algorithms generated by L-AutoDA

The generate function output by L-AutoDA is illustrated in Algorithm 2. The
algorithm starts by taking the difference between the original example
$\mathbf{x}_{0}$ and the adversarial starting point $\mathbf{x}_{1}$. By
moving along this vector, one can generate examples that are in between the
original and the adversarial, which may help in exploring the space around
known data points. Furthermore, efficient search is enabled through inclusion
of another normalized vector $\frac{\mathbf{d}}{norm}$. Then two scales of
noise are added to the example, one with the same direction as the difference
vector $\mathbf{d}$ and the other with the same direction as the normalized
difference vector $\frac{\mathbf{d}}{norm}$. The noise is further scaled by a
hyperparameter $s$ to control the magnitude of the perturbation. Combined with
these search vectors, L-AutoDA is able to generate adversarial examples that
are both effective and efficient.

Algorithm 2 generate()

1:Input: original example $\mathbf{x}_{0}$, adversarial starting point
$\mathbf{x}_{1}$, standard normal noise $\mathbf{n}$, hyperparameter $s$

2:Output: A new proposed example $\mathbf{x}$

3:$\mathbf{d}\leftarrow\mathbf{x}_{0}-\mathbf{x}_{1}$

4:$norm=\max(\|\mathbf{d}\|_{2},\|\mathbf{n}\|_{2})$

5:$\mathbf{x}\leftarrow\mathbf{x}_{1}+s(\mathbf{d}+\frac{\mathbf{d}}{norm})+s(\mathbf{n}+s\frac{\mathbf{n}}{norm})$

##  VI Discussion

Expanded Experimental Validation. Although our experimental framework,
consisting of 20 generations with 10 individuals per generation, has yielded
results surpassing those of manually-designed state-of-the-art algorithms, it
has not fully tested the boundaries of our framework or LLMs. We will increase
the number of generations and individuals to see if we can obtain better
results. We aim to test these limits by increasing the population size and the
number of generations. Additionally, initializing the search process with
existing algorithms and subsequently refining them represents a promising
avenue for further experimentation.

Broader Algorithm Search Space. or expediency and as an initial attempt for
automated attack algorithm design using LLMs into the automated design of
attack algorithms using LLMs, we confined the search space to that defined by
the generate() function. However, this narrow scope may restrict the discovery
of optimal algorithms. Future work will seek to exploit the full potential of
LLMs by allowing them to craft comprehensive algorithms without such
constraints.

Enhancing Prompt Adaptability. Our methodology employed a set of static
prompts to assist LLMs in algorithm generation. However, the fixed prompts may
not be the best prompts for LLMs to generate algorithms. The effectiveness of
these prompts, however, may not represent an optimal use of LLM capabilities.
The concept of chain-of-reasoning, which underpins our work and AEL, suggests
a close relationship with adaptive prompt generation. Investigating methods of
dynamically generating prompts is an objective of our ongoing research.

Addressing Limitations. While the synthesis of programs using large language
models is the focus of our research, it is not without its drawbacks. These
models may occasionally yield unsatisfactory outcomes, albeit at a lower rate
than traditional approaches. Improving the specificity of constraints within
the prompts to ensure the validity of the algorithms produced will be an
integral part of our forthcoming efforts.

##  VII Conclusion

In this paper, we have successfully demonstrated the innovative application of
LLMs for the automatic design of decision-based adversarial attack algorithms.
By leveraging the AEL framework, we have not only streamlined the algorithmic
design process, but also achieved a significant reduction in the time and
expertise required to develop effective adversarial attacks. Our approach,
encapsulated in the L-AutoDA framework, represents a paradigm shift in the
field of adversarial machine learning, showcasing the untapped potential of
LLMs in the realm of security and algorithm synthesis.

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# Interaction of divergence-free deceleration parameter in Weyl-type $f(Q,T)$
gravity

Gaurav N. Gadbail 0000-0003-0684-9702<EMAIL_ADDRESS>Department of
Mathematics, Birla Institute of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India. Simran Arora 0000-0003-0326-8945
<EMAIL_ADDRESS>Department of Mathematics, Birla Institute of
Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India. Praveen Kumar 0000-0002-8201-6019
<EMAIL_ADDRESS>Department of Mathematics, G H Raisoni College of
Engineering, Nagpur-440016, India. P.K. Sahoo 0000-0003-2130-8832
Corresponding author<EMAIL_ADDRESS>Department of
Mathematics, Birla Institute of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India.

###### Abstract

Abstract: We study an extension of symmetric teleparallel gravity i.e. Weyl-
type $f(Q,T)$ gravity and the divergence-free parametrization of the
deceleration parameter $q(z)=q_{0}+q_{1}\frac{z(1+z)}{1+z^{2}}$ ($q_{0}$ and
$q_{1}$ are free constants) to explore the evolution of the universe. By
considering the above parametric form of $q$, we derive the Hubble solution
and further impose it in the Friedmann equations of Weyl-type $f(Q,T)$
gravity. To see whether this model can challenge the $\Lambda$CDM limits, we
computed the constraints on the model parameters using the Bayesian analysis
for the Observational Hubble data ($OHD$) and the Pantheon sample ($SNe\,Ia$).
Furthermore, the deceleration parameter depicts the accelerating behavior of
the universe with the present value $q_{0}$ and the transition redshift
$z_{t}$ (at which the expansion transits from deceleration to acceleration)
with $1-\sigma$ and $2-\sigma$ confidence level. We also examine the evolution
of the energy density, pressure, and effective equation of state parameters.
Finally, we demonstrate that the divergence-free parametric form of the
deceleration parameter is consistent with the Weyl-type $f(Q,T)$ gravity.

Keywords: Deceleration parameter, EoS parameter, $f(Q,T)$ gravity,
Observations.

##  I Introduction

The most significant challenge theoretical physics has faced is the late-time
acceleration expansion of the universe. In accordance with different
observational evidence, such as Type $Ia$ supernovae $(SNe\,Ia)$ [1, 2, 3],
cosmic microwave background radiation (CMBR) [4], and several large-scale
structural measurements [5, 6], it is now widely believed that the universe is
undergoing a phase of accelerated expansion. The most popular explanation is
that dark energy now dominates the universe. Nevertheless, none of the current
dark energy models is entirely acceptable. The most competitive cosmological
dark energy model is a $\Lambda$CDM model, often known as the cosmological
constant model with the equation of state $\omega_{\Lambda}=-1$. The
cosmological constant DE has proven to be a valuable model to account for and
explain many cosmological data. Some theoretical arguments suggest that it
could be necessary to consider dynamical dark energy. In particular, the
coincidence problem- a situation where the cosmological constant dominates the
dynamics of the universe at late times, cannot be explained by any fundamental
hypothesis. Another significant issue is the ”fine-tuning problem”, which
refers to the disparity between the observed value and the theoretically
predicted value of the cosmological constant [7, 8]. Numerous theories for
dark energy with a time-evolving energy density have also been proposed in the
literature to solve or at least lessen the aforementioned cosmological
challenge [9, 10, 11, 12, 13, 14, 15].
Another way to explain the present acceleration of the universe is to modify
the geometry of spacetime. We can do this by modifying the Einstein-Hilbert
general relativity (GR) action. One of the best theories in this context is
$f(R)$ gravity [16, 17], which extends the gravitational action of general
relativity to include a generic scalar curvature function. Modified gravity
based on a non-minimal interaction between matter and geometry, such as
$f(R,T)$ theory [18, 19, 20, 21, 22], $f(R,G)$ theory [23, 24], are another
extensions proposed in the literature.
The Levi-Civita connection is the affine connection specified in the standard
description of General Relativity on the spacetime manifold, which utilizes
Riemannian geometry. However, there are several choices for affine connections
on any manifold and may result in various equivalent descriptions of gravity
[25, 26], which may offer multiple perspectives on understanding. One can
define the Teleparallel equivalent of GR (TEGR) by selecting a connection that
relaxes the constraint on torsion while requiring both curvature and non-
metricity to disappear [27]. Additionally, a new theory named Symmetric
Teleparallel equivalent of GR (STEGR) has been proposed by choosing a
connection that relaxes the constraint on non-metricity while necessitating
the annihilation of both curvature and torsion [28, 29]. In an extension of
STEGR, one may formulate the $f(Q)$ gravity [30], where $Q$ is a non-metricity
representing the fundamental geometrical variable characterizing how a
vector’s length changes when transported. Moreover, a recent extension of
$f(Q)$ gravity known as the $f(Q,T)$ gravity [31] is based on the non-minimal
coupling between the non-metricity $Q$ and the trace of the energy-momentum
tensor $T$. The $f(Q,T)$ theory is constructed in a manner similar to the
$f(R,T)$ theory. Here, the standard Ricci scalar $R$ is replaced by the non-
metricity $Q$, representing the symmetric teleparallel gravity formulation.

Numerous studies have shown that $f(Q,T)$ gravity is useful in explaining the
present cosmic acceleration of the universe and offering a feasible solution
to the dark energy problem [32, 33]. To review $f(Q,T)$ gravity in different
aspects, one can check references [34, 35, 36]. Furthermore, Yixin et al. [37]
explored $f(Q,T)$ gravity in the content of proper Weyl geometry called Weyl-
type $f(Q,T)$ gravity theory. However, in the present approach to $f(Q,T)$
type gravity theories, we have formulated the nonmetricity $Q$ employing the
prescriptions of the Weyl geometry, in which the covariant divergence of the
metric tensor is given by the product of the metric tensor and Weyl vector
$w_{\mu}$. The scalar nonmetricity is related simply to the square of the Weyl
vector as $Q=-6w^{2}$. Thus, the Weyl vector and the metric tensor describe
all of the geometric properties of the theory. Yixin et al. [37] analyzed the
cosmological implications of the Weyl-type $f(Q,T)$ theory for three classes
of specific models. The resulting solutions represent both accelerating and
decelerating evolutionary phases of the universe. It is seen that Weyl type
$f(Q,T)$ gravity may be considered a possible candidate for characterizing the
early and late phases of cosmic evolution. Yang et al. [38] also derived the
geodesic and Raychaudhuri equations using the Weyl $f(Q,T)$ theory. To get in
touch with works in Weyl-type $f(Q,T)$ gravity theory, one can refer [39, 40].
In this work, we consider the divergence-free parametric form of deceleration
parameter, and investigate the FLRW universe in the Weyl-type $f(Q,T)$ gravity
theory by a functional form $f(Q,T)$ as $f(Q,T)=\alpha
Q+\frac{\beta}{6\kappa^{2}}T$, where $\alpha$ and $\beta$ are free parameters.
The present model is based on the divergence-free parametric form of the
deceleration parameter to get the exact solutions to the field equations. The
present article is outlined as follows: In section II, we present the Weyl
Type $f(Q,T)$ gravity formalism. The solution of the field equation and the
Hubble parameter is presented in section III. In section IV, we discuss the
observational data and the methodology used to constrain the model parameters.
The behavior of cosmological parameters, including the deceleration parameter,
pressure, density, and EoS parameter, is covered in section V. Finally, in the
last section VI, we briefly discuss our results.

##  II Field equations

We start with the gravitational action in the Weyl-type $f(Q,T)$ given by:
[37]

$S=\int\left[\kappa^{2}f(Q,T)-\frac{1}{4}W_{\alpha\beta}W^{\alpha\beta}-\frac{1}{2}m^{2}w_{\alpha}w^{\alpha}+\right.\\\
\left.\lambda\tilde{R}+\mathcal{L}_{m}\right]\sqrt{-g}d^{4}x,$ (1)

where, $\tilde{R}=(R+6\nabla_{\mu}w^{\mu}-6w_{\mu}w^{\mu})$,
$\kappa^{2}=\frac{1}{16\pi G}$, a particle associated with a vector field has
a mass of $m$, $\mathcal{L}_{m}$ is the matter Lagrangian. Furthermore, $f$
can be defined as an arbitrary function of the non-metricity $Q$ and the trace
of the matter-energy-momentum tensor $T$. Also, the ordinary kinetic term and
the mass term of the vector field are represented by the second and third
terms in the action, respectively.
We can define the non-metricity scalar as

$Q\equiv-g^{\alpha\beta}\left(L^{\mu}_{\nu\beta}L^{\nu}_{\beta\mu}-L^{\mu}_{\nu\mu}L^{\nu}_{\alpha\beta}\right),$
(2)

where, $L^{\lambda}_{\alpha\beta}$ is the deformation tensor

$L^{\lambda}_{\alpha\beta}=-\frac{1}{2}g^{\lambda\gamma}\left(Q_{\alpha\gamma\beta}+Q_{\beta\gamma\alpha}-Q_{\gamma\alpha\beta}\right).$
(3)

One can relate that the Levi-Civita connection and the metric in Riemannian
geometry, i.e., $\nabla_{\mu}g_{\alpha\beta}=0$ are both compatible. Whereas,
this seems different in Weyl geometry by

$Q_{\mu\alpha\beta}\equiv\widetilde{\nabla}_{\mu}g_{\alpha\beta}=\partial_{\mu}g_{\alpha\beta}-\widetilde{\Gamma}^{\rho}_{\mu\alpha}g_{\rho\beta}-\widetilde{\Gamma}^{\rho}_{\mu\beta}g_{\rho\alpha}=2w_{\mu}g_{\alpha\beta},$
(4)

where,
$\widetilde{\Gamma}^{\lambda}_{\alpha\beta}\equiv\Gamma^{\lambda}_{\alpha\beta}+g_{\alpha\beta}w^{\lambda}-\delta^{\lambda}_{\alpha}w_{\beta}-\delta^{\lambda}_{\beta}w_{\alpha}$
and $\Gamma^{\lambda}_{\alpha\beta}$ is the Christoffel symbol with respect to
the metric $g_{\alpha\beta}$.
Using Eqs. (2)-(4), we obtain the following relation:

$Q=-6w^{2},$ (5)

with the effective dynamical mass of the vector field as
$m^{2}_{eff}=m^{2}+12\kappa^{2}f_{Q}+12\lambda$. The variation principle with
respect to the metric tensor and Weyl vector on (1) yields the field equation.

$\frac{1}{2}\left(T_{\alpha\beta}+S_{\alpha\beta}\right)-\kappa^{2}f_{T}\left(T_{\alpha\beta}+\Theta_{\alpha\beta}\right)=-\frac{\kappa^{2}}{2}g_{\alpha\beta}f\\\
-6\kappa^{2}f_{Q}w_{\alpha}w_{\beta}+\lambda\left(R_{\alpha\beta}-6w_{\alpha}w_{\beta}+3g_{\alpha\beta}\nabla_{\rho}w^{\rho}\right)\\\
+3g_{\alpha\beta}w^{\rho}\nabla_{\rho}\lambda-6w_{(\alpha}\nabla_{\beta)}\lambda+g_{\alpha\beta}\square\lambda-\nabla_{\alpha}\nabla_{\beta}\lambda,$
(6)

where

$T_{\alpha\beta}\equiv-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}L_{m})}{\delta
g^{\alpha\beta}},$ (7) $f_{T}\equiv\frac{\partial f(Q,T)}{\partial
T},f_{Q}\equiv\frac{\partial f(Q,T)}{\partial Q}.$ (8)

respectively. Also the expression for $\Theta_{\alpha\beta}$ is defined as

$\Theta_{\alpha\beta}=g^{\mu\nu}\frac{\delta T_{\mu\nu}}{\delta
g_{\alpha\beta}}=g_{\alpha\beta}L_{m}-2T_{\alpha\beta}-2g^{\mu\nu}\frac{\delta^{2}L_{m}}{\delta
g^{\alpha\beta}\delta g^{\mu\nu}}.$ (9)

Here, $S_{\alpha\beta}$ is the re-scaled energy momentum tensor of the free
Proca field

$S_{\alpha\beta}=-\frac{1}{4}g_{\alpha\beta}W_{\rho\sigma}W^{\rho\sigma}+W_{\alpha\rho}W^{\rho}_{\beta}-\frac{1}{2}m^{2}g_{\alpha\beta}w_{\rho}w^{\rho}+m^{2}w_{\alpha}w_{\beta},$
(10)

with $W_{\alpha\beta}=\nabla_{\beta}w_{\alpha}-\nabla_{\alpha}w_{\beta}$ .

##  III Cosmological Model and Solutions

Consider a flat Friedmann-Lemaitre-Robertson-Walker (FLRW) universe, which is
represented by the isotropic, homogeneous, and spatially flat metric

$ds^{2}=-dt^{2}+a^{2}(t)\delta_{ij}dx^{i}dx^{j},$ (11)

where, $a(t)$ is the scale factor. The vector field due to spatial symmetry is
assumed to be of the form $w_{\alpha}=\left[\psi(t),0,0,0\right]$. Hence, we
have $w^{2}=w_{\alpha}w^{\alpha}=-\psi^{2}(t)$ with $Q=-6w^{2}=6\psi^{2}(t)$.
Here, we adapt the comoving coordinates $u^{\alpha}=\left(-1,0,0,0\right)$
along with $u^{\alpha}\nabla_{\alpha}=\frac{d}{dt}$ and $H=\frac{\dot{a}}{a}$.
We also assume the Lagrangian of the perfect fluid as $\mathcal{L}_{m}=p$.

The corresponding energy momentum tensor for the perfect fluid $T_{\mu\nu}$
and $\Theta^{\mu}_{\nu}$ lead to $T^{\mu}_{\nu}=diag\left(-\rho,p,p,p\right)$
and
$\Theta^{\mu}_{\nu}=\delta^{\mu}_{\nu}p-2T^{\mu}_{\nu}=diag\left(2\rho+p,-p,-p,-p\right)$,
where $\rho$ and $p$ are the energy density and the pressure, respectively.

The generalised Proca equation and the flat space restriction for the
cosmological case read as

$\displaystyle\dot{\psi}$ $\displaystyle=$
$\displaystyle\dot{H}+2H^{2}+\psi^{2}-3H\psi,$ (12)
$\displaystyle\dot{\lambda}$ $\displaystyle=$
$\displaystyle\left(-\frac{1}{6}m^{2}-2\kappa^{2}f_{Q}-2\lambda\right)\psi-\frac{1}{6}m^{2}_{eff}\psi,$
(13) $\displaystyle\partial_{i}\lambda$ $\displaystyle=$ $\displaystyle 0.$
(14)

Imposing an FLRW metric in equation (6), we get the two Friedmann equations as

$\kappa^{2}f_{T}\left(\rho+p\right)+\frac{1}{2}\rho=\frac{\kappa^{2}}{2}f-\left(6\kappa^{2}f_{Q}+\frac{1}{4}m^{2}\right)\psi^{2}\\\
-3\lambda\left(\psi^{2}-H^{2}\right)-3\dot{\lambda}\left(\psi-H\right),$ (15)

$-\frac{1}{2}p=\frac{\kappa^{2}}{2}f+\frac{m^{2}\psi^{2}}{4}+\lambda\left(3\psi^{2}+3H^{2}+2\dot{H}\right)\\\
+\left(3\psi+2H\right)\dot{\lambda}+\ddot{\lambda}.$ (16)

where dot $(.)$ represents the time derivative and $f_{Q}$, and $f_{T}$
defines the derivative with respect to $Q$ and $T$, respectively.
In this work, we assume the functional form $f(Q,T)=\alpha
Q+\frac{\beta}{6\kappa^{2}}T$, where $\alpha$ and $\beta$ are model
parameters. It is important to note that $\beta=0$ and $\alpha=-1$ corresponds
to $f(Q,T)=-Q$, i.e., a case of the successful General Relativity theory (GR).
Furthermore, at $T=0$, the theory simplifies to $f(Q)=\alpha Q$ gravity, which
is equivalent to GR and passes all Solar System tests. Further, using the
relation $\nabla_{\lambda}g_{\mu\nu}=-w_{\lambda}g_{\mu\nu}$, we can obtain
$\psi(t)=-6H(t)$, where $H(t)=\frac{\dot{a}}{a}$. For the choice of $f(Q,T)$
and solving equations (15) and (16), we obtain the expression for pressure $p$
and energy density $\rho$ as follows:

$p=-k_{1}\,H^{2}-k_{2}\,\dot{H},$ (17)

where

$\displaystyle k_{1}$
$\displaystyle=\left(36\left(\frac{18}{\beta+3}\left(\alpha+1\right)+\frac{3M^{2}}{2\left(\beta+3\right)}\right)+\frac{18}{2\beta+3}\right),$
$\displaystyle k_{2}$
$\displaystyle=\frac{18\left(\beta+2\right)}{\left(2\beta+3\right)\left(\beta+3\right)}.$

and

$\rho=k_{3}\,H^{2}-k_{4}\,\dot{H}.$ (18)

where

$\displaystyle k_{3}$
$\displaystyle=\left(\frac{-9\left(11\beta+24\right)\left(24\alpha+25\right)}{4\left(\beta+2\right)\left(\beta+3\right)}+\frac{29\beta+72}{2\left(2\beta+3\right)\left(\beta+2\right)}\right),$
$\displaystyle k_{4}$
$\displaystyle=\frac{9\beta}{2\left(2\beta+3\right)\left(\beta+3\right)}.$

The effective equation of state $\omega_{eff}=\frac{p}{\rho}$ becomes

$\omega_{eff}=\frac{-k_{1}\,H^{2}-k_{2}\,\dot{H}}{k_{3}\,H^{2}-k_{4}\,\dot{H}}$
(19)

Considerable cosmological observations show that the expansion of the universe
has accelerated after experiencing a deceleration. Since then, the
deceleration parameter $q$ has been frequently utilized to characterize cosmic
history, at least up until this point. In this sense, some studies used
different parametric variants of $q$ while others explored non-parametric
forms. The literature has suggested several parametrization forms of $q$.
However, in this work, we will examine the divergence-free parametric form of
the deceleration [41, 42] given below:

$q(z)=q_{0}+q_{1}\frac{z(1+z)}{1+z^{2}}.$ (20)

where $q_{0}$ is the present value of $q$ and $q_{1}$ indicates the variation
of the deceleration parameter with respect to $z$.
Mathematically, this parametric form has two possible cases:

  * •

The function $q(z)$ reduces to $q(z)\sim q_{0}+q_{1}$ at high redshift (i.e.,
$z>>1$).

  * •

The function $q(z)$ reduces to $q(z)\sim q_{0}+q_{1}z$ at low redshift (i.e.,
$z<<1$), (this model is not reliable at high redshift).

Another motivation behind this parametrization is that it provides a finite
value of $q$ throughout the entire range, $z\in[-1,\infty]$, and so
encompasses the entire evolutionary history of the universe. It is important
to note that the above parametric form of $q(z)$ was inspired by one of the
most prominent divergence-free parametrizations of the dark energy equation of
state [43], and it seemed to be sufficiently adaptable to match the $q(z)$
behavior of a large class of accelerating models.
The following equation relates the deceleration parameter and the Hubble
parameter

$H(z)=H_{0}\,exp\Big{(}\int_{0}^{z}\frac{1+q(z)}{(1+z)}dz\Big{)},$ (21)

Inserting Eq. (20) into Eq. (21), we get $H(z)$ as

$H(z)=H_{0}(1+z)^{(1+q_{0})}(1+z^{2})^{\frac{q_{1}}{2}}.$ (22)

The derivative of the Hubble parameter with respect to time can be written as

$\dot{H}=\frac{dH}{dt}=-(1+z)H(z)\frac{dH}{dz}$ (23)

Using equations (22) and (23), the expressions for $p$ and $\rho$ as functions
of $z$ are the following:

$p=-H_{0}^{2}(z+1)^{2q_{0}+2}\left(z^{2}+1\right)^{q_{1}-1}\\\
\left[k_{1}\left(z^{2}+1\right)-k_{2}\left(q_{0}\left(z^{2}+1\right)+q_{1}(z+1)z+z^{2}+1\right)\right],$
(24)

and

$\rho=H_{0}^{2}(z+1)^{2q_{0}+2}\left(z^{2}+1\right)^{q_{1}-1}\\\
\left[k_{3}\left(z^{2}+1\right)+k_{4}\left(q_{0}\left(z^{2}+1\right)+q_{1}(z+1)z+z^{2}+1\right)\right].$
(25)

##  IV observational data

Now, we test the viability of the model using the recent observational data,
namely, the observational Hubble data ($OHD$) [44, 45] and Type Ia supernovae
($SNe\,Ia$) [46]. Here, we use the Pantheon sample for $SNe\,Ia$ data, which
includes of 1048 points from the Low-z, SDSS, Pan-STARSS1 (PS1) Medium Deep
Survey, SNLS, and HST surveys [47].

###  IV.1 Hubble data

We first employ a standard compilation of 31 measurements of Hubble data
obtained through differential age method (DA). This method can be used to
estimate the expansion rate of the universe at redshift $z$. Here, $H(z)$ can
be calculated as $H(z)=-\frac{dz/dt}{(1+z)}$. To perform this analysis, we
minimize

$\chi^{2}_{H}=\sum_{i=1}^{31}\frac{\left[H(z_{i},\mathcal{P})-H_{obs}(z_{i})\right]^{2}}{\sigma(z_{i})^{2}}.$
(26)

Here, $H(z_{i},\mathcal{P})$ is the theoretical value for a given model at
redshifts $z_{i}$, and $\mathcal{P}$ is the parameter space. $H_{obs}(z_{i})$
and $\sigma(z_{i})^{2}$ represents the observational value and error,
respectively.

Figure 1: Error bar plot of $H$ versus $z$ for $f(Q,T)$ model (red curve) and
the $\Lambda$CDM model (black dotted curve). The blue dots depict the 31
points of the Hubble data Figure 2: The $1-\sigma$ and $2-\sigma$ confidence
regions for the parameters corresponding to the Hubble and Pantheon data.

###  IV.2 Pantheon data

The Pantheon sample is a compilation of different supernova surveys that
identified $SNe\,Ia$ at both high and low redshift. As a result, the whole
sample spans the redshift range $0.01<z<2.26$. Provided the Tripp estimator
[48] with the light curve fitter, the normalized observational $SNe\,Ia$
distance modulus is given by $\mu=m_{B}-M_{B}+\alpha x_{1}-\beta
c+\Delta_{M}+\Delta_{B}$, where $m_{B}$, $M_{B}$, and $c$ are the observed
peak magnitude (at B-band maximum), absolute magnitude, and $SNe\,Ia$ color,
respectively. Also, $\alpha$ and $\beta$ represent the relation between
luminosity stretch and luminosity color, respectively. Further, the distance
corrections on host galaxy mass and simulation based expected biases are given
by $\Delta_{M}$ and $\Delta_{B}$.
Based on accurate $SNe\,Ia$ simulations, Scolnic et al. [46, 49] used the
BEAMS with Bias Corrections (BBC) method to account for errors due to
intrinsic scatter and selection effects. Hence, the distance modulus reduces
to $\mu=m_{B}-M_{B}$. The expressions used for our analysis are as follows:

$\displaystyle\mu^{th}$ $\displaystyle=$ $\displaystyle
5log_{10}\left(\frac{d_{L}(z)}{1Mpc}\right)+25,$ (27) $\displaystyle d_{L}(z)$
$\displaystyle=$ $\displaystyle
c(1+z)\int_{0}^{z}\frac{dz^{\prime}}{H(z^{\prime},\mathcal{P})},$ (28)
$\displaystyle\chi^{2}_{SN}$ $\displaystyle=$ $\displaystyle
min\sum_{i,j=1}^{1048}\Delta\mu_{i}(C^{-1}_{SN})_{ij}\Delta\mu_{j}.$ (29)

Figure 3: Error bar plot of $\mu$ versus $z$ for $f(Q,T)$ model (red curve)
and the $\Lambda$CDM model (black dotted curve). The blue dots depict the 1048
points of the Pantheon data Figure 4: The $1-\sigma$ and $2-\sigma$ confidence
regions for the parameters corresponding to the joint data.

###  IV.3 Hz+Pantheon

Furthermore, we use the total likelihood function to get joint constraints for
the parameters $q_{0}$ and $q_{1}$ from the Hubble and Pantheon samples.
Henceforth, the relevant likelihood and Chi-square functions are given by

$\displaystyle\mathcal{L}_{joint}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{OHD}\times\mathcal{L}_{SNeIa},$ (30)
$\displaystyle\chi^{2}_{joint}$ $\displaystyle=$
$\displaystyle\chi^{2}_{OHD}+\chi^{2}_{SNeIa}.$ (31)

The constraints on model parameters are obtained by minimizing the respective
$\chi^{2}$ using Markov Chain Monte Carlo (MCMC) and $emcee$ library. The
results are obtained in Table I. Further, figs. 1 and 3 shows the error bar
fit for the considered model and the $\Lambda$CDM with
$\Omega_{\Lambda_{0}}=0.7$, $\Omega_{m_{0}}=0.3$, and $H_{0}=69$ km/s/Mpc. The
$1-\sigma$ and $2-\sigma$ contour plots are shown for Hubble, Pantheon and the
joint observational data in figs. 2 and 4.

Table 1: Best-fit values of model parameters obtained from observational datasets Datasets | $q_{0}$ | $q_{1}$ | $z_{t}$ | $\omega_{0}$
---|---|---|---|---
$OHD$ |  $-0.553^{+0.059}_{-0.059}$ [40, 53] | $0.698^{+0.092}_{-0.092}$ |  $0.69^{+0.31}_{-0.17}$ [51] |  $-0.81^{+0.04}_{-0.04}$ [57, 58]
$SNeIa$ |  $-0.688^{+0.054}_{-0.054}$ [55, 57] | $0.635^{+0.094}_{-0.094}$ |  $1.2^{+0.1}_{-0.22}$ [53] |  $-0.91^{+0.04}_{-0.04}$ [59]
$OHD+SNeIa$ |  $-0.611^{+0.051}_{-0.051}$ [40] | $0.66^{+0.087}_{-0.087}$ |  $0.87^{+0.15}_{-0.23}$ [54, 50] |  $-0.85^{+0.03}_{-0.04}$ [57, 58]

##  V cosmological parameters

According to cosmological observational data, cosmic acceleration is a recent
phenomenon. To capture the entire evolutionary history of the universe, a
cosmological model must include both the decelerated and accelerated phases of
the expansion. Hence, it is essential to investigate the behavior of the
deceleration parameter $q$.
The behavior of $q$ for the associated values of model parameters constrained
by the $OHD$, $SNe\,Ia$, and $OHD+SNe\,Ia$ is shown in fig. 5. It is clear
that the parameter $q$ shows a transition from a decelerated to an accelerated
phase at redshift $z_{t}$. Further, the value of $z_{t}$ fluctuates in the
range $0.5-1.3$ shown by recent observations [56].
Further, from figs. 7 and 8, it is clear that the density parameter $\rho$
shows the positive behavior, whereas the pressure $p$ evolves negatively,
respectively.

Figure 5: The redshift evolution of the deceleration parameter.

The equation of state parameter can indeed helpful in categorizing the various
epochs of accelerated and decelerated expansion of the universe. The different
epochs as follows: a) $\omega=1$ represents a stiff fluid; b) $\omega=1/3$
depicts the radiation-dominated phase; and c) $\omega=0$ indicates the matter-
dominated phase. The effective EoS parameter is defined as
$\omega=\frac{p}{\rho}$. Using Eqs. (24) and (25) in the above relation, we
obtain $\omega$ in terms of $z$ as

$\omega=\frac{k_{2}\left(q_{0}\left(z^{2}+1\right)+q_{1}(z+1)z+z^{2}+1\right)-k_{1}\left(z^{2}+1\right)}{k_{3}\left(z^{2}+1\right)+k_{4}\left(q_{0}\left(z^{2}+1\right)+q_{1}(z+1)z+z^{2}+1\right)}$
(32)

Figure 6: The behavior of the EoS parameter $\omega$ versus redshift $z$.

The evolution trajectory of the parameter $\omega$ is shown in fig. (6). One
of the three possible states for the expanding universe is the cosmological
constant $(\omega=-1)$, quintessence $(-1<\omega<-1/3),$ or phantom era
$(\omega<-1)$. It is clear from fig. 6 that $\omega<0$ and shows a
quintessence dark energy, which indicates an accelerating phase. In our model,
it may be noted that the EoS parameter do not cross the phantom divide
$\omega=-1$.

Figure 7: The trajectory of the density parameter $\rho$ versus redshift $z$.
Figure 8: The trajectory of the pressure $p$ versus redshift $z$.

##  VI Conclusion

With time, the present scenario of the accelerating expansion of the universe
has become even more fascinating. Numerous dynamical DE models and modified
theories of gravity have been used to address this issue in various ways.
Still, there is a search of suitable description of the accelerating universe.
In the present work, we attempt to describe dark energy by assuming the
parametric form of the deceleration parameter in the recently proposed Weyl-
type $f(Q,T)$ gravity, where the non-metricity $Q$ is coupled to the trace $T$
of the energy-momentum tensor. From a geometric perspective, the Weyl type
$f(Q,T)$ gravity may illustrate an alternative to dark energy and even dark
matter.
Hence, we started with the simplest functional form $f(Q,T)=\alpha
Q+\frac{\beta}{6\kappa^{2}}T$, where $\alpha$ and $\beta$ are model
parameters, and the parametric form of deceleration parameter
$q(z)=q_{0}+q_{1}\frac{z(1+z)}{1+z^{2}}$ as a function of $z$, where $q_{0}$
and $q_{1}$ are constants. With the help of Eq. (21), we find the Hubble
parameter in terms of redshift $z$. Furthermore, we constrained the parameters
$q_{0}$ and $q_{1}$ by the MCMC technique using the $OHD$, $SNe\,Ia$, and
$OHD+SNe\,Ia$ datasets (mentioned in table I). We have obtained the best-fit
values for the $q_{0}$ and $q_{1}$ parameters in figs. 2 and 4. Since, the
model parameters $\alpha$ and $\beta$ are not explicitly present in the
expression of the Hubble parameter, we attempt to fix them to study the
evolution of density, pressure, and effective EoS parameters. We considered
the values as $\alpha=-1.07$ and $\beta=0.88$ from the reference [39], that
are constrained using the observational datasets.

Finally, it is seen that the deceleration parameter $q(z)$ smoothly switches
from a decelerated to an accelerated phase of expansion. Also, in figs. 7 and
8, the energy density parameter decreases as the universe continues to expand
in the far future, and pressure increases in the negative behavior. The
effective EoS parameter exhibits the negative behavior indicating the
accelerating universe and the quintessence dark energy. However, the present
values of the deceleration and effective EoS parameters are more compatible
with the most recent values of cosmological parameters observed for $OHD$
data. It is work mentioning that the solution of the Hubble parameter obtained
from a divergence-free parametric form of deceleration parameter is in good
agreement with Weyl-type $f(Q,T)$ gravity.

## Data Availability Statement

There are no new data associated with this article.

## Acknowledgments

GNG acknowledges University Grants Commission (UGC), New Delhi, India for
awarding Junior Research Fellowship (UGC-Ref. No.: 201610122060). SA
acknowledges CSIR, Govt. of India, New Delhi, for awarding Senior Research
Fellowship. PKS acknowledges CSIR, New Delhi, India for financial support to
carry out the Research project [No.03(1454)/19/EMR-II Dt.02/08/2019].

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