# Resource Allocation and Secure Wireless Communication in the Large Model-
based Mobile Edge Computing System

Zefan Wang  Nanyang Technological UniversitySingapore<EMAIL_ADDRESS>,
Yitong Wang  Nanyang Technological UniversitySingapore<EMAIL_ADDRESS>and  Jun Zhao  Nanyang Technological UniversitySingapore<EMAIL_ADDRESS>
(2018)

###### Abstract.

With the rapid advancement of large models and mobile edge computing, transfer
learning, particularly through fine-tuning, has become crucial for adapting
models to downstream tasks. Traditionally, this requires users to share their
data with model owners for fine-tuning, which is not only costly but also
raises significant privacy concerns. Furthermore, fine-tuning large-scale
models is computationally intensive and often impractical for many users. To
tackle these challenges, we introduce a system that combines offsite-tuning
with physical-layer security, which provides local data owners with a
lightweight adapter and a compressed emulator. Data owners then fine-tune the
adapter locally and securely send it back to the model owners through a
confidential channel for integration, ensuring privacy and resource
conservation. Our paper focuses on optimizing computational resource
allocation among data owners and the large model owner deployed on edge, and
on the compression ratio of adapters. We incorporate a secrecy uplink channel
to maximize the utility that we defined while minimizing system costs like
energy consumption and delay. The optimization uses the Dinkelbach algorithm,
fractional programming, successive convex approximation and alternating
optimization. Experiments demonstrate our algorithm’s superiority over
existing methods.

Large Model, Physical Layer Security, Mobile Edge Computing, Resource
Allocation.

††copyright: acmlicensed††journalyear: 2018††doi: XXXXXXX.XXXXXXX††conference:
Make sure to enter the correct conference title from your rights confirmation
emai; October 14–17, 2024; Athens, Greece††isbn: 978-1-4503-XXXX-X/18/06††ccs:
Networks††ccs: Networks Network resources allocation

##  1\. Introduction

Large models (LM)111Although many studies simply say “foundation models”, we
adopt the name “large models” to highlight both the sheer size and the
foundation nature of the models. Such a name is also used in prior work (Rawte
et al., 2023). are generative machine learning models pre-trained on large-
scale unlabeled datasets (e.g., BookCorpus or English Wikipedia (Bommasani et
al., 2021)). Pre-trained models have the capability to acquire generalized
representations that can be effectively utilized in fine-tuning by using
smaller, task-specific datasets supplied by various downstream users to cater
to a diverse array of downstream tasks (Hu et al., 2021), where models like
GPT-4 by OpenAI stand out with their sophisticated understanding and creation
of human-like text (Devlin et al., 2018). Meanwhile, LMs like DALL-E and CLIP
demonstrate remarkable abilities in understanding and generating visual
content (Radford et al., 2021), and Google-USM achieving new levels of
accuracy and naturalness in speech-to-text conversion (Zhang et al., 2023).

Challenges and motivation. There are several challenges in the process of
fine-tuning LMs. On the one hand, there is a desire to exploit the full
capabilities of LMs, on the other, the sensitive information has potential
risks of exposure to LM service providers. This risk is non-neglectable in
many applications such as healthcare or finance (Behnia et al., 2022; Xu et
al., 2023). A potential solution could be localized deployment, where the
large model owner (LMO) deploys the model on the client. However, due to the
large structures of the LMs, deployment and inference locally are essentially
impossible, for instance, GPT-4 consists of an astonishing 175 billion
parameters (Achiam et al., 2023). Moreover, local deployment risks may leak
the models’ intellectual property (IP). Offsite-tuning has been proposed as a
viable Parameter Efficient Fine-Tuning (PEFT) solution (Ding et al., 2023;
Xiao et al., 2023). In this approach, the LMO sends a lightweight adapter and
a lossy compressed emulator to the data owner (DO). The DO then fine-tunes the
adapter using the emulator. Once fine-tuned, the adapter is returned to the
LMO and integrated into the full model to create an adapted FM. Under offsite-
tuning, the DO does not need to share their training data, and the LMO does
not share the full model weights.

A significant security challenge in the context of offsite-tuning is the
potential eavesdropping on the communication channel, where eavesdroppers
could intercept the adapter during transmission. If successful, they might use
the data within the adapter to infer sensitive information about the DO’s
local data or objectives. Eavesdroppers can passively gather intel without
altering the data, such as in VoIP eavesdropping, or actively insert
themselves into the network, manipulating the data as in man-in-the-middle
attacks. These attacks can lead to the theft of intellectual property,
passwords, and other sensitive data, significantly compromising the security
and privacy of the data owner (Ma et al., 2018; Kapetanovic et al., 2015). In
response to these challenges, our primary motivation is to develop secure
wireless communication in an LM-based edge computing system, which cooperates
with the PEFT technique for resource-saving.

Contributions. The major contributions of our paper are summarized as follows:

  * •

We formulate the problem of maximizing the users’ utility-consumption ratio
(UCR) under physical-layer security for the offsite-tuning process of the LM.
To our knowledge, we first study and investigate UCR in the context of the
increasingly critical and emergent domain of offsite-tuning for LMs.

  * •

We develop an advanced optimization algorithm by integrating the Dinkelbach
algorithm, fractional programming, Successive Convex Approximation (SCA), and
alternating optimization techniques. This innovative approach enables our
algorithm to efficiently address and solve complex UCR optimization
challenges, establishing a new benchmark for effectiveness in this field.

  * •

We demonstrate through simulations that our algorithm significantly
outperforms existing baselines, adeptly coordinating the optimization of
adapter parameter offloading for users, resource allocation, and the balance
between user utility and cost. This evidence showcases our algorithm’s
superior capability to navigate and optimize the multifaceted aspects of the
offsite-tuning process, thereby advancing both operational efficiency and
security.

Roadmap. The organization of this paper is as follows. Section 2 presents
related work. Section 3 introduces the proposed system model and the
formulated problem. Our proposed solution, encompassing the algorithmic
approach, is meticulously expounded in Section 4. Section 5 provides the
numerical results, and Section 6 summarizes our paper.

##  2\. Related Work

This section explores relevant research across three focused areas pertinent
to our study. Section 2.1 is mainly devoted to current research in solving
resource allocation problems over wireless communication. Section 2.2 focuses
on integrating large models (LMs) with edge computing, and Section 2.3 focuses
on the physical layer security in mobile communication.

###  2.1. Resource allocation over wireless communication

Recently, several studies have successfully addressed resource allocation
challenges through a variety of approaches. As for convex optimization, Luo et
al. (Luo et al., 2020) considered the hierarchical federated edge learning
(HFEL) in wireless communication, the minimization of system overhead is
addressed separately using the Karush-Kuhn-Tucker (KKT) method in convex
optimization for both single and multi-edge server scenarios. Yu et al. (Yu et
al., 2020) utilized another classical method in optimization, successive
convex approximation (SCA) transforming the non-convex problem, that performs
resource allocation under constraints on UAV’s battery size and service
quality, aiming to minimize system latency by transforming the non-convex
problem. Additionally, some works also tried to use deep learning strategies
for non-convex or complex sequential problems. For instance, He et al. (He et
al., 2019) solved the problem of channel assignment under Non-orthogonal
multiple access (NOMA) with deep reinforcement learning (DRL), which further
utilized attention-based neural network for better performance. Furthermore,
Guo et al. (Guo et al., 2020) considered a more sophisticated approach to
orchestrate multiple agents for jointly solving the handover control and power
allocation problem. However, many previous works address the resource
allocation problem in various scenarios, but none of them jointly consider
such a problem based on LM and related fine-tuning issues.

###  2.2. Large model with mobile edge computing.

Integrating LM with mobile edge computing (MEC) has become a pivotal area of
research. Liu et al. (Shen et al., 2023) explored the use of Generative
Pretrained Transformers (GPT) in edge AI systems. Their framework employs a
cloud-edge-client hierarchical architecture, where GPT, located in the cloud,
coordinates with AI models on edge servers and devices. This system
efficiently processes user requests in natural language, demonstrating GPT’s
potential in autonomously organizing and optimizing edge AI models. Tian et
al. (Tian et al., 2022) proposed a novel learning approach named FedBERT. This
method combines federated learning with split learning for pre-training BERT
in a federated manner. FedBERT addresses the challenge of pre-training LMs and
ensures that sensitive local data of clients are not communicated.
Furthermore, Jiang et al. (Jiang et al., 2023) focused on enhancing the
efficiency of large models in edge computing scenarios by balancing
computational loads between edge devices and cloud servers and Dong et al.
(Dong et al., 2023) highlighted the use of these models in improving the
performance and capabilities of edge computing systems. Although many studies
consider deploying LM at the edge, they don’t address the potential resource
allocation issues with optimization accordingly, like the ones specified in
our work.

###  2.3. Secure wireless communication under physical layer security

Several studies have explored physical layer security with varying numbers of
eavesdroppers. Cui et al. (Cui et al., 2019) implemented the Alternative
Optimization algorithm to jointly optimize the transmit beamforming vector at
the base station and the phase elements at the Intelligent Reflecting Surface
(IRS), aiming to maximize the secrecy rate. Considering multi-eavesdroppers,
Yang et al. (Yang et al., 2020) proposed a RL-based secure beamforming
approach to jointly optimize users secrecy rate and quality of service (QoS).
However, to our knowledge, none specifically address secure communication
during the fine-tuning process to safeguard both the model and user security,
as articulated in our paper.

##  3\. System Model and Problem Formulation

In this section, we formally describe our system model from the holistic to
the particular. We will first introduce the offsite-tuning model as the
foundation of the entire system, followed by the specific implementation model
on the large model owner (LMO) and data owners (DOs) sides, in addition to the
transmission model. At the end of this section, we give the comprehensive
objective function for the communication model.

###  3.1. System Model

We consider a large model (LM) based secure mobile edge computing system, as
shown in Fig. 1, where the LMO, which also be referred to as the LM server,
positioned at the edge side managing its downlink channel with all $N$ single-
antenna mobile users, also designated as DOs, with each DO denoted by
$n\in\mathcal{N}=\\{1,2,\dots,N\\}$.

Offsite-tuning Model. To preserve both parties’ privacy and enhance efficiency
and effectiveness without disrupting the operational system, we utilize
offsite-tuning in our system (Xiao et al., 2023). Given a pre-trained
Transformer backbones LM $\mathcal{M}$ deployed on the edge server side,
encompassing a total of $\mathcal{L}$ layers (i.e., model depth). We divide
the $\mathcal{M}$ into two distinct components: a small, trainable adapter
$\mathcal{A}_{n}$, which is used for downstream adaptation for $n$-th DO, and
a remaining frozen portion of the model, denoted as emulator
$\mathcal{E}_{n}$. In the selection of adapters, the paramount consideration
for the adapter transmitted to the downstream tasks is its suitability across
various downstream tasks, which is influenced by the extraction of diverse
combinations of shallow and deep layers from the Transformer. We employ the
sandwich structure proven superior to single-layer configurations in (Xiao et
al., 2023), where $l_{n}^{a}$ layers are selected simultaneously from both the
top and bottom layers of $\mathcal{M}$ (i.e.,
$\mathcal{M}=\mathcal{A}_{n}^{1}\circ\mathcal{E}_{n}\circ\mathcal{A}_{n}^{2}$
and $\mathcal{A}_{n}^{1}+\mathcal{A}_{n}^{2}=l_{n}^{a}$). A variable called
adapter extraction rate denoted as
$\boldsymbol{\phi}=[\phi_{1},\phi_{2},\dots,\phi_{N}]$ for each DO,
$n\in\mathcal{N}$ is introduced to denote the number of layers selects from
both top and bottom layers of $\mathcal{M}$. To ensure the feasibility of
offsite tuning and the overall system efficiency,
$\phi_{n}\in(0,\frac{2}{L}\times\chi^{a}_{n}]$, where
$\chi^{a}_{n},n\in\mathcal{N}$ is the confidentiality coefficient for
different downstream tasks, $l_{n}^{a}=2\phi_{n}$ and
$\frac{2}{L}\times\chi^{a}_{n}=\phi_{n,max}$. For convenience,
$\mathcal{A}_{n}^{1},\mathcal{A}_{n}^{2}$ are consolidated into
$\mathcal{A}_{n}$, such that $\forall
n\in\mathcal{N},\mathcal{M}=[\mathcal{A}_{n},\mathcal{E}_{n}]$ is the
concatenation of these two parts.

For the remaining frozen component emulator $\mathcal{E}_{n}$, it is
imperative that the data contained within $\mathcal{E}_{n}$ are not overly
precise, as such precision could inadvertently reveal information about the
original model. To address this, we utilize the layer drop technique (Sajjad
et al., 2023) to uniformly select a subset of layers from $\mathcal{E}_{n}$,
the dropped emulator denoted as $\mathcal{E}^{*}_{n}$, for utilization. A
discrete variable layer retention rate $\chi^{e}_{n},n\in\mathcal{N}$, is
introduced and selected from the interval $[0.1,0.3,0.5,0.7,0.9]$ to govern
the number of layers $l_{n}^{e}$ present in $\mathcal{E}^{*}_{n}$, where
$l_{n}^{e}=\chi^{e}_{n}(1-2\phi_{n})$. Emulator $\mathcal{E}^{*}_{n}$
encompasses the gradient information pertinent to the adapter updates, while
these gradient details must not be excessively precise, lest they lead to the
inadvertent disclosure of information pertaining to the original model.
Furthermore, an emulator characterized by a smaller volume of data is more
conducive to system efficiency (Xiao et al., 2023). We aim to find a balance
between these requirements. First, we calculate the loss between two emulators
using the Mean Square Error (MSE) to ensure $\mathcal{E}^{*}_{n}$ encapsulates
as much of the gradient information necessary for training as possible. Given
that the discrepancy between $\mathcal{E}_{n}$ and $\mathcal{E}^{*}_{n}$ is
primarily influenced by $\chi^{e}_{n}$, we define the loss as a function
solely dependent $\chi^{e}_{n}$, which can be expressed as
$\mathcal{L}_{MSE}(\chi^{e}_{n})=\frac{1}{N}\sum_{i=1}^{N}||\mathcal{E}^{*}_{n}-\mathcal{E}_{n}||^{2}$.
Furthermore, the efficiency can be defined as a function that decreases with
an increase in the number of layers within $\mathcal{E}^{*}_{n}$, which is
correlated with $\phi_{n}$ and $\chi^{e}_{n}$, the efficiency can be
represented as
$\mathcal{G}(\phi_{n},\chi^{e}_{n})=\log(1+\frac{\phi_{n}}{\chi^{e}_{n}})$,
with a designed minimal fine-tuning efficiency
$\mathcal{G}(\phi_{n},\chi^{e}_{n})\geq E^{\text{eff}}_{n,\text{min}}$.

FDMA. In our paper, we implement Frequency Division Multiple Access (FDMA) for
communication between devices and the base station. FDMA is known for its
simplicity and suitability for mobile devices with limited computational
capabilities, which aids in managing the bandwidth and transmission power
allocation without the complications of interference. We consider uplink
communication where each DO trains and uploads adapter $\mathcal{A}_{n}$ back
to LMO. We define $\boldsymbol{b}=[b_{1},b_{2},\dots,b_{N}]$,
$\boldsymbol{p}=[p_{1},p_{2},\dots,p_{N}]$ as the bandwidth and transmission
power for each DO communication to LMO and $b_{n,\text{max}}$,
$p_{n,\text{max}}$ are used for fair resource allocation (Wang et al., 2016;
Chen et al., 2019).

User training adapter $\mathcal{A}_{n}$. Based on the above discussion, in our
system, we account for the varying GPU frequencies of each user as their
computational resources allocated to fine-tuning the adapter
$\mathcal{A}_{n}$, denoted as
$\boldsymbol{f^{\text{DO}}}=[f^{\text{DO}}_{1},f^{\text{DO}}_{2},\dots,f^{\text{DO}}_{N}]$
and the user’s computational capabilities directly impact the training
efficiency of the model’s adapter. We posit that for any given model
considered in our paper, the model’s parameter size is invariably positively
correlated with the number of layers within the model. In the Transformer
models we employ, each layer typically exhibits a similar structure (Vaswani
et al., 2017) (i.e., the model’s parameter size can be estimated using the
number of layers $\times$ the average parameter count per layer).
Consequently, we can directly utilize the number of layers of each model as a
representation of the corresponding size of the model. The computational time
for each DO can be represented as

(1)
$t^{\text{DO:cmp}}_{n}(f^{\text{DO}}_{n},\phi_{n})=\frac{\mathcal{C}_{n}(\phi_{n})}{f^{\text{DO}}_{n}},$

where $\mathcal{C}_{n}(\phi_{n})$ is the total GPU cycles required for user
$n$’s fine-tuning. For computation function $\mathcal{C}_{n}(\phi_{n})$, the
number of computation cycles required for user training increases with
$\phi_{n}$ since more trainable parameters in $\mathcal{A}_{n}$. In our
definition, the computation function $\mathcal{C}_{n}(\phi_{n})$ can be
formalized as

(2) $\mathcal{C}_{n}(\phi_{n})=C_{1}\phi_{n}^{C_{2}},\forall n\in\mathcal{N},$

where $C_{1}>0$, $C_{2}>1$. Based on the user’s computation discussion (Shen
et al., 2019), the energy consumed for DO training can be represented as

(3)
$E^{\text{DO:cmp}}_{n}(f^{\text{DO}}_{n},\phi_{n})=k_{n}\mathcal{C}_{n}(\phi_{n})(f^{\text{DO}}_{n})^{2},$

where $k_{n}$ is the effective switched capacitance for the user.

User sending back adapter to server. After DO’s processing, the user transmits
the fine-tuned adapter back to the LMO deployed at the edge server side.
According to Shannon’s formula, the transmission rate for each user can be
expressed as

(4)
$r_{n}(b_{n},p_{n})=b_{n}\log_{2}(1+\frac{g_{n}p_{n}}{\sigma_{n}^{2}b_{n}}),$

where $\sigma_{n}^{2}$ is the power spectral density of Gaussian noise, and
$g_{n}$ is the channel attenuation from DO to LMO.

Figure 1. Optimizing the UCR of a large model system with $N$ data owners and
a model owner through joint resource allocation optimization.

We account for a potential unauthorized eavesdropper $E^{e}$ who aims to
eavesdrop on any of the data streams from the DOs and threaten our model.
Given the utilization of FDMA in our system, the eavesdropper’s ability to
intercept the user information is linked to the user’s allocated bandwidth and
power (i.e., the eavesdropper can disrupt the user’s communication by
leveraging the bandwidth allocated to the user and the eavesdrop power equals
the transmission power of DOs (Yang et al., 2020).) The eavesdropping rate for
user $n$ is represented as

(5)
$r_{e}(b_{n},p_{n})=b_{n}\log_{2}(1+\frac{g_{n,e}p_{n}}{\sigma_{e}^{2}b_{n}}),$

where $\sigma_{e}^{2}$ is the power spectral density of Gaussian noise of
$E^{e}$ and $g_{n,e}$ is the channel gain from each DO to $E^{e}$,
respectively.

We use $r_{s,n}$ to represent the rate at which data can be securely
transmitted without being intercepted by $E^{e}$ (i.e., the individual secrecy
rate). Meanwhile, we can quantify the impact of eavesdropping on the user’s
transmission. The secrecy rate of user $n$ is defined as

(6)
$r_{s,n}(b_{n},p_{n})=\bigg{[}r_{n}(b_{n},p_{n})-r_{e}(b_{n},p_{n})\bigg{]}^{+},$

where $[z]^{+}=\text{max}(0,z)$. Once $r_{s,n}(b_{n},p_{n})=0$, the
transmission is considered as failure with no subsequent calculations. With
the valid secrecy rate, the transmission time for uplink communication is

(7)
$t^{\text{com}}_{n}(b_{n},p_{n},\phi_{n})=\frac{\phi_{n}\mathcal{L}w}{r_{s,n}(b_{n},p_{n})},$

where $w$ is the number of bits used to represent each parameter. The energy
consumed for transmission can be represented as
$E^{\text{com}}_{n}=p_{n}T^{\text{com}}_{n}$, which is

(8)
$E^{\text{com}}_{n}(b_{n},p_{n},\phi_{n})=p_{n}\frac{\phi_{n}\mathcal{L}w}{r_{s,n}(b_{n},p_{n})}.$

Server integrating and processing the received adapter. At the server side,
let
$\boldsymbol{f^{\text{LMO}}}=[f^{\text{MO}}_{1},f^{\text{LMO}}_{2},\dots,f^{\text{DO}}_{N}]$
represent the GPU frequency (i.e., the computational resources) processing the
transmitted back adapters from user $n$. Considering the parameter size of the
server received adapter $\mathcal{A}_{n}$, the processing time at the server
is

(9)
$t^{\text{LMO:cmp}}(f^{\text{LMO}}_{n},\phi_{n})=\frac{\mathcal{B}_{n}(\phi_{n})}{f^{\text{LMO}}_{n}},$

where $\mathcal{B}_{n}(\phi_{n})$ is the total GPU cycles required for
processing user $n$’s adapter. On the server side, the LMO integrated the
adapter into the original model. Given that the parameter volume in the
adapter $\mathcal{A}_{n}$ is much smaller than that in the emulator
$\mathcal{E}_{n}$ (Sajjad et al., 2023), the computational load at the model
owner’s end is primarily determined by the parameters in the $\mathcal{E}_{n}$
(not $\mathcal{E}^{*}_{n}$, since $\mathcal{E}^{*}_{n}$ is a subset of
$\mathcal{E}_{n}$ and mainly used for DO to fine-tuning). Consequently, the
computation function $\mathcal{B}_{n}(\phi_{n})$ represents the number of
computation cycles required for the LMO to integrate and process the data,
decreases with $\phi_{n}$. Same as (2), $\mathcal{B}_{n}(\phi_{n})$ can be
formalized as

(10) $\mathcal{B}_{n}(\phi_{n})=C_{3}{\phi_{n}}^{-C_{4}},$

where $C_{3},C_{4}>0$. The energy consumed for LMO processing is

(11)
$E^{\text{LMO:cmp}}_{n}(f^{\text{LMO}}_{n},\phi_{n})=k_{m}\mathcal{B}_{n}(\phi_{n})(f^{\text{LMO}}_{n})^{2}.$

In conclusion, the overall complete time of LMO completes user $n$’s task,
including both computation and computation is

(12) $\displaystyle
t_{n}(b_{n},p_{n},\phi_{n},f^{\text{DO}}_{n},f^{\text{LMO}}_{n})$
$\displaystyle=t^{\text{DO:cmp}}_{n}(f^{\text{DO}}_{n},\phi_{n})+t^{\text{com}}_{n}(b_{n},p_{n},\phi_{n})+t^{\text{LMO:cmp}}(f^{\text{LMO}}_{n},\phi_{n}).$

Then, the system delay is defined as the maximum delay experienced by any user
in the network, which can be represented as

(13)
$\mathcal{T}(\boldsymbol{b},\boldsymbol{p},\boldsymbol{\phi},\boldsymbol{f^{\text{DO}}},\boldsymbol{f^{\text{MO}}})=\text{max}_{n\in\mathcal{N}}\
t_{n}(b_{n},p_{n},\phi_{n},f^{\text{DO}}_{n},f^{\text{MO}}_{n}).$

The total communication and computation energy consumption of the system is

(14)
$\displaystyle\mathcal{E}(\boldsymbol{b},\boldsymbol{p},\boldsymbol{\phi},$
$\displaystyle\boldsymbol{f^{\text{DO}}},\boldsymbol{f^{\text{LMO}}})=\sum_{n\in\mathcal{N}}E^{\text{DO:cmp}}_{n}(f^{\text{DO}}_{n},\phi_{n})$
$\displaystyle+\sum_{n\in\mathcal{N}}E^{\text{com}}_{n}(b_{n},p_{n},\phi_{n})+\sum_{n\in\mathcal{N}}E^{\text{LMO:cmp}}_{n}(f^{\text{LMO}}_{n},\phi_{n}).$

The total cost of the system is determined by a weighted sum, encompassing
both the system delay Eq. (13), and the energy consumption Eq. (14):

(15) $\displaystyle\mathcal{S}(\boldsymbol{b},$
$\displaystyle\boldsymbol{p},\boldsymbol{\phi},\boldsymbol{f^{\text{DO}}},\boldsymbol{f^{\text{LMO}}})$
$\displaystyle=c_{t}\mathcal{T}(\boldsymbol{b},\boldsymbol{p},\boldsymbol{\phi},\boldsymbol{f^{\text{DO}}},\boldsymbol{f^{\text{LMO}}})+c_{e}\mathcal{E}(\boldsymbol{b},\boldsymbol{p},\boldsymbol{\phi},\boldsymbol{f^{\text{DO}}},\boldsymbol{f^{\text{LMO}}}).$

Utility. Based on the definition of (Zhao et al., 2023), for each user $n$, we
formulate the service experience score, which represents the ‘effectiveness’
of the users as a function of $b_{n},\phi_{n}$ and $f_{n}^{\text{DO}}$:
$U_{n}(b_{n},\phi_{n},f_{n}^{\text{DO}})$, satisfying Assumption 1 below.

###### Assumption 1.

Utility function $U_{n}(x)$ is non-decreasing in $x$ and concave in $x$, also
twice differentiable.

###### Proof.

See Appendix A. ∎

Following the above assumption, we can formulate the utility function as the
normalization of three parameters, which is relative to their maximum values.
The user score (utility) function can be formulated as

(16)
$U_{n}(b_{n},\phi_{n},f_{n}^{\text{DO}})=\varpi\ln\big{(}{1+\phi_{n}+\frac{f_{n}^{\text{DO}}}{f_{n,\text{max}}^{\text{DO}}}}+\frac{b_{n}}{b_{n,\text{max}}}\big{)},$

where $\varpi$ is used for normalization. This utility function demonstrates
efficacy and sensitivity across the entire spectrum of optimal value ranges.
The system utility, defined as the sum of all $N$ users’ utilities, is given
by

(17)
$\mathcal{U}(\boldsymbol{b},\boldsymbol{\phi},\boldsymbol{f^{\text{DO}}})=\sum_{n\in\mathcal{N}}U_{n}(b_{n},\phi_{n},f_{n}^{\text{DO}}).$

To accomplish maximum economic efficiency of the system, we aim to maximize
the system’s utility while minimizing the cost, the utility-consumption ratio
(UCR) problem $\mathbb{P}_{1}$ can be formulated as

(18) $\displaystyle\mathbb{P}_{1}:\ $
$\displaystyle\max_{\boldsymbol{b},\boldsymbol{p},\boldsymbol{\phi},\boldsymbol{f^{\text{DO}}},\boldsymbol{f^{\text{MO}}}}\frac{\mathcal{U}(\boldsymbol{b},\boldsymbol{\phi},\boldsymbol{f^{\text{DO}}})}{\mathcal{S}(\boldsymbol{b},\boldsymbol{p},\boldsymbol{\phi},\boldsymbol{f^{\text{DO}}},\boldsymbol{f^{\text{MO}}})}$
subject to: (18a) $\displaystyle b_{n}\leq b_{n,\text{max}},\forall
n\in\mathcal{N},$ (18b) $\displaystyle p_{n}\leq p_{n,\text{max}},\forall
n\in\mathcal{N},$ (18c)
$\displaystyle\phi_{n}\in(0,\phi_{n,\text{max}}],\forall n\in\mathcal{N},$
(18d) $\displaystyle f^{\text{DO}}_{n}\leq
f^{\text{DO}}_{n,\text{max}},\forall n\in\mathcal{N},$ (18e)
$\displaystyle\sum_{n\in\mathcal{N}}f^{\text{MO}}_{n}\leq
f^{\text{MO}}_{\text{max}},$ (18f)
$\displaystyle\mathcal{G}(\phi_{n},\chi^{e}_{n})\geq
E^{\text{eff}}_{n,\text{min}},\forall n\in\mathcal{N},$ (18g)
$\displaystyle\mathcal{L}_{MSE}(\chi^{e}_{n})\leq\text{Loss}_{n,\text{max}},\forall
n\in\mathcal{N},$ (18h) $\displaystyle b_{n},p_{n}\geq
0,\chi^{e}_{n}\in[0.1,0.3,0.5,0.7,0.9],\forall n\in\mathcal{N}.$

In Problem $\mathbb{P}_{1}$, (18a), (18b), and (18d) ensures the bandwidth,
power, and GPU frequencies for each transmit user must not exceed their
respective maximum limits, which means the users operate within their resource
capacities. (18c) gives the limitation of the extraction rate. (18e) limits
the total resources LMO used for computation, and (18f) (18g) achieved a
balance between the effectiveness and the computational efficiency of
$\mathcal{E}^{*}_{n}$.

##  4\. Solution of the joint optimization problem

The problem defined in (18) is a non-convex fractional programming problem
classified as NP-hard (Freund and Jarre, 2001). Since in (18), the numerator
maintains non-negativity, the denominator is strictly positive, and both are
continuous functions, we use the Dinkelbach transform (Shen and Yu, 2018) to
convert problem $\mathbb{P}_{1}$ to $\mathbb{P}_{2}(y)$ with introduce an
additional variable $y$. Let $H_{\mathbb{P}_{2}}$ denote the objective
function, then $H_{\mathbb{P}_{2}}$ and $\mathbb{P}_{2}$ are

(19) $\displaystyle H_{\mathbb{P}_{2}}(\boldsymbol{b},$
$\displaystyle\boldsymbol{p},\boldsymbol{\phi},\boldsymbol{f^{\text{DO}}},\boldsymbol{f^{\text{MO}}}\
|\ y):$ $\displaystyle=\text{numerator of
(\ref{prob_1})}-y\cdot\text{denominator of (\ref{prob_1})}$ (20)
$\displaystyle\mathbb{P}_{2}(y):$
$\displaystyle\max_{\boldsymbol{b},\boldsymbol{p},\boldsymbol{\phi},\boldsymbol{f^{\text{DO}}},\boldsymbol{f^{\text{MO}}}}H_{\mathbb{P}_{2}}(\boldsymbol{b},\boldsymbol{p},\boldsymbol{\phi},\boldsymbol{f^{\text{DO}}},\boldsymbol{f^{\text{MO}}},T\
|\ y)$ subject to:
$\displaystyle~{}\text{(\ref{cons_band})}-\text{(\ref{cons_geq}).}$

To address problem $\mathbb{P}_{2}(y)$, we define the variables
[$\boldsymbol{b},\boldsymbol{p},\boldsymbol{\phi},\boldsymbol{f^{\text{DO}}},\\\
\boldsymbol{f^{\text{MO}}},T$] as $\boldsymbol{z}$ and express the objective
function of $\mathbb{P}_{2}(y)$ as $U(\boldsymbol{z})-y\cdot
S(\boldsymbol{z})$. At the beginning of the iteration, we initiate with a
feasible $\boldsymbol{z}^{(0)}$ and set
$y^{(0)}=\frac{U(\boldsymbol{z}^{(0)})}{S(\boldsymbol{z}^{(0)})}$.
Subsequently, we solve $\mathbb{P}_{2}(y^{(0)})$, obtaining the solution
$\boldsymbol{z}^{(1)}$, and update
$y^{(1)}=\frac{U(\boldsymbol{z}^{(1)})}{S(\boldsymbol{z}^{(1)})}$. This
iterative method proceeds as follows: for the $(k+1)$-th iteration, $y^{(k)}$
is set as $\frac{U(\boldsymbol{z}^{(k)})}{S(\boldsymbol{z}^{(k)})}$, and
$\boldsymbol{z}^{(k+1)}$ is derived by solving $\mathbb{P}_{2}(y^{(k)})$. This
procedure ensures convergence and maintains optimality.

After addressing the Dinkelbach transformation, $H_{\mathbb{P}_{2}}$ still
contains the ‘max’ function of delay, we introduce an auxiliary variable $T$
to circumvent this difficulty. Furthermore, with known $y$, Problem
$\mathbb{P}_{2}(y)$ still contains six variables, we decompose the problem
into two subproblems, one has
$\boldsymbol{b},\boldsymbol{p},\boldsymbol{f^{\text{DO}}},\boldsymbol{f^{\text{MO}}},T$
as optimization variables and the other has $\boldsymbol{\phi}$ as
optimization variables.

#### Subproblem 1

(21)
$\displaystyle\max_{\boldsymbol{\phi}}H_{\mathbb{P}_{2}}(\boldsymbol{b},\boldsymbol{p},\boldsymbol{\phi},\boldsymbol{f^{\text{DO}}},\boldsymbol{f^{\text{MO}}},T\
|\ y)$ (18c),(18f).

#### Subproblem 2

(22)
$\displaystyle\max_{\boldsymbol{b},\boldsymbol{p},\boldsymbol{f^{\text{DO}}},\boldsymbol{f^{\text{MO}}},T}H_{\mathbb{P}_{2}}(\boldsymbol{b},\boldsymbol{p},\boldsymbol{\phi},\boldsymbol{f^{\text{DO}}},\boldsymbol{f^{\text{MO}}},T\
|\ y)$ (18a),(18b),(18d),(18e),(18h). (22a) $\displaystyle T\geq
t_{n}(b_{n},p_{n},\phi_{n},f^{\text{DO}}_{n},f^{\text{MO}}_{n}),\forall
n\in\mathcal{N}.$

The methodology for solving the problem is outlined as follows. At the
beginning of each iteration, first calculate the minimal $\chi^{e^{-}}_{n}$
following constraint (18g), then we utilize Alternating Optimization (AO) with
an initial value of $\boldsymbol{\phi}$ and alternately optimize (21) and
(22). Additionally, the maximal value of $\chi^{e}_{n}$ can be calculated
using
$\chi^{e^{+}}_{n}=\big{(}\mathcal{G}(\phi_{n},\chi^{e^{+}}_{n})=E_{n,\text{min}}^{\text{eff}}\big{)}^{-1}$.
Ultimately, the optimal
$\chi^{e}_{n}=\text{ROUND}\big{(}\text{min}\big{[}\chi^{e^{-}}_{n},\chi^{e^{+}}_{n}\big{]}$)
to the nearest discrete value, as under identical transmission conditions, our
primary concern lies with data security rather than exhaustive fine-tuning,
which can be compensated for in subsequent rounds of transmission and
training.

###  4.1. Solution to Subproblem 1

With fixed
$\boldsymbol{b},\boldsymbol{p},\boldsymbol{f^{\text{DO}}},\boldsymbol{f^{\text{MO}}},T$
, we first optimize $\phi_{n}$ and $y$. With fixed initial $\chi^{e^{-}}_{n}$
and the monotonic increasing property of $\mathcal{G}(\phi_{n},\chi^{e}_{n})$
with respect to $\phi_{n}$, a minimal extraction rate of each DO can be
derived from (18f), which is

(23) $\phi_{n}\geq\phi_{n,\text{min}},\
\mathcal{G}(\phi_{n,\text{min}},\chi^{e^{-}}_{n})=E^{\text{eff}}_{n,\text{min}}$

Subproblem (21) can be reformulated as

(24) $\displaystyle\mathbb{P}_{3}:\min_{\boldsymbol{\phi}}$
$\displaystyle-\mathcal{U}(\boldsymbol{b},\boldsymbol{\phi},\boldsymbol{f^{\text{DO}}})+y\cdot(c_{e}\mathcal{E}(\boldsymbol{b},\boldsymbol{p},\boldsymbol{\phi},\boldsymbol{f^{\text{DO}}},\boldsymbol{f^{\text{MO}}})+c_{t}T)$
subject to: (24a)
$\displaystyle\phi_{n,\text{min}}\leq\phi_{n}\leq\phi_{n,\text{max}},\forall
n\in\mathcal{N},$ (22a)

Since $\mathbb{P}_{3}$ is convex, the Karush-Kuhn-Tucker (KKT) conditions are
the sufficient and necessary optimality conditions for finding the optimal
solution. We can write down the partial Lagrangian function of problem
$\mathcal{L}_{1}(\boldsymbol{\phi},\boldsymbol{\eta}\ |\
y,\boldsymbol{b},\boldsymbol{p},\boldsymbol{f^{\text{DO}}},\boldsymbol{f^{\text{MO}}},T)=$

(25)
$\displaystyle-\sum_{n\in\mathcal{N}}\ln\big{(}{\theta_{n}+\frac{f_{n}^{\text{DO}}}{f_{n,\text{max}}^{\text{DO}}}}+\frac{b_{n}}{b_{n,\text{max}}}\big{)}+yc_{e}\sum_{n\in\mathcal{N}}[k_{n}C_{1}\phi_{n}^{C_{2}}(f^{\text{DO}}_{n})^{2}$
$\displaystyle+p_{n}\frac{\phi_{n}\mathcal{L}w}{r_{s,n}}+k_{m}C_{3}\phi_{n}^{-C_{4}}(f^{\text{MO}}_{n})^{2}]$
$\displaystyle+\sum_{n\in\mathcal{N}}\eta_{n}\
[\text{max}\\{\frac{C_{1}\phi_{n}^{C_{2}}}{f^{\text{DO}}_{n}}+\frac{\phi_{n}\mathcal{L}w}{r_{s,n}}+\frac{C_{3}\phi_{n}^{-C_{4}}}{f^{\text{MO}}_{n}}\\}-T],$

where $\boldsymbol{\eta}=[\eta_{1},\eta_{2},\dots,\eta_{N}]$ are Lagrange
multipliers associate with constraint. Theorem 1 is given to find the optimal
$\boldsymbol{\phi}$.

###### Theorem 1.

The optimal solution of problem $\mathbb{P}_{3}$ is

(26)
$\phi^{*}_{n}=\text{min}\\{\text{max}\\{\phi_{n,\text{min}},\widetilde{\phi}_{n}(\eta_{n}\
|\
y,\boldsymbol{b},\boldsymbol{p},\boldsymbol{f^{\text{DO}}},\boldsymbol{f^{\text{MO}}},T)\\},\phi_{n,\text{max}}\\}$

where $\eta_{n}$ needs to satisfy

(27) max
$\displaystyle\\{\frac{C_{1}(\widetilde{\phi}_{n}(\eta_{n})|_{\phi_{n,\text{min}}}^{\phi_{n,\text{max}}})^{C_{2}}}{f^{\text{DO}}_{n}}+\frac{(\widetilde{\phi}_{n}(\eta_{n})|_{\phi_{n,\text{min}}}^{\phi_{n,\text{max}}})\mathcal{L}w}{r_{s,n}}$
$\displaystyle+\frac{C_{3}(\widetilde{\phi}_{n}(\eta_{n})|_{\phi_{n,\text{min}}}^{\phi_{n,\text{max}}})^{-C_{4}}}{f^{\text{MO}}_{n}}\\}=T$

with
$\widetilde{\phi}_{n}(\eta_{n})|_{a}^{b}:=\text{max}(a,\text{min}(\widetilde{\phi}_{n}(\eta_{n}),b))$.

###### Proof.

See Appendix B. ∎

###  4.2. Solution to Subproblem 2

With fixed $\boldsymbol{\phi}$, we need to handle the sum-of-ratio function
$\sum_{n\in\mathcal{N}}\frac{p_{n}\phi_{n}\mathcal{L}w}{r_{s,n}}$. We utilize
the fractional programming (FP) technique proposed in (Zhao et al., 2023),
Problem (22) can be converted into

(28) $\displaystyle\mathbb{P}_{4}(\boldsymbol{x},\boldsymbol{\phi},y):$
$\displaystyle\max_{\boldsymbol{b},\boldsymbol{p},\boldsymbol{f^{\text{DO}}},\boldsymbol{f^{\text{MO}}},T}F(\boldsymbol{b},\boldsymbol{p},\boldsymbol{f^{\text{DO}}},\boldsymbol{f^{\text{MO}}},T)$
$\displaystyle-
yc_{e}\cdot\sum\bigg{\\{}[p_{n}\phi_{n}\mathcal{L}w]^{2}x_{n}+\frac{1}{4[r_{s,n}]^{2}x_{n}}\bigg{\\}}$
subject to: (18a),(18b),(18d),(18e),(18h),(22a),

where
$F(\boldsymbol{b},\boldsymbol{p},\boldsymbol{f^{\text{DO}}},\boldsymbol{f^{\text{MO}}},T)=\mathcal{U}(\boldsymbol{b},\boldsymbol{\phi},\boldsymbol{f^{\text{DO}}})-y\bigg{[}c_{e}\sum_{n\in\mathcal{N}}\bigg{(}k_{n}\mathcal{C}_{n}(\phi_{n})\\\
(f^{\text{DO}}_{n})^{2}+k_{m}\mathcal{B}_{n}(\phi_{n})(f^{\text{MO}}_{n})^{2}+c_{t}T\bigg{]}$.
The introduced auxiliary variables
$\boldsymbol{x}=[x_{1},x_{2},\dots,x_{N}]>0$, and satisfy
$[p_{n}\phi_{n}\mathcal{L}w]^{2}x_{n}+\frac{1}{4[r_{s,n}]^{2}x_{n}}=[\mathcal{M}_{n}(\boldsymbol{z})x_{n}+\frac{\mathcal{P}_{n}(\boldsymbol{z})}{x_{n}})]$,
where
$x_{n}=\sqrt{\frac{\mathcal{P}_{n}(\boldsymbol{z})}{\mathcal{M}_{n}(\boldsymbol{z})}}$.
The iteration algorithm of finding $\boldsymbol{x}$ is the same as finding $y$
in (20).

Since the secrecy rate (6) is neither jointly convex nor concave with respect
to $b_{n}$ and $p_{n}$ (i.e., the expression of the Hessian matrix of (6)
involves a mixture of multi-squared terms and differences of squares,
indicating that the expression’s sign can vary based on the relative sizes of
parameters in $r_{s,n}$, and the presence of both positive and negative
components in the numerator suggests that the function may not be uniformly
convex or concave across its entire domain). We utilize the Successive Convex
Approximation (SCA) method to approximate (6) at ($b_{n}^{(k+1)}$,
$p_{n}^{(k+1)}$). Assume $b_{n}^{(k)}$, $p_{n}^{(k)}$ are the current estimate
of $b_{n}$ and $p_{n}$ at the $k$-th iteration, we can linearize (6) using the
first-order Taylor expansion, the linear approximation can be expressed as

(29) $\displaystyle\widetilde{r}_{s,n}(b_{n},p_{n})=$ $\displaystyle
r_{s,n}(b_{n}^{k},p_{n}^{k})+\frac{\partial}{\partial
b_{n}}\big{[}r_{s,n}(b_{n}^{k},p_{n}^{k})\big{]}(b_{n}-b_{n}^{(k)})$
$\displaystyle+\frac{\partial}{\partial
p_{n}}\big{[}r_{s,n}(b_{n}^{k},p_{n}^{k})\big{]}(p_{n}-p_{n}^{(k)}),$

where $b_{n}^{(k)}$, $p_{n}^{(k)}$ are intermediate values and regraded as
constant. The SCA process involves iteratively solving this optimization
problem, updating the estimates ($b_{n}^{(k+1)}$, $p_{n}^{(k+1)}$), and
repeating until convergence. Then we can say $r_{n,s}(b_{n},p_{n})$ is convex.

After the FP and SCA, $\mathbb{P}_{4}$ is a convex problem. The Lagrange
function
$\mathcal{L}_{2}(\boldsymbol{b_{e}},\boldsymbol{p},\boldsymbol{f^{\text{DO}}},\boldsymbol{f^{\text{MO}}},T,\boldsymbol{\alpha},\boldsymbol{\beta},\boldsymbol{\delta},\zeta,\boldsymbol{\rho}\
|\ \boldsymbol{x},\boldsymbol{\phi},y)=$

(30)
$\displaystyle-F(\boldsymbol{b},\boldsymbol{p},\boldsymbol{f^{\text{DO}}},\boldsymbol{f^{\text{MO}}},T)+yc_{e}\cdot\sum\bigg{\\{}[p_{n}\phi_{n}\mathcal{L}w]^{2}x_{n}+\frac{1}{4[\widetilde{r}_{s,n}]^{2}x_{n}}\bigg{\\}}$
$\displaystyle+\sum[\alpha_{n}\cdot(b_{n}-b_{n,\text{max}})]+\sum[\beta_{n}\cdot(p_{n}-p_{n,\text{max}})]$
$\displaystyle+\sum[\delta_{n}\cdot(f_{n}^{\text{DO}}-f_{n,\text{max}}^{\text{DO}})]+\zeta\cdot(\sum_{n\in\mathcal{N}}f^{\text{MO}}_{n}-f^{\text{MO}}_{\text{max}})$
$\displaystyle+\sum[\rho_{n}\cdot(t_{n}(b_{n},p_{n},f^{\text{DO}}_{n},f^{\text{MO}}_{n})-T)].$

We present the KKT conditions as

  * •

Stationary: (The value of each equation = 0)

(31) $\frac{\partial\mathcal{L}_{2}}{\partial
b_{n}}=-\frac{b_{n,\text{max}}}{b_{n}}-\frac{yc_{e}}{2x_{n}(\widetilde{r}_{s,n})^{3}}\frac{\partial\widetilde{r}_{s,n}}{\partial
b_{n}}+\alpha_{n}-\rho_{n}\frac{\phi_{n}\mathcal{L}w}{(\widetilde{r}_{s,n})^{2}}\frac{\partial\widetilde{r}_{s,n}}{\partial
b_{n}}$ (32)
$2yc_{e}x_{n}(\phi_{n}\mathcal{L}w)^{2}p_{n}-\bigg{(}\frac{\partial\widetilde{r}_{s,n}}{\partial
p_{n}}+\rho_{n}\frac{\phi_{n}\mathcal{L}w}{(\widetilde{r}_{s,n})^{2}}\bigg{)}\frac{yc_{e}}{2x_{n}(\widetilde{r}_{s,n})^{3}}+\beta_{n}$
(33) $\frac{\partial\mathcal{L}_{2}}{\partial
f_{n}^{\text{DO}}}=-\frac{f_{n,\text{max}}^{\text{DO}}}{f_{n}^{\text{DO}}}+2yc_{e}k_{n}\mathcal{C}_{n}(\phi_{n})f_{n}^{\text{DO}}+\delta_{n}-\rho_{n}\frac{\mathcal{C}_{n}(\phi_{n})}{(f_{n}^{\text{DO}})^{2}}$
(34) $\frac{\partial\mathcal{L}_{2}}{\partial
f_{n}^{\text{MO}}}=2yc_{e}k_{m}\mathcal{B}_{n}(\phi_{n})f_{n}^{\text{MO}}+\zeta-\rho_{n}\frac{\mathcal{B}_{n}(\phi_{n})}{(f_{n}^{\text{MO}})^{2}}$
(35) $\frac{\partial\mathcal{L}_{2}}{\partial
T}=\sum_{n\in\mathcal{N}}\rho_{n}-yc_{t}$

  * •

Complementary Slackness: $\forall\in\mathcal{N}$

(36) $\displaystyle\alpha_{n}\cdot(b_{n}-b_{n,\text{max}})=0$ (37)
$\displaystyle\beta_{n}\cdot(p_{n}-p_{n,\text{max}})=0,$ (38)
$\displaystyle\delta_{n}\cdot(f_{n}^{\text{DO}}-f_{n,\text{max}}^{\text{DO}})=0,$
(39)
$\displaystyle\zeta\cdot(\sum_{n\in\mathcal{N}}f^{\text{MO}}_{n}-f^{\text{MO}}_{\text{max}})=0$
(40)
$\displaystyle\rho_{n}\cdot(t_{n}(b_{n},p_{n},f^{\text{DO}}_{n},f^{\text{MO}}_{n})-T)=0,$

  * •

Dual Feasibility: (18a), (18b), (18d), (18e), (18h), (22a).

  * •

Primal Feasibility: $\forall n\in\mathcal{N}$

(43) $\displaystyle\begin{array}[]{ll}&\text{(\ref{kkt_11}a): }\alpha_{n}\geq
0;\text{(\ref{kkt_11}b): }\beta_{n}\geq 0;\text{(\ref{kkt_11}c):
}\delta_{n}\geq 0;\\\ &\text{(\ref{kkt_11}d): }\zeta\geq
0;\text{(\ref{kkt_11}e): }\rho_{n}\geq 0,\end{array}$

where (32) is $\frac{\partial\mathcal{L}_{2}}{\partial p_{n}}$. Using
(31)-(43), we develop a three-step algorithm with known
“$y,x,\boldsymbol{\phi}$” (we denote as $\Delta$ below). Step 1: Find
$(\boldsymbol{f^{\text{DO}}},\boldsymbol{f^{\text{DO}}},\boldsymbol{\delta},\zeta\
|\ \Delta)$ with respect to $\boldsymbol{\rho}$. Step 2: Find
$(\boldsymbol{b},\boldsymbol{p},\boldsymbol{\alpha},\boldsymbol{\beta}\ |\
\Delta)$ with respect to $\boldsymbol{\rho}$. Step 3:
Find$(T,\boldsymbol{\rho}\ |\ \Delta)$.

Step 1. From (33), we could derive the solution of $f_{n}^{\text{DO}}$ with
represented by $(\delta_{n},\rho_{n})$ defined as
$\widehat{f}_{n}^{\text{DO}}(\delta_{n},\rho_{n}\ |\ \Delta)$, based on the
value of $\widehat{f}_{n}^{\text{DO}}(\delta_{n},\rho_{n}\ |\ \Delta)$, we
have the following discussion:

  * •

Case 1: If $\widehat{f}_{n}^{\text{DO}}(\delta_{n},\rho_{n}\ |\
\Delta)<f_{n,\text{max}}^{\text{DO}}$

The condition of this case can also be represented as
$\widehat{f}_{n}^{\text{DO}}(\delta_{n},\rho_{n}\ |\
\Delta)\geq\sqrt[3]{\frac{\rho_{n}}{2yc_{e}k_{n}}}$. In this case, based on
(38), we can simply set $\delta_{n}=0$. We have
$f_{n}^{\text{DO}}=\widehat{f}_{n}^{\text{DO}}(0,\rho_{n}\ |\ \Delta)$

  * •

Case 2: If $\widehat{f}_{n}^{\text{DO}}(\delta_{n},\rho_{n}\ |\ \Delta)\geq
f_{n,\text{max}}^{\text{DO}}$

In this case, the condition can be written as
$\widehat{f}_{n}^{\text{DO}}(\delta_{n},\rho_{n}\ |\
\Delta)<\sqrt[3]{\frac{\rho_{n}}{2yc_{e}k_{n}}}$. Since $\delta>0$, according
to (38), $f_{n}^{\text{DO}}=f_{n,\text{max}}^{\text{DO}}$, substituting it
into (33), we can obtain the equation of $\delta_{n}$ represented by
$\rho_{n}$, specifically $\widehat{\delta}_{n}(\rho_{n}\ |\
\Delta)|_{{f_{n}^{\text{DO}}=f_{n,\text{max}}^{\text{DO}}}}=1+\rho_{n}\frac{\mathcal{C}_{n}(\phi_{n})}{(f_{n,\text{max}}^{\text{DO}})^{2}}-2yc_{e}k_{n}\mathcal{C}_{n}(\phi_{n})f_{n,\text{max}}^{\text{DO}}$.

Summarize both cases, and the conclusion could be derived:

(44) $\widetilde{f}_{n}^{\text{DO}}(\rho_{n}\ |\
\Delta)=\text{min}\\{f_{n,\text{max}}^{\text{DO}},\widehat{f}_{n}^{\text{DO}}(0,\rho_{n}\
|\ \Delta)\\},$ (47) $\displaystyle\widetilde{\delta}_{n}(\rho_{n}\ |\
\Delta)=\left\\{\begin{array}[]{ll}0,&\widetilde{f}_{n}^{\text{DO}}(0,\rho_{n}\
|\ \Delta)<f_{n,\text{max}}^{\text{DO}}\\\ \widehat{\delta}_{n}(\rho_{n}\ |\
\Delta),&\text{others}\end{array}\right.$

Similarly, The solution of
$f_{n}^{\text{MO}}=\widehat{f}_{n}^{\text{MO}}(\zeta,\rho_{n}\ |\ \Delta)$ can
be obtained from (34) as a function of $(\zeta,\rho_{n})$, the discussion is

  * •

Case 1. If $\sum_{n\in\mathcal{N}}\widehat{f}_{n}^{\text{MO}}(\zeta,\rho_{n}\
|\ \Delta)<f_{\text{max}}^{\text{MO}}$

The condition can also be written as
$\sum_{n\in\mathcal{N}}\sqrt[3]{\frac{\rho_{n}}{2yc_{e}k_{m}}}\leq
f_{\text{max}}^{\text{MO}}$. In this case, according to (34) (39), $\zeta=0$.
We have $f_{n}^{\text{MO}}=\widehat{f}_{n}^{\text{MO}}(0,\rho_{n}\ |\
\Delta)$.

  * •

Case 2. If $\sum_{n\in\mathcal{N}}\widehat{f}_{n}^{\text{MO}}(\zeta,\rho_{n}\
|\ \Delta)\geq f_{\text{max}}^{\text{MO}}$

The condition means
$\sum_{n\in\mathcal{N}}\sqrt[3]{\frac{\rho_{n}}{2yc_{e}k_{m}}}>f_{\text{max}}^{\text{MO}}$,
since $\zeta>0$, according to (39),
$\sum_{n\in\mathcal{N}}f^{\text{MO}}_{n}=f^{\text{MO}}_{\text{max}}$,
substituting it into (34), we can obtain the equation of $\zeta$ represented
by $(\rho_{n})$, $\widehat{\zeta}(\rho_{n}\ |\
\Delta)|_{\sum_{n\in\mathcal{N}}f_{n}^{\text{MO}}=f_{\text{max}}^{\text{MO}}}$,
then use (34) get the value of $f_{n}^{\text{MO}}$,
$f_{n}^{\text{MO}}=\widehat{f}_{n}^{\text{MO}}(\widehat{\zeta}(\rho_{n}\ |\
\Delta),\rho_{n}\ |\ \Delta)$

Summarize both cases, and the conclusion could be derived:

(48) $\displaystyle\widetilde{f}_{n}^{\text{MO}}(\rho_{n}\ |\ \Delta)$
$\displaystyle=\text{min}\\{\widehat{f}_{n}^{\text{MO}}(\widehat{\zeta}(\rho_{n}\
|\ \Delta),\rho_{n}\ |\ \Delta),\widehat{f}_{n}^{\text{MO}}(0,\rho_{n}\ |\
\Delta)\\}$ (51) $\displaystyle\widetilde{\zeta}(\rho_{n}\ |\
\Delta)=\left\\{\begin{array}[]{ll}0,&\sum_{n\in\mathcal{N}}\widetilde{f}_{n}^{\text{MO}}(0,\rho_{n}\
|\ \Delta)<f_{\text{max}}^{\text{MO}}\\\ \widehat{\zeta}_{n}(\rho_{n}\ |\
\Delta),&\text{others}\end{array}\right.$

Step 2. First, from (31), since all the functions except $\alpha_{n}$ are
negative and the result is $0$, $\alpha_{n}>0$ and which result in
$b_{n}=b_{n,\text{max}}$ from (36). Substituting $b_{n}=b_{n,\text{max}}$ into
(31), we have

(52)
$\bigg{[}-\frac{yc_{e}}{2x_{n}(\widetilde{r}_{s,n})^{3}}\frac{\partial\widetilde{r}_{s,n}}{\partial
b_{n}}-\rho_{n}\frac{\phi_{n}\mathcal{L}w}{(\widetilde{r}_{s,n})^{2}}\frac{\partial\widetilde{r}_{s,n}}{\partial
b_{n}}\bigg{]}_{b_{n}=b_{n,\text{max}}}+\alpha_{n}=1$

The solution of $\alpha_{n}$ in (52) can be denoted as a function of $p_{n}$
and Lagrange multipliers $\rho_{n}$ which is
$\widehat{\alpha}_{n}(p_{n},\rho_{n}\ |\ \Delta)$. Substituting
$b_{n}=b_{n,\text{max}}$ into (32), we can obtain the following equation:

(53)
$\displaystyle\bigg{[}-\frac{yc_{e}}{2x_{n}(\widetilde{r}_{s,n})^{3}}\frac{\partial\widetilde{r}_{s,n}}{\partial
p_{n}}-\rho_{n}\frac{\phi_{n}\mathcal{L}w}{(\widetilde{r}_{s,n})^{2}}\frac{\partial\widetilde{r}_{s,n}}{\partial
p_{n}}\bigg{]}_{b_{n}=b_{n,\text{max}}}$
$\displaystyle+2yc_{e}x_{n}(\phi_{n}\mathcal{L}w)^{2}p_{n}+\beta_{n}=0.$

From (53), we could derive the solution of $p_{n}$ represented by
$(\beta_{n},\rho_{n})$ defined as $\widehat{p}_{n}(\beta_{n},\rho_{n}\ |\
\Delta)$. Summarize (52)(53), we can obtain the optimal value of
$\boldsymbol{b}^{*}$, and the expression of
$\boldsymbol{p},\boldsymbol{\alpha}$ are as follows

(57) $\displaystyle\left\\{\begin{array}[]{l}b_{n}^{*}=b_{n,max}\\\
\widehat{p}_{n}(\beta_{n},\rho_{n}\ |\
\Delta)=max\\{\check{p}_{n}(\beta,\rho_{n}\ |\ \Delta),0\\},\\\
\widehat{\alpha}_{n}(p_{n},\rho_{n}\ |\
\Delta)=\check{\alpha}_{n}(\widetilde{p}_{n}(\beta_{n},\rho_{n}\ |\
\Delta),\rho_{n}\ |\ \Delta),\end{array}\right.$

Since the expression of $\widehat{\alpha}_{n}(p_{n},\rho_{n}\ |\ \Delta)$
contains $\boldsymbol{p}$, which means we can only analyze and derive the
value of $\boldsymbol{\alpha}$ after we get $\boldsymbol{p}$, for
$\widehat{p}_{n}(\beta_{n},\rho_{n}\ |\ \Delta)$, we can have the following
discussion

  * •

Case 1. If $\widehat{p}_{n}(\beta_{n},\rho_{n}\ |\ \Delta)<p_{n,\text{max}}$

In this case, according to (37), we can simply set $\beta_{n}=0$. We have
$p_{n}=\widehat{p}_{n}(0,\rho_{n}\ |\ \Delta)$ and constraints (32), (18b),
(43b) are satisfied.

  * •

Case 2. If $\widehat{p}_{n}(\beta_{n},\rho_{n}\ |\ \Delta)\geq
p_{n,\text{max}}$

In this case, we can denote $\beta_{n}>0$ from (37), and
$p_{n}=p_{n,\text{max}}$. Substituting $p_{n}=p_{n,\text{max}}$ into (53), we
can derive the solution of $\beta_{n}$ with represented by $\rho_{n}$ defined
as $\widehat{\beta}_{n}(\rho_{n}\ |\
\Delta)=\bigg{[}\frac{yc_{e}}{2x_{n}(\widetilde{r}_{s,n})^{3}}\frac{\partial\widetilde{r}_{s,n}}{\partial
p_{n}}-\rho_{n}\frac{\phi_{n}\mathcal{L}w}{(\widetilde{r}_{s,n})^{2}}\frac{\partial\widetilde{r}_{s,n}}{\partial
p_{n}}\bigg{]}_{(b_{n},p_{n}=b_{n,\text{max}},p_{n,\text{max}})}\\\
-2yc_{e}x_{n}(\phi_{n}\mathcal{L}w)^{2}p_{n}$.

Summarize both cases, and the conclusion could be derived:

(59) $\widetilde{p}_{n}(\rho_{n}\ |\
\Delta)=\text{min}\\{p_{n,\text{max}},\text{max}\\{0,\widehat{p}_{n}(0,\rho_{n}\
|\ \Delta)\\}\\}$ (62) $\displaystyle\widetilde{\beta}(\rho_{n}\ |\
\Delta)=\left\\{\begin{array}[]{ll}0,&\widehat{p}_{n}(\beta_{n},\rho_{n}\ |\
\Delta)<p_{n,\text{max}}\\\ \widehat{\beta}_{n}(\rho_{n}\ |\
\Delta),&\text{others}\end{array}\right.$

So $\widehat{\alpha}_{n}(p_{n},\rho_{n}\ |\ \Delta)$ can be represented using
(52), $\widetilde{\alpha}_{n}(\rho_{n}\ |\
\Delta)=1+\bigg{[}\frac{yc_{e}}{2x_{n}(\widetilde{r}_{s,n})^{3}}\frac{\partial\widetilde{r}_{s,n}}{\partial
b_{n}}+\rho_{n}\frac{\phi_{n}\mathcal{L}w}{(\widetilde{r}_{s,n})^{2}}\frac{\partial\widetilde{r}_{s,n}}{\partial
b_{n}}\bigg{]}_{(b_{n}=b_{n,\text{max}},p_{n}=\widetilde{p}_{n}(\rho_{n}\ |\
\Delta))}$.

After Step 1 and Step 2, we derive the solutions of all the optimal variables
except $T$ related to the Lagrange multipliers $\boldsymbol{\rho}$, and we
continue to handle the last constraint.

Step 3. With constraint and our previous results, we can denote the updated
constraints represented by $(T,\boldsymbol{\rho})$ are as follows

From (22a): $\displaystyle t_{n}(b_{n,\text{max}},\widetilde{p}_{n}(\rho_{n}\
|\ \Delta),\phi^{*}_{n},\widetilde{f}_{n}^{\text{DO}}(\rho_{n}\ |\ \Delta),$
(63) $\displaystyle\widetilde{f}_{n}^{\text{MO}}(\rho_{n}\ |\ \Delta))\leq T$
From (40):
$\displaystyle\rho_{n}\cdot(t_{n}(b_{n,\text{max}},\widetilde{p}_{n}(\rho_{n}\
|\ \Delta),\phi^{*}_{n},\widetilde{f}_{n}^{\text{DO}}(\rho_{n}\ |\ \Delta),$
(64) $\displaystyle\widetilde{f}_{n}^{\text{MO}}(\rho_{n}\ |\ \Delta))-T)=0.$

We can have the following discussion

  * •

Case 1. If $\rho_{n}=0$ satisfies constraint (63)

In this case, we can simply set $\rho_{n}=0$ and we can derive the value of
$p_{n}$, $f_{n}^{\text{DO}}$ and $f_{n}^{\text{MO}}$ (i.e., all the
optimization variables except $T$), a lower bound of $T$ can be derived since
$t_{n}(b_{n,\text{max}},\widetilde{p}_{n}(\rho_{n}\ |\
\Delta),\phi^{*}_{n},\widetilde{f}_{n}^{\text{DO}}(\rho_{n}\ |\
\Delta),\widetilde{f}_{n}^{\text{MO}}(\rho_{n}\ |\ \Delta))<T$.

  * •

Case 2. If $\rho_{n}=0$ violates constraint (63)

In this case, $\rho_{n}>0$ and
$t_{n}(b_{n,\text{max}},\widetilde{p}_{n}(\rho_{n}\ |\
\Delta),\phi^{*}_{n},\widetilde{f}_{n}^{\text{DO}}(\rho_{n}\ |\ \Delta),\\\
\widetilde{f}_{n}^{\text{MO}}(\rho_{n}\ |\ \Delta))-T=0$.

Summarize both cases, and the conclusion could be derived

(68) $\displaystyle\widetilde{\rho}_{n}(T\ |\
\Delta)=\left\\{\begin{array}[]{ll}0,&t_{n}(b_{n,\text{max}},\widetilde{p}_{n}(\rho_{n}\
|\ \Delta),\phi^{*}_{n},\\\ &\widetilde{f}_{n}^{\text{DO}}(\rho_{n}\ |\
\Delta),\widetilde{f}_{n}^{\text{MO}}(\rho_{n}\ |\ \Delta))<T\\\
\widehat{\rho}_{n}(T\ |\ \Delta),&\text{others}\end{array}\right.$

Since parameter $T$ will be the last parameter to derive, which means we
already obtained the value of $T$ when we discuss according to the different
situation of $\boldsymbol{\rho}$. In the above summarization, the unknown
parameter $\boldsymbol{\rho}$ occurs on both sides of the equation, we can
propose a new function $R_{n}(\rho_{n}\ |\ T,\Delta)$ which satisfies

(69) $\displaystyle R_{n}(\rho_{n}\ |\ T,\Delta)=-\rho_{n},$
$\displaystyle\text{if}\ t_{n}(b_{n,\text{max}},\widetilde{p}_{n}(\rho_{n}\ |\
\Delta),\phi^{*}_{n},\widetilde{f}_{n}^{\text{DO}}(\rho_{n}\ |\
\Delta),\widetilde{f}_{n}^{\text{MO}}(\rho_{n}\ |\ \Delta))<T$ (70)
$\displaystyle R_{n}(\rho_{n}\ |\ T,\Delta)$
$\displaystyle=t_{n}(b_{n,\text{max}},\widetilde{p}_{n}(\rho_{n}\ |\
\Delta),\phi^{*}_{n},\widetilde{f}_{n}^{\text{DO}}(\rho_{n}\ |\ \Delta),$
$\displaystyle\widetilde{f}_{n}^{\text{MO}}(\rho_{n}\ |\
\Delta))-T,~{}\text{if}\ \text{others}.$

Since (Zhao et al., 2023) proves that $R_{n}(\rho_{n}\ |\ T,\Delta)$ is non-
increasing with the increment of $\rho_{n}$. We can then utilize the
multivariate bisection algorithm proposed in (Galván, 2017) and (Zhao et al.,
2023) to derive the optimal $\boldsymbol{\rho}$ and $T$.

The complete resource allocation algorithm in Algorithm 1.

1 Initialize
$sol^{(0)}=(\boldsymbol{b}^{(0)},\boldsymbol{p}^{(0)},\boldsymbol{\phi}^{(0)},\boldsymbol{(f^{\text{DO}}})^{(0)},\boldsymbol{(f^{\text{MO}}})^{(0)},T^{(0)})$,
iteration number $j=0$.

2 Calculate
$\boldsymbol{x}^{(0)}=\sqrt{\frac{\mathcal{P}_{n}(\boldsymbol{z})^{(0)}}{\mathcal{M}_{n}(\boldsymbol{z})^{(0)}}}$,
$y^{(0)}=\frac{U(\boldsymbol{z}^{(0)})}{S(\boldsymbol{z}^{(0)})}$,
$\boldsymbol{\chi}^{e^{-}(0)}$

3 repeat

4 Solve Subproblem 1. Obtain $\boldsymbol{\phi}^{(j+1)}$.

5 Solve Subproblem 2 and obtain
$(\boldsymbol{b}^{(j+1)},\boldsymbol{p}^{(j+1)},\boldsymbol{(f^{\text{DO}}})^{(j+1)},\boldsymbol{(f^{\text{MO}}})^{(j+1)},T^{(j+1)})$.

6
$sol^{(j+1)}=(\boldsymbol{p}^{(j+1)},\boldsymbol{B}^{(j+1)},\boldsymbol{f}^{(j+1)})$.

7 Calculate $\boldsymbol{\chi}^{e^{+}(j+1)}$ and obtain
$\boldsymbol{\chi}^{e(j+1)}$.

8 Update $\mathbb{x}^{(j+1)}$, $y^{(j+1)}$

9 Set $j\leftarrow j+1$.

10until  _Convergence or the number of iterations achieves maximum $J$_;

Algorithm 1 Resource Allocation Algorithm

##  5\. Experimental results

In this section, we report the experimental results in detail.

###  5.1. Parameter Setting

In configuring the parameters for our communication system, we use specific
settings based on established models and empirical data. The path loss between
the large model server and each downstream user is quantified by the equation
$128.1+37.6\log(\text{distance})$ dB, where the distance is measured in
kilometers. This model also incorporates a standard deviation of 8 dB for
shadow fading. The power spectral density of Gaussian noise is set at $-174$
dBm/Hz. Focusing on computational capacities, the maximum GPU computation
frequency for each user, $f_{n,\text{max}}^{\text{DO}}$, is set at 7 GHz
(utilizing four NVIDIA GeForce RTX 3060 units). For the server, the maximum
GPU computation frequency, $f_{\text{max}}^{\text{MO}}$ is set as 100 GHz. The
effective switched capacitance parameters, $k_{n}$ and $k_{m}$, are both fixed
at $10^{-27}$. As for the transmission capabilities, the maximum transmit
power of mobile users, $p_{n,\text{max}}$, is 0.2W, while the extraction rates
$\phi_{n}$ vary within the range of $[0,1]$. The total parameter size
considered in our model is $14M$. Lastly, the sum of the cost coefficients
$c_{e}+c_{t}=1$.

###  5.2. Performance Comparison with baselines

We consider three different baseline methods in the experiment compared with
our proposed method, 1\. Average allocation. Set each
$b_{n}=b_{n,\text{max}},p_{n}=p_{n,\text{max}},f_{n}^{\text{DO}}=f_{n,\text{max}}^{\text{DO}},f_{n}^{\text{MO}}=\frac{f_{\text{max}}^{\text{MO}}}{N}$
and $\phi_{n}=0.5$. 2\. Optimize
$\boldsymbol{b},\boldsymbol{p},\boldsymbol{s}$ only. Set each
$f_{n}^{\text{DO}}=f_{n,\text{max}}^{\text{DO}},f_{n}^{\text{MO}}=\frac{f_{\text{max}}^{\text{MO}}}{N}$.
3\. Optimize $\boldsymbol{f^{\text{DO}}},\boldsymbol{f^{\text{MO}}}$ only. Set
each $b_{n}=b_{n,\text{max}},p_{n}=p_{n,\text{max}}$ and $\phi_{n}=0.5$.

(a) Time consumption versus user number

(b) Energy consumption versus user number

(c) UCR versus user number

Figure 2. Metrics concerning the number of users.

(a) UCR versus max transmission power

(b) UCR versus max user frequency

(c) UCR versus max server frequency

Figure 3. The system utility-cost ratio (UCR) versus various parameters.

(a) Total time consumption

(b) Total energy consumption

Figure 4. Experiments with different weight parameters.

Impact of user number on UCR. In Fig. 2, we observe total energy consumption
increase as the number of users increases. The difference in energy
consumption between the optimal (our algorithm) and the least effective
(average allocation) performance is approximately 44%, which demonstrates the
advantage of our algorithm compared to the average allocation method. However,
allocating fewer resources to each user might reduce the efficiency of the
fine-tuning process, reflecting a constraint in resources as the user base
expands. As for time consumption, the time efficiency can also be impacted as
the number of users increases. Our result shows that the difference in time
consumption between our algorithm and the average allocation method is
approximately 38.18%. This implies that our algorithm is more time-efficient
than the average allocation method, although an increase in users might lead
to longer times the server requires to fine-tune each user’s model.

Furthermore, a decreasing trend can be shown in the average UCR value as the
number of users increases, further corroborating the negative impact of
resource dilution on user utility. With more users sharing the server’s
computational resources, each user experiences a reduction in the
effectiveness of the offsite-tuning process. The difference in UCR values
between our algorithm and the average allocation method is about -42.86%,
indicating that our algorithm is more effective in maintaining higher UCR
values, although the specific implications of the UCR values and their impact
on the overall system performance must be considered.

The results indicate that our algorithm outperforms the average allocation
method regarding energy consumption, UCR value, and time efficiency. As the
number of users increases, the impact of resource allocation efficiency on
each user becomes more pronounced, potentially leading to a decrease in the
efficiency of the fine-tuning process. Our algorithm effectively mitigates
these challenges, demonstrating higher energy and time efficiency and an
advantage in maintaining UCR values.

UCR versus transmission power. In Fig. 3(a), the UCR increases as the
transmission power grows for all algorithms since a higher transmission power
expands the search space for optimization, allowing for more efficient use of
the available power. At lower transmission powers (e.g., 0.5W and 1W), our
algorithm has a slightly higher UCR than the others, indicating its
effectiveness even in low-power scenarios. However, as the transmission power
increases, the performance of our algorithm significantly surpasses the
others. This is evident in the UCR values where our algorithm reaches 3.8 at
5W, while others are below this mark, and other algorithms always hold more
significant fluctuations and generally lower UCR values compared to ours. For
instance, average allocation shows the least improvement and remains below 1
UCR even at higher power levels.

UCR versus computation resource. Fig. 3(b) and Fig. 3(c) shows a clear trend
where the UCR increases as the GPU frequency grows. This suggests that higher
GPU frequencies provide a wider search space for optimization algorithms,
leading to better performance in terms of UCR. In both two figures, our
algorithm consistently outperforms other algorithms. For example, at 8GHz user
GPU frequency and 160 GHz server GPU frequency, the difference in UCR compared
to the average allocation is 600% and 700%, respectively.

###  5.3. Performance When Adapting Weight Parameters

In our model, $c_{e}$ and $c_{t}$, are employed to modulate the optimization’s
focus. A higher value of $c_{e}$ compared to $c_{t}$ signifies an emphasis on
minimizing energy consumption within the optimization framework (resp,
indicates a prioritization of time efficiency over energy efficiency in the
optimization process). To elucidate the impact of these weight parameters, we
perform a series of experiments with varied $c_{e}$ and $c_{t}$ pairings,
assessing our approach against a standard baseline: uniform allocation. This
baseline approach allocates resources uniformly among users, setting the
weight parameters at an equal balance of $c_{e}=0.5,c_{t}=0.5$.

Figures 4(a) and 4(b) present the outcomes of total energy and time
consumption, respectively, evaluated across five distinct $(c_{e},c_{t})$
pairings under varying maximum transmission power thresholds. The data
indicates as $c_{e}$ increases and $c_{t}$ diminishes, there is a noticeable
reduction in total energy usage while total time expenditure escalates. This
pattern emerges because an augmented $c_{e}$ (resp, $c_{t}$) shifts the focus
of our optimization approach towards minimizing energy expenses (resp,
correspondingly, time consumption).

##  6\. Conclusion

In conclusion, our work presents an impactful approach to the offsite fine-
tuning of large models in the mobile edge computing environment under the
physical layer security and addresses both the privacy concerns and resource
allocation problems inherent in the conventional fine-tuning process. We
formulated the problem of maximizing the utility-consumption ratio, which
balances between maximizing user utility and minimizing system costs. Our
proposed optimization algorithm blends the Dinkelbach algorithm, Successive
Convex Approximation, fractional programming and alternating optimization
techniques, and the algorithm converges effectively. Additionally, the
simulated evaluations of our algorithm underscore its superiority over
existing methods.

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##  Appendix A Proof of Assumption 1

In Assumption 1, the concavity of $U_{n}(x)$, represented by a non-positive
second derivative ($U^{{}^{\prime\prime}}_{n}(x)\leq 0$ for $x>0$, is
indicative of diminishing marginal returns. A concave utility function ensures
that any local maximum is also a global maximum, simplifying the search for
optimal solutions. Also, for $x>0$, both $U^{{}^{\prime\prime}}_{n}(x)$ and
$U^{{}^{\prime}}_{n}(x)$ are well-defined and continuous, which means
$U_{n}(x)$ being twice differentiable. For $U_{n}(x)$, we do not define a
value at $x=0$, if $U_{n}(x)$ approaches a finite limit as $x\rightarrow
0^{+}$, this limit can be used to define the function at $x=0$. If no such
limit exists, $U_{n}(x)$ remains undefined at $x=0$. The non-decreasing aspect
of $U_{n}(x)$ ensures that the utility increases, or at the very least remains
constant, as $x$ (the secrecy rate) increases.

##  Appendix B Proof of Theorem 1

After applying the KKT conditions to Problem $\mathbb{P}_{3}$, we have

(71)
$\displaystyle\frac{\partial\mathcal{L}_{1}(\boldsymbol{\phi},\boldsymbol{\eta})}{\partial\phi_{n}}=-\frac{1}{\phi_{n}}+yc_{e}\big{(}k_{n}C_{1}C_{2}(f^{\text{DO}}_{n})^{2}\phi_{n}^{(C_{2}-1)}+\frac{p_{n}\mathcal{L}w}{r_{s,n}}$
$\displaystyle-
k_{m}C_{3}C_{4}(f^{\text{MO}}_{n})^{2}\phi_{n}^{(-C_{4}-1)}\big{)}$
$\displaystyle+\eta_{n}\big{(}\frac{C_{1}C_{2}\phi_{n}^{(C_{2}-1)}}{f^{\text{DO}}_{n}}+\frac{\mathcal{L}w}{r_{s,n}}-\frac{C_{3}C_{4}\phi_{n}^{(-C_{4}-1)}}{f^{\text{MO}}_{n}}-T\big{)}=0.$

And

(72)
$\eta_{n}\cdot(\text{max}\\{\frac{C_{1}\phi_{n}^{C_{2}}}{f^{\text{DO}}_{n}}+\frac{\phi_{n}\mathcal{L}w}{r_{s,n}}+\frac{C_{3}\phi_{n}^{-C_{4}}}{f^{\text{MO}}_{n}}\\}-T)=0$

From (71), we can denote the solution by $\widetilde{\phi}_{n}(\eta_{n}|\
y,\boldsymbol{b},\boldsymbol{p},\boldsymbol{f^{\text{DO}}},\boldsymbol{f^{\text{MO}}},T)$.
Note that the right-hand side of (71) is monotonically increasing to
$\phi_{n}$, the optimal solution $\widetilde{\phi}_{n}(\eta_{n}\ |\
y,\boldsymbol{b},\boldsymbol{p},\boldsymbol{f^{\text{DO}}},\boldsymbol{f^{\text{MO}}},T)$
can be obtained using bisection method. Considering (24a), the Lagrange
multiplier should meet the KKT conditions.