# Music mood detection based on audio and lyrics with Deep Neural Net

###### Abstract

We consider the task of multimodal music mood prediction based on the audio
signal and the lyrics of a track. We reproduce the implementation of
traditional feature engineering based approaches and propose a new model based
on deep learning. We compare the performance of both approaches on a database
containing 18,000 tracks with associated valence and arousal values and show
that our approach outperforms classical models on the arousal detection task,
and that both approaches perform equally on the valence prediction task. We
also compare the a posteriori fusion with fusion of modalities optimized
simultaneously with each unimodal model, and observe a significant improvement
of valence prediction. We release part of our database for comparison
purposes.

MFCCs

    Mel-Frequency Cepstral Coefficients
SVM

    Support Vector Machine
GMM

    Gaussian Mixture Model
ConvNet

    convolutional neural network
RNN

    recurrent neural network
LSTM

    Long Short-Term Memory
GRU

    Gated Recurrent Unit
biLSTM

    Bidirectional Long Short-Term Memory
MIR

    Music Information Retrieval
MIREX

    Music Information Retrieval Evaluation eXchange
TF

    Term-Frequency
TFIDF

    Term-Frequency Inverse Document Frequency
MSD

    Million Song Dataset
MDS

    Multi-Dimensional Scaling
POS

    Part-of-speech

##  1 Introduction

Music Information Retrieval (MIR) has been an ever growing field of research
in recent years, driven by the need to automatically process massive
collections of music tracks, an important task to, for example, streaming
companies. In particular, automatic music mood detection has been an active
field of research in MIR for the past twenty years. It consists of
automatically determining the emotion felt when listening to a track.111We use
the words emotion and mood interchangeably, as done in the literature (see
[15]). In this work, we focus on the task of multimodal mood detection based
on the audio signal and the lyrics of the track. We apply deep learning
techniques to the problem and compare our approach to classical feature
engineering-based ones on a database of 18,000 songs labeled with a continuous
arousal/valence representation. This database is built on the Million Song
Dataset (MSD) [2] and the Deezer catalog. To our knowledge this constitutes
one of the biggest datasets for multimodal mood detection ever proposed.

###  1.1 Related work

Music mood studies appeared in the first half of the $20$th century, with the
work of Hevner [7]. In this work, the author defines groups of emotions and
studies classical music works to unveil correlations between emotions and
characteristics of the music. A first indication that music and lyrics should
be jointly considered when analyzing musical mood came from a psychological
study exposing independent processing of these modalities by the human brain
[3]. For the past 15 years, different approaches have been developed with a
wide range of datasets and features. An important fraction of them was put
together by Kim et al. in [15]. Li and Ogihara [18] used signal processing
features related to timbre, pitch and rhythm. Tzanetakis et al. [28] and
Peeters [22] also used classical audio features, such as Mel-Frequency
Cepstral Coefficients (MFCCs), as input to a Support Vector Machine (SVM).
Lyrics-based mood detection was most often based on feature engineering. For
example, Yang and Lee [31] resorted to a psycho-linguistic lexicon related to
emotion. Argamon et al. [1] extracted stylistic features from text in an
author detection task. Multimodal approaches were also studied several times.
Laurier et al. [16] compared prediction level and feature level fusion,
referred to as late and early fusion respectively. In [26], Su et al.
developed a sentence level fusion. An important part of the work based on
feature engineering was compiled into more complete studies, among which the
one from Hu and Downie [9] is one of the most exhaustive, and compares many of
the previously introduced features.

Influenced by advances in deep learning, notably in speech recognition or
machine translation, new models began to emerge, based on fewer feature
engineering. Regarding audio-based methods, the Music Information Retrieval
Evaluation eXchange (MIREX) competition [5] has monitored the evolution of the
state of the art. In this framework, Lidy et al. [19] have shown the promise
of audio-based deep learning. Recently, Jeon et al. [14] presented the first
multimodal deep learning approach using a bimodal convolutional recurrent
network with a binary mood representation. However, they neither compared
their work to classical approaches, nor evaluated the advantage of their mid-
level fusion against simple late fusion of unimodal models. In [12], Huang et
al. resorted to deep Boltzmann machines to unveil early correlations between
audio and lyrics, but their method was limited by the incompleteness of their
dataset, which made impossible the use of temporally local layers, e.g.
recurrent or convolutional ones. To our knowledge, there is no clear answer as
to whether feature engineering yields better results than more end-to-end
systems for the multimodal task, probably because of the lack of easily
accessible large size datasets.

###  1.2 Mood representation

A variety of mood representations have been used in the literature. They
either consist of monolabel tagging with either simple tags (e.g. in [9]),
clusters of tags (e.g. in the MIREX competition) or continuous representation.
In this work, we resort to the latter option. Russell [24] defined a
2-dimensional continuous space of embedding for emotions. A point in this
space represents the valence (from negative to positive mood) and arousal
(from calm to energetic mood) of an emotion. This representation was used
multiple times in the literature [12, 27, 29], and presents the advantage of
being satisfyingly exhaustive. It is worth noting that this representation has
been validated by embedding emotions in a 2-dimensional space based on their
co-occurrences in a database [10]. Since we choose this representation we
formulate mood estimation as a 2-dimensional regression problem based on a
track’s lyrics and/or audio.

###  1.3 Contributions of this work

We study end-to-end lyrics-based approaches to music mood detection and
compare their performance with classical lyrics-based methods performance, and
give insights on the performing architectures and networks types. We show that
lyrics-based networks show promising results both in valence and arousal
prediction.

We describe our bimodal deep learning model and evaluate the performance of a
mid-level fusion, compared to unimodal approaches and to late fusion of
unimodal predictions. We show that arousal is highly correlated to the audio
source, whereas valence requires both modalities to be predicted significantly
better. We also see that the latter task can be notably improved by resorting
to mid-level fusion.

Finally, we compare our model to traditional feature engineering methods and
show that deep-learning-based approaches outperform classical models, when it
comes to multimodal arousal detection, and we show that both systems are
equally performing on valence prediction. For future comparison purposes, we
also release part of our database consisting of valence/arousal labels and
corresponding song identifiers.

##  2 Classical feature engineering-based approaches

We compare our model to classical approaches based on feature engineering.
These methods were iteratively deepened over the years: for audio-based
models, a succession of works [18, 22, 28] indicated the top performing audio
features for mood detection tasks ; for lyrics-based approaches, a series of
studies [1, 31, 10] investigated a wide variety of text-based features.
Finally, fusion methods were also studied multiple times [16, 29, 9]. Hu and
Downie compiled and deepened these works in a series of papers [10, 9, 8],
which is the most accomplished feature-engineering-based approach of the
subject. We reimplement this work and compare its performance to ours. This
model consists in the choice of the optimal weighted average of the
predictions of two unimodal models: an SVM on top of MFCCs, spectral flux,
rolloff and centroid, for audio; and an SVM on top of basic, linguistic and
stylistic features (n-grams, lexicon-based features, etc.) for lyrics.

##  3 Deep learning-based approach

We first explore unimodal deep learning models and then combine them into a
multimodal network. In each case, the model simultaneously predicts valence
and arousal. Inputs are subdivided in several segments for training, so that
each input has the same length. Output is the average of the predictions
computed by the model on several segments of the input. For the bimodal
models, subdivision of audio and lyrics requires synchronization of the
modalities.

###  3.1 Audio only

We use a mel-spectrogram as input, which are 2-dimensional. We choose a
convolutional neural network (ConvNet) [17], the architecture is shown in Fig.
1 (a). It is composed of two consecutive $1$-dimensional convolution layers
(convolutions along the temporal dimension) with $32$ and $16$ feature maps of
size $8$, stride $1$, and max pooling of size $4$ and stride $4$. We resort to
batch normalization [13] after each convolutional layer. We use two fully
connected layers as output to the network, the intermediate layer being of
size $64$.

(a) Audio

(b) Lyrics

(c) Bimodal

Figure 1: Architecture of unimodal and bimodal models

###  3.2 Lyrics only

We use a word embedding as input to the network, i.e. each word is embedded in
a continuous space and the vectors corresponding to each word are stacked, the
input being consequently 2-dimensional. We choose to resort to a word2vec [21]
embedding trained on 1.6 million lyrics, as first results seemed to indicate
that this specialized embedding performs better than embedding pretrained on
an unspecialized, albeit bigger, dataset. We compare several architectures,
with recurrent and convolutional layers. One of them is shown in Fig. 1 (b).
We also compare this approach with a simple continuous bag-of-words method
that acts as a feature-free baseline. The models that were tested are
described in Table 1.

Model name | Description
---|---
CBOW |  Continuous bag-of-words: random forest on top of means of input words embedding
GRU |  Single Gated Recurrent Unit (GRU) [4], size $40$, dense layers of size $64$ and $2$, preceded by dropout layers of parameter $0.5$
LSTM |  Single Long Short-Term Memory (LSTM) [6], size $80$, dense layers of size $64$ and $2$, preceded by dropout layers of parameter $0.5$
biLSTM |  Single LSTM, size $40$, dense layers of size $64$ and $2$, preceded by dropout layers of parameter $0.5$
2LSTMs |  Two LSTM layers, of size $40$, dense layers of size $64$ and $2$, preceded by dropout layers of parameter $0.5$
ConvNet+LSTM |  Convolutional layer with $16$ features maps of size ($2$,$2$), stride $1$, max-pooling of size $2$, stride $2$, an LSTM layer of size $40$ and dense layers of size $32$ and $2$, preceded by dropout layers of parameter $0.5$
2ConvNets+2LSTMs |  Two convolutional layers with $16$ features maps of size ($2$,$2$), stride $1$, max-pooling of size $2$, stride $2$, two LSTM layers of size $40$ and dense layers of size $32$ and $2$, preceded by dropout layers of parameter $0.5$

Table 1: Description of lyrics-based models.

###  3.3 Fusion

For the fusion model, we reuse the unimodal architecture from which we remove
the fully connected layers and concatenate the outputs of each network. On top
of this concatenation, we use two fully connected layers with an intermediate
vector length of size $100$. This architecture is presented in Fig. 1(c). This
allows for detection of more complex correlations between modalities. We
choose to compare this with a simple late fusion, which is a weighted average
of the outputs of the unimodal models, the weight being grid-searched. The
mid-level fusion model is referred to as middleDL and the late fusion model as
lateDL.

##  4 Experiment

###  4.1 Dataset

The MSD [2] is a large dataset commonly used for MIR tasks. The tracks are
associated with tags from LastFM222http://www.last.fm/, some of which are
related to mood. We apply the procedure described by Hu and Downie in [11] to
select the tags that are akin to a mood description. We then make use of the
dataset published by Warriner et al. [30] which associates 14,000 English
words with their embedding in Russell’s valence/arousal space. We use it for
embedding previously selected tags into the valence/arousal space. When
several tags are associated with the same track, we retain the mean of the
embedding values. Finally, we normalize the database by centering and reducing
valence and arousal. It would undoubtedly be more accurate to have tracks
directly labeled with valence/arousal values by humans, but no database with
sufficient volume exists. An advantage of this procedure is its applicability
to different mood representations, and thus to different existing databases.

The raw audio signal and lyrics are not provided in the MSD. Only features are
available, namely MFCCs for audio, word-counts for lyrics. For this reason, we
use a mapping between the MSD and the Deezer catalog using the song metadata
(song title, artist name, album title) and have then access to raw audio
signals and original lyrics for a part of the songs. As a result, we collected
a dataset of 18,644 annotated tracks. We note that lyrics and audio are not
synchronized. Automatic synchronization being outside of the scope of this
work, we resort to a simple heuristic for audio-lyrics alignment. It consists
of aligning both modalities proportionally based on their respective length,
i.e. for a certain audio segment, we extract words from the lyrics that are at
the corresponding location relatively to the length of the lyrics. We release
the labels, along with Deezer song identifiers, MSD identifiers, artist and
track name333https://github.com/deezer/deezer_mood_detection_dataset. More
data can be retrieved using the Deezer
API444https://developers.deezer.com/api. Unfortunately, we cannot release the
lyrics and music, due to rights restrictions.

We train the models on approximately $60\%$ of the dataset, and validate their
parameters with another $20\%$. Each model is then tested on the remaining
$20\%$. We refer to these three sets as training, validation and test set,
respectively. We split the dataset randomly, with the constraint that songs by
the same artist must not appear in two different sets (since artist and moods
may be correlated).

###  4.2 Implementation details

For audio, we use a mel-spectrogram as input to the network, with $40$ mel-
filters and $1024$ sample-long Hann window with no overlapping, with a
sampling frequency of $44.1$kHz, computed with YAAFE [20]. We use data
augmentation, that was investigated for audio and proven useful in [25], in
order to grow our dataset. First, we decide to extract $30$ second long
segments from the original track. The input of the network is consequently of
size $40$*$1292$. We choose to sample seven extracts per track: we draw them
uniformly from the song. We also use pitch shifting and lossy encoding, which
are transformations with which emotion is invariant, and get three extra
segments per original sample. In the end, we get a $28$-fold increase in the
size of the training set.

For lyrics, the input word embedding was computed with gensim’s implementation
of word2vec [23] and we used $100$-dimensional vectors. We use data
augmentation for lyrics as well by extracting seven $50$-word segments from
each track. Consequently, the input of each neural network is of size
$100$*$50$.

###  4.3 Results

mode | model | valence | arousal
---|---|---|---
audio | CA | 0.118 | 0.197
ConvNet | 0.179 | 0.235
lyrics | CA | 0.140 | 0.032
CBOW | 0.080 | 0.031
LSTM | 0.117 | 0.027
GRU | 0.106 | 0.017
biLSTM | 0.076 | 0.017
2LSTMs | 0.128 | 0.024
| ConvNet+LSTM | 0.134 | 0.026
| 2ConvNets+2LSTMs | 0.127 | 0.022
bimodal | CA | 0.219 | 0.216
LateDL | 0.194 | 0.235
middleDL | 0.219 | 0.232
Table 2: $R^{2}$ scores of the different tested approaches.

| coefficient* | 0.0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0
---|---|---|---|---|---|---|---|---|---|---|---|---
Feature engineering approaches | valence | 0.133 | 0.163 | 0.186 | 0.201 | 0.211 | 0.211 | 0.207 | 0.192 | 0.174 | 0.147 | 0.112
arousal | 0.034 | 0.081 | 0.121 | 0.152 | 0.178 | 0.199 | 0.211 | 0.217 | 0.218 | 0.212 | 0.201
Deep learning approaches | valence | 0.118 | 0.136 | 0.152 | 0.165 | 0.175 | 0.182 | 0.186 | 0.188 | 0.187 | 0.183 | 0.177
arousal | 0.025 | 0.065 | 0.102 | 0.135 | 0.164 | 0.19 | 0.212 | 0.231 | 0.246 | 0.257 | 0.265

Table 3: $R^{2}$ scores of the late fusion of unimodal models for classical
approaches and deep learning approaches, for different values of weighting.
*This coefficient is the weight of the audio prediction. The weight of the
lyrics prediction is its complementary to one.

We present the results and compare in particular deep learning approaches with
classical ones. The results are presented in Tab. 2 and 3. In the latter, CA
refers to classical models (described in Sect. 2).

modalities | BWC* | CA and DL | CA | DL
---|---|---|---|---
mean
audio | 0.7 | 0.193 | 0.118 | 0.179
lyrics | 0.5 | 0.177 | 0.140 | 0.134
fused | 0.5 | 0.243 | 0.219 | 0.219
Table 4: $R^{2}$ scores of the optimal weighted mean of classical and deep
learning approaches for valence prediction for different modalities. *BWC:
best weighting coefficient. This coefficient is the optimal weight of the deep
learning-based prediction. CA and DL respectively refers to classical
approaches and deep learning methods.

Unimodal approaches. The results of each unimodal model are given in Table 2.
For lyrics-based ones, we have tested several models without feature
engineering. The highest performing method, on both validation and test set,
is based on both recurrent and convolutional layers. In the following, we
choose this model as the one to be compared with classical models.

For both unimodal models, one can see a similar trend for classical and deep
learning approaches: lyrics and audio achieve relatively similar performance
on valence detection, whereas audio clearly outperforms lyrics when it comes
to arousal prediction. This is unsurprising, as arousal is closely related to
rhythm and energy, which are essentially induced by the audio signal. On the
contrary, valence is explained by both lyrics and audio, indicating that the
positivity of an emotion can be conveyed through the text as well as through
the melody, the harmony, the rhythm, etc. Similar observations were made by
Laurier et al. [16], where angry and calm songs were classified significantly
better by audio than by lyrics, and happy and sad songs were equally well-
classified by both modalities. This is consistent with our observations, as
happy and sad emotions can be characterized by high and low valence, and angry
and calm emotions by high and low arousal.

When looking more closely at the results, one can observe that deep learning
approaches are much higher performing than classical ones when it comes to
prediction based on audio. On the contrary, classical lyrics-based models are
higher performing than our deep learning model, in particular when it comes to
valence detection, which is the most informative task for the study on lyrics
only (as stated above). The reason can be that classical systems resort to
several emotion related lexicons designed by psychological studies. On the
contrary, classical audio feature engineering for mood detection does not make
use of such external resources curated by experts.

Late fusion analysis. As stated earlier, the late fusion consists of a simple
optimal weighted average between the prediction of both unimodal models. We
resort to a grid-search on the value of the weighting between 0 and 1. The
result for the reimplementation of traditional approaches and for our model is
presented in Table 3. One can observe a similar phenomenon for both classical
models and ours. In both cases, the fusion of the modalities does not
significantly improve arousal detection performance compared to audio-based
models. It is as predicted, as we saw that audio-based models perform
significantly better than lyrics-based ones. For deep learning models, using
lyrics in addition to audio in a late fusion scheme leads to no improvement,
so there is no gain added by using lyrics. When it comes to valence detection,
both modalities are valuable: in both approaches, the top performing model is
a relatively balanced average of unimodal predictions. Here also, these
observations generalize to valence/arousal what was observed on the emotions
happy, sad, angry and calm in [16]. Indeed, based on this study, not only are
lyrics and audio equally performant for predicting happy and sad songs, but
they are also complementary, so that fused models can achieve notably better
accuracies. However, predicting angry and calm songs is not improved when
using lyrics in addition to audio.

Bimodal approaches comparison. Bimodal method performances are reported in
Table 2. Several interesting remarks can be made based on these results. First
of all, one can notice that if one compares late fusion for both approaches,
arousal detection is outperformed by deep learning systems, as the
corresponding unimodal approach based on audio is more performant, and we have
seen that lyrics-based arousal detection is in both cases performing poorly.
On the contrary, late fusion for valence detection yields better results for
classical systems. In this case, the lack of performance of lyrics-based
methods relying on deep learning is not compensated for by a slightly improved
audio-based performance.

However, when it comes to mid-level fusion presented in paragraph 3.3, there
is a clear improvement for valence detection. It seems to indicate that there
might be earlier correlations between both modalities, that our model is able
to detect. Concerning arousal detection, the capacity of the network to unveil
such correlations seems useless: we have seen that our lyrics-based model is
not able to bring additional information to the audio-based model.

This performing fusion, along with more accurately predicted valence thanks to
audio, is sufficient for achieving similar performance to classical
approaches, without the use of any external data designed by experts.
Interestingly, both models remain useful, as long as they learn complementary
information. For valence detection, an optimized weighted average of the
predictions of both models yields the performance presented in Table 4. We can
see a significant gain obtained for a balanced average of both predictions,
indicating that both models have different applications, in particular when it
comes to lyrics-based valence detection.

##  5 Conclusion and future work

We have shown that multimodal mood prediction can go without feature
engineering, as deep learning-based models achieve better results than
classical approaches on arousal detection, and both methods perform equally on
valence detection. It seems that this gain of performance is the results of
the capacity of our model to unveil and use mid-level correlations between
audio and lyrics, particularly when it comes to predicting valence, as we have
seen that for this task, both modalities are equally important.

The gain of performance obtained when using this fusion instead of late fusion
indicates that further work can be done for understanding correlations between
both modalities, and there is no doubt that a database with synchronized
lyrics and audio would be of great help to go further. Future work could also
rely on a database with labels indicating the degree of ambiguity of the mood
of a track, as we know that in some cases, there can be significant
variability between listeners. Such databases would be particularly helpful to
go further in understanding musical emotion. Temporally localized label in
sufficient volume can also be of particular interest. Future work could also
leverage unsupervised pretraining to deep learning models, as unlabeled data
can be easier to find in high volume. We also leave it as a future work to
pursue improvements of lyrics-based models, with deeper architectures or by
optimizing word embeddings used as input. Studying and optimizing in detail
ConvNets for music mood detection offers the opportunity to temporally
localize zones responsible for the valence and arousal of a track, which could
be of paramount importance to understand how music, lyrics and mood are
correlated. Finally, by learning from feature engineering approaches, one
could use external resources designed by psychological studies to improve
significantly the prediction accuracy, as indicated by the complementarity of
both approaches.

##  6 Acknowledgments

The authors kindly thank Geoffroy Peeters and Gabriel Meseguer Brocal for
their insights as well as Matt Mould for his proof-reading. The research
leading to this work benefited from the WASABI project supported by the French
National Research Agency (contract ANR-16-CE23-0017-01).

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  *[MIR]: Music Information Retrieval
  *[MSD]: Million Song Dataset
  *[MFCCs]: Mel-Frequency Cepstral Coefficients
  *[SVM]: Support Vector Machine
  *[MIREX]: Music Information Retrieval Evaluation eXchange
  *[ConvNet]: convolutional neural network
  *[GRU]: Gated Recurrent Unit
  *[LSTM]: Long Short-Term Memory


# A Stackelberg Game Approach to Resource Allocation for IRS-aided
Communications

Yulan Gao1,2, Chao Yong1, Zehui Xiong2, Dusit Niyato2, Yue Xiao1, Jun Zhao2
1The National Key Laboratory of Science and Technology on Communications
University of Electronic Science and Technology of China, Chengdu 611731,
China
email<EMAIL_ADDRESS>2The School of Computer Science and Engineering,
Nanyang Technological University, Singapore 639798

###### Abstract

It is known that the capacity of the intelligent reflecting surface (IRS)
aided cellular network can be effectively improved by reflecting the incident
signals from the transmitter in a low-cost passive reflecting way.
Nevertheless, in the actual network operation, the base station (BS) and IRS
may belong to different operators, consequently, the IRS is reluctant to help
the BS without any payment. Therefore, this paper investigates price-based
reflection resource (elements) allocation strategies for an IRS-aided
multiuser multiple-input and single-output (MISO) downlink communication
systems, in which all transmissions over the same frequency band. Assuming
that the IRS is composed with multiple modules, each of which is attached with
a smart controller, thus, the states (active/idle) of module can be operated
by its controller, and all controllers can be communicated with each other via
fiber links. A Stackelberg game-based alternating direction method of
multipliers (ADMM) is proposed to jointly optimize the transmit beamforming at
the BS and the passive beamforming of the active modules. Numerical examples
are presented to verify the proposed algorithm. It is shown that the proposed
scheme is effective in the utilities of both the BS and IRS.

###### Index Terms:

Intelligent reflecting surface (IRS), transmit beamforming, passive
beamforming, Stackelberg game, alternating direction method of multipliers
(ADMM).

##  I Introduction

By enabling the intelligent reflecting surface (IRS) to the wireless systems,
the IRS-aided wireless system recently has attracted significant interest due
to its potential to further improve the system capacity and spectral
efficiency [1, 2]. Specifically, IRS exploits large reflecting elements to
proactively steer the incident radio-frequency wave towards destination
terminals [3], which is a promising solution to build a programmable wireless
environment for 6G systems [4]. Thereby, the fine-grained three-dimensional
reflecting beamforming can be achieved without the need of any transmit radio
frequency (RF) chain [5].

###  I-A Related Work

The IRS-aided wireless systems refer to the scenario that a large number of
software-controlled reflection elements with adjustable phase shifts for
reflecting the incident signal. As such, the phase shifts of all reflecting
elements can be tuned adaptively according to the state of networks, e.g., the
channel conditions and the incident angle of the signal by the base station
(BS). It is commonly believed that the propagation environment can be improved
without incurring additional noise at the reflector elements. Currently, major
communication field researchers are actively involved in the research of IRS-
aided communications [6, 7, 8]. For example, [6] summarized the main
communication applications and competitive advantages of the IRS technology.
In the spirit of these works, a vast corpus of literature focused on
optimizing active-passive beamforming for unilateral spectral efficiency
maximization subject to power constraint. For instance, [7] proposed a
fractional programming based alternating optimization approach to maximize the
weighted SE in IRS-aided multiple-input multiple-output (MISO) downlink
communication systems. In particular, three assumptions for the feasible set
of reflection coefficient were consider at IRS, including the ideal reflection
coefficient constrained by peak-power, continuous phase shifter, and discrete
phase shifter. Meantime, in MISO wireless systems, the problem of minimizing
the total transmit power at the access point was considered to energy-
efficient active-passive beamforming [8]. Notably, the aforementioned studies
for IRS-aided communications were based on the premise of ignoring the power
consumption at IRS. In contrast, in [9], an energy efficiency (EE)
maximization problem was investigated by developing a realistic IRS power
consumption model, where IRS power consumption relies on the type and the
resolution of meta-element.

###  I-B Motivation and Contributions

The above resource allocation works address the joint transmit beamforming and
phase shift optimization problem in IRS-aided communication systems. These
works assume that IRS operators are all selfless, and will always participate
in the cooperative transmission despite their own energy
consumption/maintanence cost [9] and profits. However, this assumption becomes
unrealistic in practice, due to the advances in intelligent communication and
the shrinking resources. In other words, if an IRS operator cannot benefit
from the participation, it will not join in the cooperative communication.
Moreover, the common assumption in the existing studies for IRS-aided
communications is that all the reflecting elements are used to reflect the
incident signal, i.e., adjusting reflecting coefficient of each meta-element
simultaneously each time. However, along with the use of a large number of
high-resolution reflecting elements, especially with continuous phase
shifters, triggering all the reflecting elements every time may result in
significant power consumption [9]. Moreover, the hardware support for the IRS
implementation is the use of a large number of tunable metasurfaces.
Specifically, the tunability feature can be realized by introducing mixed-
signal integrated circuits (ICs) or diodes/varactors, which can vary both the
resistance and reactance, offering complete local control over the complex
surface impedance [10]. According to the IRS power consumption model presented
in [9] and the hardware support, activating the entire IRS not only incurs
increased power consumption, but also entails the increased latency of
adjusting phase-shift and accelerates equipment depreciation. Therefore,
realizing reflection resource management is significantly important for IRS-
aided communications. In this paper, for IRS-aided multiuser MISO systems, we
consider the resource allocation problem in which an IRS operator serves the
BS and prices the active modules. The problem is formulated as a Stackelberg
game, in which the IRS operator decides the price for the active modules.

The contributions of this paper are summarized as follows:

  * •

For the first time, a modular architecture of IRS is proposed that divides all
the reflecting elements into multiple modules, whose states (active and idle)
are controlled by multiple parallel switches belonging to a common controller.
We assume that each module contains multiple reflecting elements, i.e., the
size of each module is larger than the incident signal wavelength, since the
unit meta-element size is subwavelength [10]. As mentioned in [2], the IRS is
programmatically controlled by the controller, and hence, from an operational
standpoint, independent module activating can be implemented easily.
Therefore, the proposed architecture of IRS allows the realization of the
reflection resource management, since each module is independently activated
by its switch.

  * •

Based on the proposed modular architecture of IRS, this paper proposes a new
price-based resource allocation scheme for both the BS and IRS. Furthermore,
the Stackelberg game is formulated to maximize the individual revenue of the
BS and IRS for the proposed price-based resource allocation. Since the entire
game is a non-convex mixed-integer problem, which is even hard to solve in a
centralized way, the problem is transformed into a convex problem by
introducing the mixed row block $\ell_{1,2}\text{-norm}$ [11], which yields a
suitable semidefinite relaxation. To solve this problem, we apply a
Stackelberg game-based alternating direction method of multipliers (ADMM) to
identify the price, active module subsets, and subsequently both the transmit
power allocation and the corresponding passive beamforming.

##  II System Model and Problem Formulation

###  II-A System Model

Consider the downlink communication between a BS equipped with $M$ antennas
and $K$ single-antenna mobile users. The communication takes place via an IRS
with $S$ modules, and each module consisting $N$ reflection elements, and
thus, the total reflecting elements of IRS is $SN.$ Define ${\cal
K}:=\\{1,2,\ldots,K\\},$ ${\cal S}:=\\{1,2,\ldots,S\\},$ and ${\cal
I}=\\{1,2,\ldots,(SN)\\}$ as the index sets of users, the reflection modules,
and the reflecting elements, respectively. Let
${\mathbf{H}}_{0,s}\in{\mathbb{C}}^{N\times M}$ be the channel matrix from the
BS to the $s\text{th}$ module of IRS,
${\mathbf{g}}_{s,k}\in{\mathbb{C}}^{N\times 1}$ be the channel vector from the
$s\text{th}$ module of the IRS to user $k.$ The direct channel for the BS to
user $k$ is denoted as $h_{d,k}\in{\mathbb{C}}^{M\times 1}.$ Denote by
$\phi_{i},\forall i\in{\cal I}$ the $i\text{th}$ reflecting element of the
IRS. Let
${\boldsymbol{\Phi}}=\text{diag}\\{{\boldsymbol{\Phi}}_{1},{\boldsymbol{\Phi}}_{2},\ldots,{\boldsymbol{\Phi}}_{S}\\}\in{\mathbb{C}}^{(SN)\times(SN)},$
where
${\boldsymbol{\Phi}}_{s}=\text{diag}[\phi_{(s-1)N+1},\phi_{(s-1)N+2},\ldots,\phi_{sN}]\in{\mathbb{C}}^{N\times
N}.$ Define
${\boldsymbol{\phi}}=[({\boldsymbol{\phi}}_{1})^{T},({\boldsymbol{\phi}}_{2})^{T},\ldots,({\boldsymbol{\phi}}_{S})^{T}]^{T}\in{\mathbb{C}}^{(SN)\times
1},$ where
${\boldsymbol{\phi}}_{s}=[({\phi}_{(s-1)N+1})^{{\dagger}},({\phi}_{(s-1)N+2})^{{\dagger}},\ldots,({\phi}_{sN})^{{\dagger}}]^{T}\in{\mathbb{C}}^{N\times
1}.$

We assume that all the modules of IRS can potentially join the cooperative
communication, then, the channel matrix from the BS to the IRS and the IRS to
user $k$ respectively are

$\displaystyle{\mathbf{H}}$
$\displaystyle=[({\mathbf{H}}_{0,1})^{T},({\mathbf{H}}_{0,2})^{T},\ldots,({\mathbf{H}}_{0,S})^{T}]^{T}\in{\mathbb{C}}^{(SN)\times
M}$ (1) $\displaystyle{\mathbf{g}}_{k}$
$\displaystyle=[({\mathbf{g}}_{1,k})^{T},({\mathbf{g}}_{2,k})^{T},\ldots,({\mathbf{g}}_{S,k})^{T}]^{T}\in{\mathbb{C}}^{(SN)\times
1},\forall k\in{\cal K}.$

The SINR for user $k$, which is denoted by $\gamma_{k}$ can be computed by

$\gamma_{k}=\frac{|({\mathbf{h}}_{d,k}^{{\dagger}}+{\mathbf{g}}_{k}^{{\dagger}}{\boldsymbol{\Phi}}(||\boldsymbol{\Phi}||_{0,F}){\mathbf{H}}){\mathbf{w}}_{k}|^{2}}{\sum_{j\neq
k}^{K}|({\mathbf{h}}_{d,k}^{{\dagger}}+{\mathbf{g}}_{k}^{{\dagger}}{\boldsymbol{\Phi}}(||{\boldsymbol{\Phi}}||_{0,F}){\mathbf{H}}){\mathbf{w}}_{j}|^{2}+\sigma^{2}},$
(2)

where ${\mathbf{w}}_{k}\in{\mathbb{C}}^{M\times 1}$ is the transmit
beamforming vector for user $k,$ $\sigma^{2}$ is the background noise at user
$k$, and the $\ell_{0,F}\text{-norm}$ is the number of nonzero blocks of
matrix $\boldsymbol{\Phi},$ i.e.,
$||{\boldsymbol{\Phi}}||_{0,F}\triangleq\left|\left\\{s:||{\boldsymbol{\Phi}}_{s}||_{F}\neq
0\right\\}\right|.$ Moreover, $\boldsymbol{\Phi}(||\boldsymbol{\Phi}||_{0,F})$
is the corresponding phase shift for the active modules.

###  II-B Stackelberg Game Formulation

In this paper, we assume that the BS and IRS belong to different operators,
and the IRS is selfish. In the case of unfavorable propagation condition on
the direct signal path, in order to improve the sum rate of system, the BS
needs to pay for the IRS’s forwarding service. Then, the IRS’s objective is to
maximize its utility, denoted by $V$, calculated as follows:

$V={r}||\boldsymbol{\Phi}||_{0,F},$ (3)

where $r$ is the price to IRS for providing $||\boldsymbol{\Phi}||_{0,F}$
reflection modules. Notably, in IRS-aided communication, the authority to
adjust the passive beamforming,
$\boldsymbol{\Phi}(||\boldsymbol{\Phi}||_{0,F})$ is controlled by the BS. The
objective of the IRS is to solve the following problem:

$\text{Leader-Problem:}~{}~{}\max_{r}~{}V,~{}~{}~{}\text{s.t.~{}~{}}r>0.$ (4)

In response to the action of the IRS, the BS chooses a best
$||\boldsymbol{\Phi}||_{0,F}$ active modules, decides the phase shift
$\boldsymbol{\Phi}(||\boldsymbol{\Phi}||_{0,F})$ of the activated modules and
its own transmit beamforming ${\mathbf{W}}.$ Then, the BS’s utility is
designed as the sum data rate of all users excluding its cost of forwarding
service, which is formulated as

$\displaystyle
U=\sum\nolimits_{k=1}^{K}\log_{2}\left(1+\gamma_{k}\right)-{r}||{\boldsymbol{\Phi}}||_{0,F}.$
(5)

The problem of obtaining the optimal strategy for the BS can be formulated as
follows:

Follower-Problem:
$\displaystyle\max_{{\mathbf{w}}_{k},{\boldsymbol{\Phi}}}~{}~{}~{}U$ (6a) s.t.
$\displaystyle\sum\nolimits_{k=1}^{K}||{\mathbf{w}}_{k}||_{2}^{2}\leq
p^{\max}$ (6b) $\displaystyle|\phi_{i}|\leq 1,\forall i=1,2,\ldots,(SN).$ (6c)

Since optimization problems (4) and (6) is nonconvex and generally impossible
to solve as their solutions usually requires an intractable combinational
search. A common alternative is to consider the mixed
$\ell_{1,2}\text{-norm}$, defined as
$||\boldsymbol{\Phi}||_{1,F}=\sum_{s=1}^{S}||{\boldsymbol{\Phi}}_{s}||_{F}.$
Note that the $\ell_{1,F}\text{-norm}$ behaves as the $\ell_{0,F}\text{-norm}$
on $\boldsymbol{\Phi}$, which implies that each
$||{\boldsymbol{\Phi}}_{s}||_{F}$ is encouraged to be zero, therefore inducing
group-sparsity [12]. For our purpose, we will use the convex
$\ell_{1,F}\text{-norm}$ as a group-sparsity to replace the nonconvex
$\ell_{0,F}\text{-norm}$ in (4) and (6). Through all the transformation, the
objectives of the leader and follower can thus be relaxed to

F-Problem:
$\displaystyle\max_{{\mathbf{w}}_{k},\boldsymbol{\Phi}}\sum\nolimits_{k=1}^{K}\text{log}_{2}(1+\gamma_{k})-r\delta||\boldsymbol{\Phi}||_{1,F}$
(7a) s.t.
$\displaystyle(\ref{s:2}\text{b})~{}\text{and}~{}(\ref{s:2}\text{c}),$ (7b)

where $\delta$ is the positive real tuning parameter that controls the
sparsity of the solution, and thus the number of activated modules.

$\text{L-Problem:}~{}\max_{r}~{}r\delta||\boldsymbol{\Phi}||_{1,2},~{}\text{s.t.~{}}r>0.$
(8)

For the proposed Stackelberg game, the Stackelberg game equilibrium (SE) is
defined as follows.

###### Definition 1

Define
${\mathbf{W}}=[{\mathbf{w}}_{1},{\mathbf{w}}_{2},\ldots,{\mathbf{w}}_{K}]\in{\mathbb{C}}^{M\times
K}.$ Let ${r}^{*}$ be a solution of problem (8) and
$({\mathbf{W}}^{*},{\boldsymbol{\Phi}}^{*})$ be a solution for problem (7).
Then, the point $({r}^{*},{\mathbf{W}}^{*},{\boldsymbol{\Phi}}^{*})$ is the
Stackelberg equilibrium for the proposed Stackelberg game if for any
$({r},{\mathbf{W}},{\boldsymbol{\Phi}})$, the following conditions are
satisfied:

$\displaystyle U({r}^{*},{\mathbf{W}}^{*},{\boldsymbol{\Phi}}^{*})$
$\displaystyle\geq U({r}^{*},{\mathbf{W}},{\boldsymbol{\Phi}})$ (9)
$\displaystyle V({r}^{*},{\mathbf{W}}^{*},{\boldsymbol{\Phi}}^{*})$
$\displaystyle\geq V({r},{\mathbf{W}}^{*},{\boldsymbol{\Phi}}^{*}).$

##  III Game Analysis

In the proposed game, both at the BS’s and the IRS’s side, since there is only
one player, the best response of the BS and IRS can be readily obtained by
solving F-Problem and L-Problem, respectively. For the proposed game, the SE
can be obtained as follows: For a given ${r},$ F-Problem (7) is solved first.
Then, with the obtained best response functions
$({W}^{*},{\boldsymbol{\Phi}}^{*})$ of the BS, we solve L-Problem (8) for the
optimal price ${r}^{*}.$

###  III-A Strategy Analysis for the BS

If we denote the price for serving the BS as ${r},$ the optimization problem
(7) can be solved by treating parameter $r$ as constant.

To tackle the logarithm in the objective function of (7), we apply the
Lagrangian dual transform. Then, the objective function of (7) can be
equivalently written as

$\displaystyle\max_{{\mathbf{W}},{\boldsymbol{\Phi}}}~{}\sum\nolimits_{k=1}^{K}\log_{2}\left(1+\alpha_{k}\right)$
$\displaystyle-\sum\nolimits_{k=1}^{K}\alpha_{k}+\sum\nolimits_{k=1}^{K}\frac{(1+\alpha_{k})\gamma_{k}}{1+\gamma_{k}}$
(10)
$\displaystyle-{r}\delta\sum\nolimits_{s=1}^{S}||{\boldsymbol{\Phi}}_{s}||_{F}.$

In (10), when ${\mathbf{W}}$ and ${\boldsymbol{\Phi}}$ hold fixed, the optimal
$\alpha_{k}$ is $\alpha_{k}^{*}=\gamma_{k},\forall k\in{\cal K}.$ Then, for a
given price ${r}$ and a fixed $\\{\alpha_{k}\\}_{k\in{\cal K}},$ optimizing
${\mathbf{W}}$ and ${\boldsymbol{\Phi}}$ is reduced to

$\displaystyle\max_{{\mathbf{W}},{\boldsymbol{\Phi}}}\sum\nolimits_{k=1}^{K}\frac{\tilde{\alpha}_{k}\gamma_{k}}{1+\gamma_{k}}-{r}\delta\sum\nolimits_{s=1}^{S}||{\boldsymbol{\Phi}}_{s}||_{F},$
(11)

where $\tilde{\alpha}_{k}=1+\alpha_{k}.$

####  III-A1 Transmit Beamforming

In the following, we investigate how to find a better beamforming matrix
${\mathbf{W}}$ given fixed ${\boldsymbol{\Phi}}$ for (11). Denote the combined
channel for user $k$ by

${\mathbf{h}}_{k}^{{\dagger}}={\mathbf{h}}_{d,k}^{{\dagger}}+{\mathbf{g}}_{k}^{{\dagger}}{\boldsymbol{\Phi}}{\mathbf{H}},\forall
k\in{\cal K.}$ (12)

Then, the SINR $\gamma_{k}$ in (2) is given by

$\gamma_{k}=\frac{|{\mathbf{h}}_{k}^{{\dagger}}{\mathbf{w}}_{k}|^{2}}{\sum\nolimits_{j\neq
k}^{K}|{\mathbf{h}}_{k}{\mathbf{w}}_{j}|^{2}+\sigma^{2}}.$ (13)

Using $\gamma_{k}$ in (13), the objective function of (11) is written as a
function of $\\{{\mathbf{w}}_{k}\\}_{k=1}^{K}:$

$\displaystyle\sum_{k=1}^{K}\frac{\tilde{\alpha}_{k}\gamma_{k}}{1+\gamma_{k}}-$
$\displaystyle{r}\delta\sum_{s=1}^{S}||{\boldsymbol{\Phi}}_{s}||_{F}=$ (14)
$\displaystyle\sum_{k=1}^{K}\frac{\tilde{\alpha}_{k}|{\mathbf{h}}_{k}^{{\dagger}}{\mathbf{w}}_{k}|^{2}}{\sum_{j=1}^{K}|{\mathbf{h}}_{k}^{{\dagger}}{\mathbf{w}}_{j}|^{2}+\sigma^{2}}-{r}\delta\sum_{s=1}^{S}||{\boldsymbol{\Phi}}_{s}||_{F}.$

Thus, for given ${r},$ $\\{{\alpha}_{k}\\}_{k\in{\cal K}},$ and
${\boldsymbol{\Phi}}$, optimizing $\\{{\mathbf{w}}_{k}\\}_{k=1}^{K}$ becomes

$\displaystyle\max_{\\{{\mathbf{w}}_{k}\\}_{k=1}^{K}}$
$\displaystyle\sum_{k=1}^{K}\frac{\tilde{\alpha}_{k}|{\mathbf{h}}_{k}^{{\dagger}}{\mathbf{w}}_{k}|^{2}}{\sum_{j=1}^{K}|{\mathbf{h}}_{k}^{{\dagger}}{\mathbf{w}}_{j}|^{2}+\sigma^{2}}$
(15) s.t. $\displaystyle\sum_{k=1}^{K}||{\mathbf{w}}_{k}||_{2}^{2}\leq
p^{\max}.$

Using quadratic transform, the objective function of (15) is reformulated as

$\displaystyle\sum_{k=1}^{K}$
$\displaystyle\frac{\tilde{\alpha}_{k}|{\mathbf{h}}_{k}^{{\dagger}}{\mathbf{w}}_{k}|^{2}}{\sum_{j=1}^{K}|{\mathbf{h}}_{k}^{{\dagger}}{\mathbf{w}}_{j}|^{2}+\sigma^{2}}=\sum\nolimits_{k=1}^{K}2\sqrt{\tilde{\alpha}_{k}}\text{Re}\left\\{\beta_{k}^{{\ddagger}}{\mathbf{h}}_{k}^{{\dagger}}{\mathbf{w}}_{k}\right\\}$
(16)
$\displaystyle-\sum\nolimits_{k=1}^{K}|\beta_{k}|^{2}\left(\sum\nolimits_{j=1}^{K}|{\mathbf{h}}_{k}^{{\dagger}}{\mathbf{w}}_{j}|^{2}+\sigma^{2}\right),$

where $(\cdot)^{{\ddagger}}$ denotes the conjugate. $\beta_{k}\in{\mathbb{C}}$
is the auxiliary variable. Then, solving problem (15) over
$\\{{\mathbf{w}}_{k}\\}_{k=1}^{K}$ is equivalent to solving the following
problem over $\\{{\mathbf{w}}_{k}\\}_{k=1}^{K}$ and
${\boldsymbol{\beta}}=[\beta_{1},\ldots,\beta_{K}]^{T}\in{\mathbb{C}}^{K\times
1}:$

$\displaystyle\max_{\\{{\mathbf{w}}_{k}\\}_{k=1}^{K},{\boldsymbol{\beta}}}{}$
$\displaystyle~{}\sum\nolimits_{k=1}^{K}2\sqrt{\tilde{\alpha}_{k}}\text{Re}\left\\{\beta_{k}^{{\ddagger}}{\mathbf{h}}_{k}^{{\dagger}}{\mathbf{w}}_{k}\right\\}$
(17)
$\displaystyle-\sum\nolimits_{k=1}^{K}|\beta_{k}|^{2}\left(\sum\nolimits_{j=1}^{K}|{\mathbf{h}}_{k}^{{\dagger}}{\mathbf{w}}_{j}|^{2}+\sigma^{2}\right)$
s.t. $\displaystyle\sum\nolimits_{k=1}^{K}||{\mathbf{w}}_{k}||_{2}^{2}\leq
p^{\max}.$

The optimal $\beta_{k}$ for a given $\\{{\mathbf{w}}_{k}\\}_{k=1}^{K}$ is

$\beta_{k}^{*}=\frac{\sqrt{\tilde{\alpha}_{k}}{\mathbf{h}}_{k}^{{\dagger}}{\mathbf{w}}_{k}}{\sum_{j=1}^{K}|{\mathbf{h}}_{k}^{{\dagger}}{\mathbf{w}}_{j}|^{2}+\sigma^{2}}.$
(18)

Then, fixing ${\boldsymbol{\beta}},$ the optimal ${\mathbf{w}}_{k}$ is

${\mathbf{w}}_{k}^{*}=\sqrt{\tilde{\alpha}_{k}}\beta_{k}(\lambda_{0}{\mathbf{I}}_{M}+\sum\nolimits_{j=1}^{K}|\beta_{j}|^{2}{\mathbf{h}}_{j}{\mathbf{h}}_{j}^{{\dagger}})^{-1}{\mathbf{h}}_{k},$
(19)

where $\lambda_{0}$ is the dual variable introduced for the power constraint,
which is optimally determined by

$\lambda_{0}^{*}=\max\\{0,p^{\max}-\sum\nolimits_{k=1}^{K}||{\mathbf{w}}_{k}||_{2}^{2}\\}.$
(20)

####  III-A2 Optimizing Reflection Response Matrix ${\boldsymbol{\Phi}}$

Optimize ${\boldsymbol{\Phi}}$ in (11) given fixed pricing ${\mathbf{r}},$
$\\{\alpha_{k}\\}_{k\in{\cal K}},$ and $\\{{\mathbf{w}}_{k}\\}_{k=1}^{K}$.
Using $\gamma_{k}$ defined in (2), the objective function of (11) is expressed
as a function of ${\boldsymbol{\Phi}}$:

$\sum_{k=1}^{K}\frac{\tilde{\alpha}_{k}|({\mathbf{h}}_{d,k}^{{\dagger}}+{\mathbf{g}}_{k}^{{\dagger}}{\boldsymbol{\Phi}}(\mathbf{H})){\mathbf{w}}_{k}|^{2}}{\sum_{j=1}^{K}|({\mathbf{h}}_{d,k}^{{\dagger}}+{\mathbf{g}}_{k}^{{\dagger}}{\boldsymbol{\Phi}}{\mathbf{H}}){\mathbf{w}}_{j}|^{2}+\sigma^{2}}-{r}\delta\sum_{s=1}^{S}||{\boldsymbol{\Phi}}_{s}||_{F}.$
(21)

Define
${\mathbf{a}}_{j,k}=\text{diag}\\{{\mathbf{g}}_{k}^{{\dagger}}\\}{\mathbf{H}}{\mathbf{w}}_{j},b_{j,k}={\mathbf{h}}_{d,k}^{{\dagger}}{\mathbf{w}}_{j},\forall
k,j=1,2,\ldots,K.$ Combining with the definition of ${\boldsymbol{\phi}}$,
(21) can be rewritten as

$\sum_{k=1}^{K}\frac{\tilde{\alpha}_{k}|b_{k,k}+{\boldsymbol{\phi}}^{{\dagger}}{\mathbf{a}}_{k,k}|^{2}}{\sum_{j=1}^{K}|b_{j,k}+{\boldsymbol{\phi}}^{{\dagger}}{\mathbf{a}}_{j,k}|^{2}+\sigma^{2}}-{r}\delta\sum_{s=1}^{S}||{\boldsymbol{\Phi}}_{s}||_{F}.$
(22)

Note that
$\sum_{s=1}^{S}||{\boldsymbol{\Phi}}_{s}||_{F}=\sum_{s=1}^{S}||{\boldsymbol{\phi}}_{s}||_{2},$
optimizing ${\boldsymbol{\phi}}$ can be represented as follows:

$\displaystyle\max_{{\boldsymbol{\phi}}}{}$
$\displaystyle\sum_{k=1}^{K}\frac{\tilde{\alpha}_{k}|b_{k,k}+{\boldsymbol{\phi}}^{{\dagger}}{\mathbf{a}}_{k,k}|^{2}}{\sum_{j=1}^{K}|b_{j,k}+{\boldsymbol{\phi}}^{{\dagger}}{\mathbf{a}}_{j,k}|^{2}+\sigma^{2}}-{r}\delta\sum_{s=1}^{S}||{\boldsymbol{\phi}}_{s}||_{2}$
(23) s.t.
$\displaystyle{\boldsymbol{\phi}}^{{\dagger}}{\mathbf{e}}_{i}{\mathbf{e}}_{i}^{{\dagger}}{\boldsymbol{\phi}}\leq
1,\forall i=1,2,\ldots,(SN).$

Based on the quadratic transform, the new objective function of (23) is

$\displaystyle\sum\nolimits_{k=1}^{K}2\sqrt{\tilde{\alpha}_{k}}\text{Re}\left\\{\epsilon_{k}^{{\ddagger}}{\boldsymbol{\phi}}^{{\dagger}}{\mathbf{a}}_{k,k}+\epsilon_{k}^{{\ddagger}}b_{k,k}\right\\}-\sum\nolimits_{k=1}^{K}|\epsilon_{k}|^{2}$
(24)
$\displaystyle\times\left(\sum\nolimits_{j=1}^{K}|b_{j,k}+{\boldsymbol{\phi}}^{{\dagger}}{\mathbf{a}}_{j,k}|^{2}+\sigma^{2}\right)-{r}\delta\sum\nolimits_{s=1}^{S}||{\boldsymbol{\phi}}_{s}||_{2}^{2},$

and
${\boldsymbol{\epsilon}}=[\epsilon_{1},\ldots,\epsilon_{K}]^{T}\in{\mathbb{C}}^{K\times
1}$ refers to the auxiliary variable vector. Similarly, we optimize
${\boldsymbol{\phi}}$ and $\boldsymbol{\epsilon}$ alternatively [7]. The
optimal $\epsilon_{k}$ for given $\boldsymbol{\phi}$ can be obtained easily,
shown as follows:

$\epsilon_{k}^{*}=\frac{\sqrt{\tilde{\alpha}_{k}}(b_{k,k}+{\boldsymbol{\phi}}^{{\dagger}}{\mathbf{a}}_{k,k})}{\sum_{j=1}^{K}|b_{j,k}+{\boldsymbol{\phi}}^{{\dagger}}{\mathbf{a}}_{j,k}|^{2}+\sigma^{2}}.$
(25)

Then, the remaining problem is optimizing $\boldsymbol{\phi}$ for given
$\boldsymbol{\epsilon}.$ By introducing new variable
${\boldsymbol{\theta}}={\boldsymbol{\phi}}\in{\mathbb{C}}^{(SN)\times 1}.$
Likewise, ${\boldsymbol{\theta}}_{s}\in{\mathbb{C}}^{N\times 1}$ represents
the $s$th block of vector ${\boldsymbol{\theta}}.$ Thus, for the fixed
${\boldsymbol{\epsilon}},$ the optimization problem of ${\boldsymbol{\phi}}$
is given as follows:

$\displaystyle\max_{{\boldsymbol{\phi}},{\boldsymbol{\theta}}}{}$
$\displaystyle\sum_{k=1}^{K}2\sqrt{\tilde{\alpha}_{k}}\text{Re}\left\\{\epsilon_{k}^{{\ddagger}}{\boldsymbol{\phi}}^{{\dagger}}{\mathbf{a}}_{k,k}+\epsilon_{k}^{{\ddagger}}b_{k,k}\right\\}-\sum_{k=1}^{K}|\epsilon_{k}|^{2}$
(26)
$\displaystyle\left(\sum_{j=1}^{K}|b_{j,k}+{\boldsymbol{\phi}}^{{\dagger}}{\mathbf{a}}_{j,k}|^{2}+\sigma^{2}\right)-{r}\delta\sum_{s=1}^{S}||{\boldsymbol{\theta}}_{s}||_{2}^{2}$
s.t.
$\displaystyle{\boldsymbol{\phi}}^{{\dagger}}{\mathbf{e}}_{i}{\mathbf{e}}_{i}^{{\dagger}}{\boldsymbol{\phi}}\leq
1,\forall i=1,2,\ldots,(SN)$
$\displaystyle{\boldsymbol{\theta}}={\boldsymbol{\phi}}.$

Utilizing the method of augmented Lagrangian minimization, (26) can be handled
by solving

$\displaystyle\min_{{\boldsymbol{\Lambda}}}\max_{{\boldsymbol{\phi}},{\boldsymbol{\theta}}}{}$
$\displaystyle~{}L_{c}({\boldsymbol{\phi}},{\boldsymbol{\theta}},{\boldsymbol{\Lambda}})$
(27) s.t.
$\displaystyle~{}{\boldsymbol{\phi}}^{{\dagger}}{\mathbf{e}}_{i}{\mathbf{e}}_{i}^{{\dagger}}{\boldsymbol{\phi}}\leq
1,\forall i=1,2,\ldots,(SN),$

where $c>0$ is the penalty factor;
${\boldsymbol{\Lambda}}\in{\mathbb{C}}^{(SN)\times 1}$ is the Lagrangian
vector multiplier for ${\boldsymbol{\theta}}={\boldsymbol{\phi}}.$ The partial
augmented Lagrangian function is defined as

$\displaystyle L_{c}$
$\displaystyle({\boldsymbol{\phi}},{\boldsymbol{\theta}},{\boldsymbol{\Lambda}})=\sum_{k=1}^{K}2\sqrt{\tilde{\alpha}_{k}}\text{Re}\left\\{\epsilon_{k}^{{\ddagger}}{\boldsymbol{\phi}}^{{\dagger}}{\mathbf{a}}_{k,k}+\epsilon_{k}^{{\ddagger}}b_{k,k}\right\\}$
(28)
$\displaystyle-\sum_{k=1}^{K}|\epsilon_{k}|^{2}\left(\sum_{j=1}^{K}|b_{j,k}+{\boldsymbol{\phi}}^{{\dagger}}{\mathbf{a}}_{j,k}|^{2}+\sigma^{2}\right)$
$\displaystyle-{r}\delta\sum_{s=1}^{S}||{\boldsymbol{\theta}}_{s}||_{2}^{2}-\text{Re}\left\\{\text{Tr}\left[{\boldsymbol{\Lambda}}^{{\dagger}}({\boldsymbol{\theta}}-{\boldsymbol{\phi}})\right]\right\\}-\frac{c}{2}||{\boldsymbol{\theta}}-{\boldsymbol{\phi}}||_{2}^{2}.$

  * •

Updating ${\boldsymbol{\phi}}$:

By dual theory and KKT conditions, the optimal solution is given by (29).

$\displaystyle{\boldsymbol{\phi}}^{*}=\left(2\sum_{k=1}^{K}|\epsilon_{k}|^{2}\sum_{j=1}^{K}{\mathbf{a}}_{j,k}{\mathbf{a}}_{j,k}^{{\dagger}}+2\sum_{i=1}^{SN}\mu_{i}{\mathbf{e}}_{i}{\mathbf{e}}_{i}^{{\dagger}}+c{\mathbf{I}}_{SN}\right)^{-1}\left(2\sum_{k=1}^{K}\sqrt{\tilde{\alpha}_{k}}\epsilon_{k}^{{\ddagger}}{\mathbf{a}}_{k,k}+{\boldsymbol{\Lambda}}+c{\boldsymbol{\theta}}-2\sum_{k=1}^{K}|\epsilon_{k}|^{2}\sum_{j=1}^{K}b_{j,k}{\mathbf{a}}_{j,k}\right),$
(29)

The Lagrangian multiplier $\mu_{i}$ updated by

$\mu_{i}^{*}=\max\left\\{0,1-{\boldsymbol{\phi}}^{{\dagger}}{\mathbf{e}}_{i}{\mathbf{e}}_{i}^{{\dagger}}{\boldsymbol{\phi}}\right\\}.$
(30)

  * •

Updating  ${\boldsymbol{\theta}}$:

The problem of ${\boldsymbol{\theta}}$ is an unconstrained group leastabsolute
selection and shrinkage operator (group Lasso) problem [13], i.e.,

$\max_{\boldsymbol{\theta}}~{}-{r}\delta\sum_{s=1}^{S}||{\boldsymbol{\theta}}_{s}||_{2}-\text{Re}\left\\{\text{Tr}\left[{\boldsymbol{\Lambda}}^{{\dagger}}({\boldsymbol{\theta}}-{\boldsymbol{\phi}})\right]\right\\}-\frac{c}{2}||{\boldsymbol{\theta}}-{\boldsymbol{\phi}}||_{2}^{2}.$
(32)

Let ${\boldsymbol{\Lambda}}_{s}\in{\mathbb{C}}^{N\times 1}$ denote the
$s\text{th}$ row block of vector ${\boldsymbol{\theta}},s=1,2,\ldots,S.$ Then,
(32) can be divided into $S$ independent problems of
${\boldsymbol{\theta}}_{s}$ for $s=1,2,\ldots,S$

$\max_{{\boldsymbol{\theta}}_{s}}~{}-{r}\delta||{\boldsymbol{\theta}}_{s}||_{2}-\text{Re}\left\\{\text{Tr}\left[{\boldsymbol{\Lambda}}_{s}^{{\dagger}}({\boldsymbol{\theta}}_{s}-{\boldsymbol{\phi}}_{s})\right]\right\\}-\frac{c}{2}||{\boldsymbol{\theta}}_{s}-{\boldsymbol{\phi}}_{s}||_{2}^{2}$
(33)

Defining
${\mathbf{x}}_{s}=c{\boldsymbol{\phi}}_{s}-{\boldsymbol{\Lambda}}_{s},$ and
${\mathbf{x}}_{s}-c{\boldsymbol{\theta}}_{s}\in{r}\partial||{\boldsymbol{\theta}}_{s}||_{2}$,
and thus, we can easily obtain

${\boldsymbol{\theta}}_{s}=\left\\{\begin{array}[]{ccc}&{\mathbf{0}},&\text{~{}if~{}}||{\mathbf{x}}_{s}||_{2}\leq{r\delta}\\\
&\frac{(||{\mathbf{x}}_{s}||_{2}-{r}\delta){\mathbf{x}}_{s}}{c||{\mathbf{x}}_{s}||_{2}},&\text{otherwise}.\end{array}\right.$
(34)

The update of Lagrangian vector $\boldsymbol{\Lambda}_{s}$ is given by

${\boldsymbol{\Lambda}}_{s}={\boldsymbol{\Lambda}}_{s}+c({\boldsymbol{\theta}}_{s}-{\boldsymbol{\phi}}_{s}),\forall
s=1,2,\ldots,S.$ (35)

###  III-B Game Analysis for the IRS Pricing

Substituting (34) into ( L-Problem) in (8), the optimization problem at the
IRS side can be formulated as

$\max_{{r}>0}~{}\sum_{s=1}^{S}\kappa_{s}\frac{-\delta^{2}{r}^{2}+\delta||{\mathbf{x}}_{s}||_{2}{r}}{c},$
(36)

where $\kappa_{s}$ is indicate function, i.e.,

$\kappa_{s}=\left\\{\begin{array}[]{ccc}&0,&\text{~{}if~{}}||{\mathbf{x}}_{s}||_{2}\leq{r\delta}\\\
&1,&\text{~{}otherwise}.\end{array}\right.$ (37)

The optimal solution of (36) is

${r}^{*}=\frac{\sum_{s=1}^{S}\kappa_{s}||{\mathbf{x}}_{s}||_{2}}{2\delta\sum_{s=1}^{S}\kappa_{s}}.$
(38)

The entire framework including the identifying the price and the trigger
module subsets as well as the transmit beamforming and the phase shift is
summarized in Algorithm 1.

Algorithm 1 Algorithm Summary

Initial: The IRS initialize the price ${r}(1),$ and set the outer iteration
number $\tau=1$
Part I: the alternating optimization for solving (11)
(1.1) Initialize ${\mathbf{W}}(1)$ and ${\boldsymbol{\Phi}}(1)$ to feasible
values, and set the iteration number $t=1.$
Repeat
(1.2) Update the nominal SINR $\alpha_{k}(t),\forall k\in{\cal K}$;
(1.3) Update $\beta_{k}(t),\forall k\in{\cal K}$ by (18);
(1.4) Update transmit beamforming ${\mathbf{W}}(t)$ by (19); update
$\lambda_{0}(t)$ by (20)
(1.5) Update $\epsilon_{k}(t),\forall k\in{\cal K}$ by (25);
(1.6) Update ${\boldsymbol{\phi}}(t)$ by (29); update $\mu_{i}(t),\forall
i=1,2,\ldots,(SN),$ by (30);
(1.7) Update ${\boldsymbol{\theta}}_{s}(t)$ by (34) in parallel for
$s=1,2,\ldots S;$
(1.8) Update $\boldsymbol{\Lambda}_{s}(t)$ by (35) in parallel for
$s=1,2,\ldots,S;$
(1.9) Update $t=t+1;$
(1.10) Until  The value of function (10) converges.
Part II: Update price ${r}$ by solving problem (36) in the outer loop
(2.1) Solve problem (36) for given
$\\{{\boldsymbol{\theta}}_{s}(t)\\}_{s=1}^{S},\\{{\boldsymbol{\Lambda}}_{s}(t)\\}_{s=1}^{S},$
update ${r}(\tau)$ by (38)
(2.2) Until  the utility of the IRS is convergence.

##  IV Simulation Results

In this section, extensive numerical results are presented to evaluate the
performances of the proposed resource allocation strategies based on the
approach of active module pricing. For simplicity, we set the balance
parameter $\delta$ to $0.1.$ To keep the complexity of simulations tractable,
we focus on the scenario, where the $K\in\\{4,6\\}$ users are randomly
deployed within a circle cell centered at $(200,0)\text{~{}m}$, and the cell
radius is $10\text{~{}m}$, the BS and IRS are employed at $(0,0)\text{~{}m}$
and $(50,50)\text{~{}m},$ respectively, where the number of reflecting
elements of each module is set as $N=8.$ We assume that the BS is equipped
with $4$ ($6$) antennas for $K=4$ ($K=6$). From [14], we set the path loss
exponent of the direct link as $3.5$, and the path loss at the reference
distance $1\text{~{}m}$ is set as $30\text{~{}dBm}$ for each individual link.
For the IRS-aided link, $2$ is the value of the path loss exponent from the BS
to the IRS and that from the IRS to users. For simplicity, we assume the
Rayleigh fading model to account for small-scale fading.

Figure 1: Impact of $p^{\max}$ on $U$.

Figure 2: Impact of $p^{\max}$ on $V$.

The performance of the Stackelberg game-based ADMM scheme is evaluated against
two existing benchmark schemes, i.e., random pricing scheme and direct link
only scheme. In the random pricing scheme, the IRS randomly determines its
strategies, without considering the existence of the BS. The direct link only
scheme means no IRS to aid, i.e., no module is activated at the IRS. Figures 2
and 2 show the effect of the maximum transmit power $p^{\max}$ on the utility
of the BS and the IRS, respectively, when the number of modules is $8.$
Correspondingly, Figs. 4 and 4 depict the sum rate of all users and the
service prices versus the maximum transmit power at BS, respectively. For the
BS and IRS, the Stackelberg game-based ADMM scheme achieves the highest
utility value compared with random pricing and direct link schemes, which
indicates that the proposed pricing-based Stackelberg game scheme performs
best in resource allocation for IRS-aided communications. From the results, we
observe that the utility values of the BS increases as $p^{\max}$ grows from
$-5\text{~{}dBm}$ to $5\text{~{}dBm}.$ Meanwhile, the utility value of the IRS
achieved by the Stackelberg game-based ADMM scheme first decreases slowly
until the maximum transmit power increases to $0\text{~{}dBm}$ and then
decreases rapidly by increasing the value of $p^{\max}$. This is because that
the cost of power consumption is not considered in the utility of the BS, and
thereby, the BS will tend to select a small number of active modules when the
transmit power is sufficient. Meanwhile, for $p^{\max}>0\text{~{}dBm}$, the
IRS needs to incentive the BS to select reflection resource through lower
price, which can be observed from Fig. 4.

Figure 3: Sum rate vs. $P^{\max}.$

Figure 4:  Prices vs. $p^{\max}.$

##  V Conclusion

The adoption of an IRS for downlink multi-user communication from a multi-
antenna BS was investigated in this paper. Specifically, we developed a
Stackelbeg game approach to analyze the interaction between the BS and the IRS
operator considering that the IRS operator may be selfish or has its own
objective. Different from the existing studies on IRS that merely focused on
tuning the reflection coefficient of all the reflecting elements, we
considered the reflection resource allocation, which can be realized via
active module selection under the proposed modular IRS architecture that all
the modules are controlled by independent controllers. The Stackelberg game-
based ADMM was proposed to solve either the transmit beamforming at the BS or
the passive beamorming of the activated modules. Numerical examples were
presented to verify the proposed studies. It was shown that the proposed
scheme is effective in the utilities of both the BS and IRS.

## Acknowledgement

This work was supported in part by the National Science Foundation of China
under Grant number 61671131, also supported by the National Research
Foundation (NRF), Singapore, under Singapore Energy Market Authority (EMA),
Energy Resilience, NRF2017EWT-EP003-041, Singapore NRF2015-NRF-ISF001-2277,
Singapore NRF National Satellite of Excellence, Design Science and Technology
for Secure Critical Infrastructure NSoE DeST-SCI2019-0007, A*STAR-NTU-SUTD
Joint Research Grant on Artificial Intelligence for the Future of
Manufacturing RGANS1906, Wallenberg AI, Autonomous Systems and Software
Program and Nanyang Technological University (WASP/NTU) under grant M4082187
(4080), Singapore Ministry of Education (MOE) Tier 1 (RG16/20), and NTU-WeBank
JRI (NWJ-2020-004), Alibaba Group through Alibaba Innovative Research (AIR)
Program, Alibaba-NTU Singapore Joint Research Institute (JRI), Nanyang
Technological University (NTU) Startup Grant, Alibaba-NTU Singapore Joint
Research Institute (JRI), Singapore Ministry of Education Academic Research
Fund Tier 1 RG128/18, Tier 1 RG115/19, Tier 1 RT07/19, Tier 1 RT01/19, and
Tier 2 MOE2019-T2-1-176, NTU-WASP Joint Project, Singapore National Research
Foundation (NRF) under its Strategic Capability Research Centres Funding
Initiative: Strategic Centre for Research in Privacy-Preserving Technologies &
Systems (SCRIPTS), Energy Research Institute @NTU (ERIAN), Singapore NRF
National Satellite of Excellence, Design Science and Technology for Secure
Critical Infrastructure NSoE DeST-SCI2019-0012, AI Singapore (AISG) 100
Experiments (100E) programme, NTU Project for Large Vertical Take-Off &
Landing (VTOL) Research Platform.

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