Energy-Efficient Multi-RIS-Aided Rate-Splitting Multiple Access: A Graph Neural Network Approach

This letter explores energy efficiency (EE) maximization in a downlink multiple-input single-output (MISO) reconfigurable intelligent surface (RIS)-aided multiuser system employing rate-splitting multiple access (RSMA). The optimization task entails base station (BS) and RIS beamforming and RSMA common rate allocation with constraints. We propose a graph neural network (GNN) model that learns beamforming and rate allocation directly from the channel information using a unique graph representation derived from the communication system. The GNN model outperforms existing deep neural network (DNN) and model-based methods in terms of EE, demonstrating low complexity, resilience to imperfect channel information, and effective generalization across varying user numbers.


I. INTRODUCTION
E NERGY efficiency (EE) is an important system design metric as we evolve towards sustainable sixth-generation (6G) communications.To enhance EE, efficient resource allocation and interference management are crucial.Two advanced techniques, rate-splitting multiple access (RSMA) [1] and reconfigurable intelligent surfaces (RISs) [2], are pivotal in addressing interference challenges at both transmission protocol and system levels.RSMA is an advanced multiple access technique that encompasses and can outperform space division multiple access (SDMA), non-orthogonal multiple access (NOMA), and orthogonal multiple access (OMA) in diverse radio environments.It divides messages into common and private components, allowing users to initially decode the common part and subsequently decode the private part after the removal of the common part.This flexibility empowers RSMA to efficiently manage interference by allocating resources to both common and private parts.An RIS consists of numerous passive reflecting elements that configure the radio environment and serve relay-like functions.RIS technology holds promise for enhanced spectral and energy efficiency for future 6G communication [2].
The combination of RSMA and RISs for improving spectral and energy efficiency is therefore of significant interest.Previous studies [3], [4], [5], [6], [7] have explored their synergy.In [3], the combination of various RSMA and RIS architectures was outlined and the rate region of RIS-aided RSMA was illustrated.In [4], the sum-rate maximization problem in RIS-aided uplink RSMA systems was addressed by utilizing optimization techniques.In [5], a downlink multiuser RSMA system with a more generalized RIS architecture, called the fully-connected RIS, was investigated for sum-rate maximization, employing an alternating optimization-based algorithm.Studies such as [6] and [7] investigated the EE performance of RIS-assisted multiuser RSMA systems.These works applied optimization techniques to convert non-convex problems into tractable convex ones for optimizing base station (BS) beamformers, RIS phase shifts, and RSMA rate allocations.However, the approximation and convexification methods used in these approaches often yielded suboptimal solutions and required time-consuming computations.
To address these challenges, deep learning-based methods have been proposed [8], [9], [10], [11].In [8], a beamforming scheme based on a deep-unfolded iterative algorithm was introduced to optimize max-min fairness in RIS-assisted massive multiple-input multiple-output (MIMO) RSMA systems.Deep reinforcement learning (DRL) methods, including proximal policy optimization (PPO) in [9] and [10], and deep deterministic policy gradient (DDPG) in [11], were used to tackle sum-rate and EE maximization problems in various RSMA scenarios.While DRL offers superior performance compared to model-based algorithms, it encounters challenges related to convergence and generalization in dynamic environments.
In this letter, we introduce a novel approach that leverages graph neural networks (GNNs) to jointly optimize BS beamforming, common rate allocation, and RIS phase shifts, with the goal of maximizing EE in multi-RIS-aided multiuser systems employing RSMA.This letter presents the first application of GNN to RIS-aided RSMA systems.Our GNN model incorporates a unique graph representation of the transmission system and learns joint beamforming directly from channel state information (CSI) through an information exchange and update mechanism among nodes and edges.Simulation results confirm the effectiveness of our approach, the advantages of combining RSMA and RISs, and the GNN model's ability to generalize to varying user numbers.
Notation: (•) its main diagonal.vec(A) is the vectorization of matrix A.
[A] i,j represents the (i, j)-entry of matrix A.

II. SYSTEM MODEL AND PROBLEM DESCRIPTION
We consider a multi-RIS-aided downlink multiuser MISO system employing RSMA.A single BS equipped with M antennas communicates with K users each equipped with a single antenna, with the aid of L RISs, each containing N passive elements.The channels between the BS, the th RIS, and the kth user are denoted as H BS,RIS ∈ C N ×M , h RIS ,k ∈ C 1×N , and h BS,k ∈ C 1×M .The inter-RIS links are assumed to be negligible due to weak double reflections, similar to [12].In accordance with the RSMA scheme, the signal from each user k (k = 1, . . ., K ) is divided into two parts [1]: a common part, collectively encoded among all K users into a common message s 0 , and a private part encoded into a private message s k .The BS employs beamforming vectors g 0 ∈ C M ×1 and g k ∈ C M ×1 (for each user k, k = 1, . . ., K ) for the common and private messages, respectively.Thus, the transmitted signal from the BS can be represented as The received signal at user k can be expressed as where T represents the RIS phase shifts for the th RIS, and w k ∼ CN(0, σ 2 k ) is the complex additive Gaussian noise at user k.The achievable rate for decoding the common message at user k is given by To ensure the common message s 0 is successfully decoded by all users, its rate must not exceed min{a 1 , a 2 , . . ., a K }.Thus, we have where c k is the common rate allocated to user k.Upon successfully decoding the common message, each user removes the common message and subsequently decodes its private message.The achievable rate for user k to decode its individual private message can be expressed as The total transmission rate for user k is the sum of its allocated common rate and its private rate, i.e., c k + r k .
The system EE is quantified as the ratio of the system sum rate K k =1 (c k + r k ) to the system power consumption, which includes the BS's transmit power E[ x 2 ] = tr(GG H ), BS's circuit power consumption P B , and RIS power consumption L × N × P RIS , where P RIS represents the power consumption per RIS element.
Our design objective is to maximize the system EE by jointly optimizing BS beamforming, common rate allocation, and RIS phase shifts.This problem is formulated as follows: where (4b) is the BS transmit power constraint, (4c) and (4d) specify the common rate constraints in accordance with RSMA, (4e) is the individual rate constraint with R th k being the target rate for user k, and (4f) defines the discrete RIS phase shift values within the set }} with Q-bit resolution.The optimization variables in problem (4) are coupled and the problem is nonconvex.While a conventional approach involving alternate optimization can be used to decouple the variables, it often involves approximations and relaxations which result in increased complexity and suboptimal solutions.To overcome these challenges, we introduce a data-driven approach, as described in the next section.

III. THE PROPOSED GNN-BASED APPROACH
The proposed method employs a GNN framework, converting the communication model into a graph representation with nodes, edge weights, and node features that convey pertinent information which is exchanged and updated via the graph structure, as elaborated below.

A. GNN Mechanism
Our graph design consists of a total of L + K + 1 nodes, categorized into three groups (see Fig. 1): • RIS Nodes (L nodes): The RIS nodes are not interconnected with each other, assuming negligible or no communication between RISs.However, each RIS node forms weighted connections to each private user node with fixed edge weight e k , = tr(H k H H k ), where ] contains CSI for transmissions from the BS to user k through RIS . 1hese edge weights reflect the varying significance of each RIS with respect to each user.The initial features of the RIS nodes are denoted as r (0) .Each RIS node learns the RIS phase shifts θ .1/K , ∀k , conveying information from private user nodes to the common user node.These edge weights will be updated to learn common rate allocations of users, c k .
The initial features of the common user node are denoted as u (0) c .The common user node learns BS beamforming for the common message, g 0 .The GNN learning mechanism enables interactions and information exchanges among these three node categories through a layered structure, as shown in Fig. 1.In the initial layer, the initial features of the private user nodes u (0) k are obtained from H k using a neural network-based feature extractor, represented as f (0) u : R 2M (N +1)×1 → R q×1 , where q is a configurable parameter, followed by an element-wise mean operation ϕ mean to isolate the information related to user k and preserve the permutation invariance property of the GNN: , where f (0) : R q×1 → R q×1 is a feature extractor.The information from other nodes is aggregated and updated by node/edge update layers.In the dth layer (d = 1, . . ., D), the node features of private user nodes are updated by where f (d) u : R 3qd×1 → R q×1 is a node update function and ϕ max (•) is an element-wise max function.Here, private user nodes identify dominant interference from the other users via the element-wise max function, capture RIS significance through weighted averages, and maintain prior information by concatenating previous node features.Similarly, the features of the RIS nodes and the common user node are updated respectively by where f (d) : R 2qd×1 → R q×1 and f (d) c : R 2qd×1 → R q×1 are node update functions.In (7), the updated edge weights at the dth layer, ω After D layers of node updates, the readout layer, denoted as the (D + 1)th layer, produces the desired θ 's, g k 's, g 0 , and c k 's.Each RIS node computes the discrete RIS phase shifts θ by first obtaining the continuous phase shifts θ = [ θ ,1 , . . ., θ ,N ] T as follows: where f (D+1) : R q(D+1)×1 → R 2N ×1 denotes a linear readout function, followed by quantization to satisfy constraint (4f).Each private user node computes the BS beamformers for private messages, ] T , as follows: where f (D+1) u : R q(D+1)×1 → R 2M ×1 represents a linear readout function, followed by power normalization to satisfy constraint (4b).We propose a method to determine the power normalization factor μ = [0, 1] such that tr(GG H ) = μP max BS in (4b), as described in Section III-B.The common user node computes the BS beamformer for the common message, g 0 , from u (D) c in a similar fashion.The edge weights between the private user nodes and the common user node produce common rate allocations for users: which meets constraints (4c) and (4d).Finally, constraint (4e) is incorporated in the loss function as a regularization term during training.The loss function is designed as , where β is a hyperparameter.

B. Power Control
In EE maximization, unlike sum-rate maximization, setting the BS transmit power to its maximum value, i.e., tr(GG H ) = P max BS (or μ = 1), is not always the optimal strategy.Therefore, an effective method is required to control the power level in order to optimize EE.We address this in the GNN framework by first transforming the final updated graph into the graph spectral domain, which captures the inter-relationship information among all nodes, and then applying a learnable network to determine the power level based on the features embedded in this spectral domain.First, the adjacency matrix of the considered graph, structured with nodes ordered as RIS nodes, private user nodes, and common user node, is given by where O is an all-zeros matrix, J is an all-ones matrix, I is an identity matrix, 0 is an all-zeros row vector, and ω = [ω Then, the normalized weighted graph Laplacian L is defined as where A i = j [A] i,j and A j = i [A] j ,i are the sum of the ith row and jth row of A, respectively.Then, we transform the node features to the graph spectral domain by performing X = LX, where X [r Then, we collect ϕ mean ({ X i } i=1,...,L+K +1 ), where X i denotes the ith row of X, and apply a learnable fully-connected network f p : R q(D+1)×1 → R 2 and a softmax operation to generate binary outputs that represent the power level μ ∈

IV. SIMULATION RESULTS
In our simulation, we consider a topology comprising a BS located at coordinates (0, 0), L = 3 RISs located at (100, 0), (100, -10), and (100, 10), respectively, and K users randomly positioned within a circular area centered at (200, 0) with a radius of 20.All units are meters.The RIS-related channels are modeled as quasi-static Rician flat-fading, and the direct BS-to-user channels are modeled as quasi-static Rayleigh flat-fading [14].The system parameters are set as: with output sizes 2M, 2N, and 2M, respectively; and f p is a fully-connected network with one hidden layer of size q(D + 1)/2.The channel information serves as training samples for GNN, collected in batches of 32 for model parameter updating using the Adam optimizer with a learning rate of 0.001 and momentum parameters of (β 1 , β 2 ) = (0.9, 0.999).During testing, channel information is generated as testing samples and results are recorded by averaging 3200 testing samples.We compare the following schemes in our study: 1) the proposed GNN model designed for RSMA-based systems, 2) the GNN model tailored for NOMA-based systems, 3) another deep learning model (referred to as DNN) [15] adapted to our problem by incorporating a power control network of two fully-connected layers of sizes 100 and 2, respectively, in addition to the existing PhaseNet and BeamNet components, and 4) the successive convex approximation (SCA) method proposed in [6].Adapting the GNN model to the NOMA scenario involves the removal of the common user node in the graph, adjustment or elimination of common raterelated constraints specified in (4c)-(4e), and adoption of the decoding order and rate expression for each user [16], which is given by r k = log 2 (1 + min{SINR 1→k , . . ., SINR k →k }), where represents the SINR observed at user i when decoding user k for subsequent successive interference cancelation (SIC) processes at user i. Fig. 2(a) depicts the EE performance across various numbers of RIS elements.Increasing the number of RIS elements generally improves EE due to increased sum rates, until the additional power consumption from more RIS elements offsets the benefits.Thus, a tradeoff exists between power consumption and the advantages gained from increasing the number of RIS elements.The proposed GNN outperforms DNN, highlighting the effectiveness of utilizing and refining CSI through the graph representation.SCA exhibits inferior performance due to the convexification and linearization techniques employed in SCA that lead to suboptimal results.RSMA outperforms NOMA when the same GNN model is applied, due to RSMA's ability to adjust the ratio of common and private rates and its different decoding processes in multiuser scenarios.With the regularization term in the loss function, the GNN model satisfies constraint (4e) with high probability (over 99.5%).Fig. 2(b) examines the effect of imperfect CSI.Particularly, the imperfect RIS-user channels h RIS ,k = h RIS ,k + Δh RIS ,k , where Δh RIS ,k ∼ CN (0, σ 2 e ), are used to design beamforming.Both GNN and DNN exhibit robustness against imperfect CSI due to diverse training samples composed of various channel realizations, whereas SCA experiences substantial performance degradation as it heavily relies on precise channel information for beamforming optimization.
Fig. 3(a) shows the EE performance under varying P max BS values, both with and without a power control mechanism for GNN and DNN.The case without power control corresponds to setting tr(GG H ) = P max BS in (4b).Notably, the EE performance stabilizes with power control and degrades significantly without power control when P max BS is larger.When P max BS is smaller, the optimal strategy for maximizing Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.EE is to transmit at the maximum power, a strategy also learned by GNN.This confirms the effectiveness of the power control mechanism described in Section III-B.Fig. 3(b) shows the convergence of GNN, where GNN converges around 1100 iterations when the criterion of reaching within 1% of the converged value is considered.Table I assesses the GNN model's generalization across varying numbers of training and testing users.As can be seen, matching user numbers yield the best performance, while mismatches between training and testing users lead to various degrees of performance degradation, particularly with smaller training or testing user sets (e.g., K = 2), as sensitivity to channel variations is heightened with fewer users.Nevertheless, the results highlight GNN's effectiveness in training with smaller user sets while maintaining comparable performance even in scenarios with more users.Table II presents runtime results, indicating remarkable efficiency of both GNN and DNN compared to SCA.While GNN and DNN exhibit similar runtime complexity, GNN significantly outperforms DNN in the EE performance.

V. CONCLUSION
We have developed a GNN model for optimizing BS beamforming, common rate allocation, and RIS phase shifts to maximize energy efficiency in a multi-RIS-aided downlink multiuser MISO system with RSMA.The proposed GNN outperforms conventional DNN and SCA methods and is adaptable to NOMA systems.We validated the power control mechanism, assessed the model's robustness against imperfect CSI and generalization across varying user numbers, and examined its runtime complexity.Given its favorable performance and complexity, the proposed GNN is wellsuited for practical real-time applications in RIS-aided RSMA systems.

k
. The initial features of the RIS nodes r(0) and the common user node u using a permutation-invariant weighted average based on the graph structure: r are adopted for computing weighted averages, where f (d) ω : R qd×1 → R is an edge update function and | • | is the element-wise absolute operator.

1
for all users, and σ 2 k = 2 × 10 −5 .In the GNN model, the parameters are set as: D = 3, q = 128, β = 1; f (0) u and f (0) are fully-connected networks with one hidden layer of sizes 128 and 64, respectively; f (d) u , f (d) , and f (d) c for d = 1, . . ., D are fully-connected networks each with one hidden layer of size 128, and linear transformers for d = D +1

Abstract-This letter explores energy efficiency (EE) maximization in a downlink multiple-input single-output (MISO) reconfigurable intelligent surface (RIS)-aided multiuser system employing rate-splitting multiple access (RSMA). The optimization task entails base station (BS) and RIS beamforming and RSMA common rate allocation with constraints. We propose a graph neural network (GNN) model that learns beamforming and rate allocation directly from the channel information using a unique graph representation derived from the communication system. The GNN model outperforms existing deep neural network (DNN) and model-based methods in terms of EE, demonstrating low complexity, resilience to imperfect channel information, and effective generalization across varying user numbers. Index Terms-Reconfigurable intelligent surface, rate-splitting multiple access, beamforming, graph neural network.
Energy-Efficient Multi-RIS-Aided Rate-Splitting Multiple Access: A Graph Neural Network Approach Bing-Jia Chen, Graduate Student Member, IEEE, Ronald Y. Chang , Senior Member, IEEE, Feng-Tsun Chien , Senior Member, IEEE, and H. Vincent Poor , Life Fellow, IEEE H , (•) T , and tr(•) denote the Hermitian, transpose, and trace operators.(•) and (•) are real and imaginary parts of the argument.R and C are sets of real and complex numbers.diag(a) creates a diagonal matrix with vector a on c 2024 The Authors.This work is licensed under a Creative Commons Attribution 4.0 License.
For more information, see https://creativecommons.org/licenses/by/4.0/Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.