Paper deep dive
A General Deep Learning Framework for Wireless Resource Allocation under Discrete Constraints
Yikun Wang, Yang Li, Yik-Chung Wu, Rui Zhang
Intelligence
Status: succeeded | Model: google/gemini-3.1-flash-lite-preview | Prompt: intel-v1 | Confidence: 93%
Last extracted: 3/23/2026, 12:02:10 PM
Summary
The paper proposes a general deep learning framework for mixed-discrete wireless resource allocation problems. It introduces a 'support set' representation for discrete variables and a Discrete Variable Learning Network (DVLN) that uses sequential decoding and dynamic masking to enforce constraints and handle non-SPSD properties, while avoiding zero-gradient issues. The DVLN is trained jointly with a Continuous Variable Learning Network (CVLN) in an unsupervised manner, demonstrating superior performance and efficiency in cell-free and movable antenna-aided systems.
Entities (5)
Relation Signals (3)
DVLN → isjointlytrainedwith → CVLN
confidence 95% · The DVLN is jointly trained with a continuous variable learning network (CVLN)
DVLN → uses → Support set
confidence 95% · proposes a general DL framework by introducing the support set to represent the discrete variables
DVLN → solves → Mixed-discrete wireless resource allocation
confidence 90% · propose a general DL framework for a broad class of mixed-discrete wireless resource allocation problems
Cypher Suggestions (0)
No Cypher suggestions yet.
Abstract
Abstract:While deep learning (DL)-based methods have achieved remarkable success in continuous wireless resource allocation, efficient solutions for problems involving discrete variables remain challenging. This is primarily due to the zero-gradient issue in backpropagation, the difficulty of enforcing intricate constraints with discrete variables, and the inability in generating solutions with non-same-parameter-same-decision (non-SPSD) property. To address these challenges, this paper proposes a general DL framework by introducing the support set to represent the discrete variables. We model the elements of the support set as random variables and learn their joint probability distribution. By factorizing the joint probability as the product of conditional probabilities, each conditional probability is sequentially learned. This probabilistic modeling directly tackles all the aforementioned challenges of DL for handling discrete variables. By operating on probability distributions instead of hard binary decisions, the framework naturally avoids the zero-gradient issue. During the learning of the conditional probabilities, discrete constraints can be seamlessly enforced by masking out infeasible solutions. Moreover, with a dynamic context embedding that captures the evolving discrete solutions, the non-SPSD property is inherently provided by the proposed framework. We apply the proposed framework to two representative mixed-discrete wireless resource allocation problems: (a) joint user association and beamforming in cell-free systems, and (b) joint antenna positioning and beamforming in movable antenna-aided systems. Simulation results demonstrate that the proposed DL framework consistently outperforms existing baselines in terms of both system performance and computational efficiency.
Tags
Links
- Source: https://arxiv.org/abs/2603.19322v1
- Canonical: https://arxiv.org/abs/2603.19322v1
Trouble viewing inline? Open PDF directly →
Full Text
93,446 characters extracted from source content.
Expand or collapse full text
A General Deep Learning Framework for Wireless Resource Allocation under Discrete Constraints Yikun Wang, Yang Li, Yik-Chung Wu, and Rui Zhang Y. Wang is with the Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong, and also with the School of Computing and Information Technology, Great Bay University, Dongguan 523000, China (email: ykwang@e.hku.hk).Y. Li is with the School of Computing and Information Technology, Great Bay University, Dongguan 523000, China (email: liyang@gbu.edu.cn). Y.-C. Wu is with the Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong (email: ycwu@e.hku.hk).R. Zhang is with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore (email: elezhang@nus.edu.sg). Abstract While deep learning (DL)-based methods have achieved remarkable success in continuous wireless resource allocation, efficient solutions for problems involving discrete variables remain challenging. This is primarily due to the zero-gradient issue in backpropagation, the difficulty of enforcing intricate constraints with discrete variables, and the inability in generating solutions with non-same-parameter-same-decision (non-SPSD) property. To address these challenges, this paper proposes a general DL framework by introducing the support set to represent the discrete variables. We model the elements of the support set as random variables and learn their joint probability distribution. By factorizing the joint probability as the product of conditional probabilities, each conditional probability is sequentially learned. This probabilistic modeling directly tackles all the aforementioned challenges of DL for handling discrete variables. By operating on probability distributions instead of hard binary decisions, the framework naturally avoids the zero-gradient issue. During the learning of the conditional probabilities, discrete constraints can be seamlessly enforced by masking out infeasible solutions. Moreover, with a dynamic context embedding that captures the evolving discrete solutions, the non-SPSD property is inherently provided by the proposed framework. We apply the proposed framework to two representative mixed-discrete wireless resource allocation problems: (a) joint user association and beamforming in cell-free systems, and (b) joint antenna positioning and beamforming in movable antenna-aided systems. Simulation results demonstrate that the proposed DL framework consistently outperforms existing baselines in terms of both system performance and computational efficiency. I Introduction Mixed-discrete optimization problems are pervasive in wireless resource allocation, particularly in emerging paradigms such as cell-free (CF) systems [1], holographic multiple-input multiple-output (MIMO) systems[2], and fluid antenna (FA) or movable antenna (MA)-aided systems[3, 4]. In these applications, optimization involves not only continuous variables such as beamforming vectors and transmit powers, but also discrete decisions including user scheduling, antenna selection, or antenna positioning. Mathematically, these problems are commonly NP-hard, making it difficult to find the optimal solution [5]. A common heuristic approach is to solve the problem in two steps, i.e., the discrete variables are first determined using rule-based or greedy methods [6], and then the continuous variables are optimized under the fixed discrete variable configuration. However, such two-stage strategies suffer from severe performance loss due to the absence of joint optimization. To address this issue, several joint optimization algorithms have been proposed [7, 8, 9, 10, 11, 12]. Many of these methods rely on continuous relaxation of discrete variables to enable gradient-based optimization [7, 9, 10]. However, such relaxation inevitably introduces a performance loss due to the relaxation gap and the need for post-hoc projection onto the discrete feasible set. Moreover, these approaches commonly require iterative updates [7, 8, 9, 10, 11, 12], which incur prohibitive computational latency [13, 14] and render them unsuitable for real-time deployment. Recently, learning-to-optimize has emerged as a promising alternative, leveraging neural networks (NNs) to learn end-to-end mappings from environmental inputs to continuous resource allocation decisions. Representative architectures such as graph neural networks (GNNs) [14, 15, 16] and transformers [17, 18, 19] have demonstrated the ability to achieve near-optimal performance with remarkably low inference latency. However, directly extending these approaches to mixed-discrete optimization problems remains non-trivial, primarily due to the following three challenges. The first challenge is due to the conflict between discrete outputs and gradient-based training. Standard NNs rely on backpropagation, which fails when outputs are discrete because the resulting gradients are zero almost everywhere [20]. Several methods have been proposed to address this challenge. One popular technique is the straight-through estimator (STE) [21, 22], which bypasses the non-differentiability by copying gradients from the discrete outputs back to the continuous representation. However, such approach may provide misleading information about the true optimization landscape, causing performance degradation, especially when discrete variables are highly structured or coupled [22, 23]. Alternative approaches approximate the discrete outputs using differentiable surrogates such as softmax or Gumbel-softmax [23, 24, 25]. While these methods enable gradient flow, they introduce approximation errors that deviate from the true objective. Although a temperature parameter governs how well the relaxed output approximates a true discrete decision, it requires careful specification of the temperature schedule to balance the model learnability and approximation error. The second challenge lies in difficulty in enforcing the constraints involving discrete variables in deep learning. For instance, in MA-aided systems, to avoid mutual coupling between antennas, the distance between any pair of MAs must exceed a given threshold [4]. This yields a highly non-convex and combinatorial feasible set that is challenging for NNs to deal with. While penalty-based methods [26] can incorporate such constraints into the loss function, they require meticulous tuning of penalty coefficients. Moreover, the penalty-based methods cannot guarantee strict feasibility, which is a critical limitation in applications with stringent reliability requirements. The third challenge stems from the struggle of typical deep learning in achieving non-same-parameter-same-decision (non-SPSD) property on the discrete variable solutions [27, 24]. In wireless resource allocation, the problem is deemed non-SPSD when identical or nearly identical system parameters are assigned different solutions. For example, consider a multi-user scheduling problem, where two users experience nearly identical channel conditions. However, the optimal scheduler may activate only one of them due to the strong interference between them. This inherent asymmetry in the discrete solution cannot be captured by current DL approaches [14, 15, 16, 21, 25, 17, 18, 19], resulting in severe performance degradation [27, 24]. To address the aforementioned challenges, we propose a general DL framework for a broad class of mixed-discrete wireless resource allocation problems. The proposed approach leverages a reformulation that expresses discrete decisions using a support set to enhance learnability. Building on this, we design a discrete variable learning network (DVLN) to learn the probability distribution over discrete solutions given system parameters, addressing the zero-gradient problem of discrete outputs. Crucially, to enforce the constraints with discrete variables, the DVLN adopts an encoder-decoder architecture with sequential decoding process: rather than predicting all discrete variables in parallel, it generates the solutions of different variables sequentially, dynamically masking infeasible candidates at each step to guarantee strict feasibility. Notably, the solution is generated by an attention mechanism based on a context embedding, which takes into account the current discrete variable solution and evolves dynamically with each step, thereby enabling our DL framework to effectively capture the non-SPSD property. The DVLN is jointly trained with a continuous variable learning network (CVLN), which is specialized for outputting continuous resource allocation solutions. The two networks are optimized end-to-end in an unsupervised manner by directly maximizing the underlying system performance metric, without requiring a precomputed dataset of optimal solutions. The main contributions of this paper are summarized as follows: • A general formulation for mixed-discrete resource allocation in wireless systems is proposed, where discrete variables are represented using the support set. • A general DL framework comprising a DVLN and a CVLN is proposed. With a novel sequential and dynamic structure, the DVLN directly resolves the challenges of deep learning when handling discrete variables and constraints, namely zero-gradient problem, incapable of enforcing complex constraints involving discrete variables, and lack of non-SPSD property. With the CVLN being flexible to tailor various continuous resource variables, the DVLN and CVLN are jointly trained in an unsupervised manner. • The proposed DL framework is illustrated with two typical wireless examples: (a) joint user equipment (UE)–access point (AP) association and beamforming in CF systems; and (b) joint antenna positioning and beamforming in MA-aided systems. In both cases, simulations show that the proposed DL framework achieves superior performance and strict constraint satisfaction with significantly lower inference latency than existing DL-based and model-based approaches. The remaining paper is organized as follows. Section I introduces a general problem formulation and two application examples. Section I describes the general DL framework, while Section IV introduces the training algorithm. In Sections V and VI, the DL framework is demonstrated with two examples, with performance evaluated through simulations. Finally, the conclusion is drawn in Section VII. Notations: We use a, a, A, and A to denote scalar, vector, matrix, and set, respectively; TA^T and HA^H to denote transpose and Hermitian transpose; ∪ and ∩ to denote the set union and intersection; |⋅||·| and ∥⋅∥\|·\| to denote the modulus and l-2 norm; (⋅,⋅)CN(·,·) to denote complex Gaussian distribution. I General Problem Formulation and Application Examples I-A Problem Formulation Many wireless resource allocation problems appear as a mixed-discrete optimization problem in the following form: (P1): max∈0,1Nb,∈ℂNw _b∈\0,1\^N_b,w ^N_w U(,;) U(b,w;h) (1a) s.t. fm(;)≤0,∀m=1,2,…,Ω, f_m(b;h)≤ 0,~∀ m=1,2,…, , (1b) gm(;)≤0,∀m=1,2,…,Ψ, g_m(w;h)≤ 0,~∀ m=1,2,…, , (1c) where ∈0,1Nbb∈\0,1\^N_b and ∈ℂNww ^N_w denote the discrete and continuous variables, respectively, and ∈ℂNhh ^N_h represents the known system parameters. Although the discrete variable b is written as a binary vector with each entry bn∈0,1b_n∈\0,1\, this representation can handle vectors taking any finite discrete values via binary encoding schemes like one-hot encoding (e.g., see Example 2 in Subsection C). Beside the discrete variable b, the general problem formulation (P1) also includes continuous variable w, thus (P1) is a “mixed” optimization. For instance, b may represent the indicator of user scheduling or antenna selection, w may denote the beamforming or power allocation, and h represents the channel state information or user locations. The objective of (P1) is to maximize a utility function U(⋅,⋅;⋅)U(·,·;·), which may depend on all the variables and system parameters. Moreover, fm(⋅;⋅)f_m(·;·) and gm(⋅;⋅)g_m(·;·) impose possible restrictions on b and w, respectively111For constraints involving coupled discrete and continuous variables, our proposed DL framework can be integrated with a previously proposed penalty-dual learning framework [28]. However, this extension is beyond the scope of the present paper.. This formulation (P1) covers applications in many previous works as special cases[1, 7, 8, 9, 10, 21, 24, 25]. Specific examples are given in Subsection C. I-B Equivalent Problem Transformation In this subsection, we transform (P1) into an equivalent form by leveraging the concept of support set, which enables the N to better learn the desired solution mapping. Specifically, the binary variable b can be fully captured by its support set A, defined as the set of indices corresponding to the non-zeros entries in b. Instead of optimizing over the full NbN_b-dimensional vector b, the problem becomes identifying the support set A. The reformulation is formally given by the following property. Property 1 (Reformulation of (P1)): Any mixed-discrete optimization problem in the form of (P1) is equivalent to (P2): max∈2b,∈ℂNw _A∈2^N_b,w ^N_w U(,;) U(b,w;h) (2a) s.t. (1b), (1c), bn=1,∀n∈,0,∀n∉, b_n= \ array[]l1, ∀ n ,\\ 0, ∀ n , array . (2d) where b≜1,2,…,NbN_ b \1,2,…,N_b\, and 2b2^N_ b denotes the power set of bN_b, i.e., the collection of its all 2Nb2^N_b possible subsets. The discrete variable b is fully determined by A through the relationship in (2d). For example, if =∅A= , we have =b=0; while if =bA=N_b, we have =b=1. In order to avoid repeatedly solving (P2) in a case-by-case manner for different h, we strive to design a mapping function given by ,=ℱ(),\A,w\=F(h), (3) where ℱ(⋅):ℂNh→2b×ℂNwF(·):C^N_h→ 2^N_b×C^N_w is represented by a N, such that the average objective function (2a) is maximized. The corresponding DL-based optimization problem can be written as (P3): maxℱ(⋅) _F(·) [U(,;)] _h[U(b,w;h)] (4) s.t. (1b),(1c),(2d),(3), ( binb),~( binc),~( disd),~( totalDNN), where the expectation is taken over the probability distribution of h. I-C Examples in Wireless Communications Here, we present two typical examples of mixed-discrete optimization problems in wireless communications and demonstrate how they can be expressed in the forms of (P2) and (P3). Example 1 (Joint UE-AP Association and Beamforming in CF Systems): Consider a downlink CF system comprising L APs and K UEs. Each UE is equipped with a single antenna and each AP is with M antennas. All APs are connected to a central process unit (CPU) for joint signal processing (see Fig. 1(a)). Define the association variable as bkl∈0,1b_kl∈\0,1\, where bkl=1b_kl=1 denotes that the k-th UE is served by the l-th AP, and bkl=0b_kl=0 otherwise. Additionally, due to the limitation of fronthaul capacity, the following constraints are imposed on the association variables [29, 1]. First, each AP can at most serve Kmax≪K_max K UEs, which is mathematically described by ∑k=1Kbkl≤Kmax,∀l∈ℒ≜1,2,…,L.Σ _k=1^Kb_kl≤ K_max,~~~∀ l \1,2,…,L\. (5) Second, each UE is served by at most Lmax≪L_max L APs, which is given by ∑l=1Lbkl≤Lmax,∀k∈≜1,2,…,K.Σ _l=1^Lb_kl≤ L_max,~~~∀ k \1,2,…,K\. (6) (a) Joint UE-AP association and beamforming in CF systems. (b) Joint antenna positioning and beamforming for MA-aided systems. Figure 1: Illustrations of two typical examples of mixed-discrete wireless resource allocation problems. We adopt a block-fading channel model in which the channels are assumed to be constant within a coherence period. We define the channel between the l-th AP and the k-th UE as kl∈ℂMh_kl ^M. Consequently, the received signal at the k-th UE can be expressed as yk=∑l=1LbklklHklsk+∑k=1,k′≠kK∑l=1Lbk′lklHk′lsk′+nk,y_k= _l=1^Lb_klh_kl^Hw_kls_k+ _k=1,k ≠ k^K _l=1^Lb_k lh_kl^Hw_k ls_k +n_k, (7) where sk∈ℂs_k denotes the desired symbol of the k-th UE that satisfies [|sk|2]=1E[|s_k|^2]=1, kl∈ℂMw_kl ^M is the beamforming vector of the k-th UE at the l-th AP, and nk∼(0,σk2)n_k (0, _k^2) is the additive white Gaussian noise (AWGN) with σk2 _k^2 being the noise power. From (7), the achievable sum rate of all K users can be expressed as RCF(,;)= R^CF(b,w;h)= (8) ∑k=1Klog2(1+|∑l=1LbklklHkl|2∑k=1,k′≠kK|∑l=1Lbk′lklHk′l|2+σk2), _k=1^K _2 (1+ | _l=1^Lb_klh_kl^Hw_kl|^2 _k=1,k ≠ k^K| _l=1^Lb_k lh_kl^Hw_k l|^2+σ^2_k ), where =[b11,b12,…,b1L,b21,…,bKL]T∈0,1KLb=[b_11,b_12,…,b_1L,b_21,…,b_KL]^T∈\0,1\^KL, =[11T,12T,…,1LT,21T,…,KLT]T∈ℂKLMw=[w_11^T,w_12^T,…,w_1L^T,w_21^T,…,w_KL^T]^T ^KLM, and =[11T,12T,…,1LT,21T,…,KLT]T∈ℂKLMh=[h_11^T,h_12^T,…,h_1L^T,h_21^T,…,h_KL^T]^T ^KLM. Consequently, the joint UE-AP association and beamforming problem can be formulated as (P4): max∈0,1KL,∈ℂKLM _b∈\0,1\^KL,w ^KLM RCF(,;) R^CF(b,w;h) (9a) s.t. (5),(6), ( equ6),~( equ7), ∑k=1K‖kl‖2≤Pmax,∀l∈ℒ, _k=1^K\|w_kl\|^2≤ P_max,∀ l , (9b) where PmaxP_max denotes the maximum transmit power of each AP. The problem (P4) shares the same form with (P1), with Nb=KLN_b=KL, Nw=KLMN_w=KLM, Nh=KLMN_h=KLM, Ω=K+L =K+L, and Ψ=L =L. Using Property 1, problem (P4) can be rewritten as (P5): max∈2×ℒ,∈ℂKLM _A∈ 2^K×L,w ^KLM RCF(,;) R^CF(b,w; h) (10a) s.t. (5), (6), (9b), ( equ6), ( equ7), ( ccon), bkl=1,∀(k,l)∈,0,∀(k,l)∉, b_kl= \ array[]l1,~~∀(k,l) ,\\ 0,~~∀(k,l) , array . (10d) where ×ℒK×L denotes the Cartesian product of set K and ℒL, i.e., ×ℒ=(k,l)|k∈,l∈ℒK×L=\(k,l)|k ,~l \. Let ℱCF(⋅):ℂKLM→2×ℒ×ℂKLMF^CF(·):C^KLM→ 2 K×L×C^KLM represent the to-be-designed joint UE-AP association and beamforming mapping, the corresponding DL problem is expressed in the form of (P3) as (P6): maxℱCF(⋅) ~ _F^CF(·) [RCF(,;)] _ h[R^CF(b,w; h)] (11a) s.t. (5), (6), (9b), (10d), ,=ℱCF(). \A,w\=F^CF(h). (11b) Example 2 (Joint Antenna Positioning and Beamforming for MA-aided Systems): Consider an MA-aided downlink system with one base station (BS) serving K UEs. The BS is equipped with M MAs and each UE is equipped with a single fixed-position antenna. The positions of the M MAs can be adjusted simultaneously within a predefined two-dimensional rectangular area, which is divided into N≫MN M candidate positions (CPs) for placing the M MAs (see Fig. 1(b)), with the coordinate of the n-th CP denoted by n=[xn,yn]Tp_n=[x_n,y_n]^T, where xnx_n and yny_n represent the coordinates along the x-axis and y-axis, respectively. Define a binary positioning vector as ≜[b1,b2,…,bN]T∈0,1Nb [b_1,b_2,…,b_N]^T∈\0,1\^N, where bn=1b_n=1 indicates an MA is placed at the n-th CP, and bn=0b_n=0 otherwise. The vector b is subject to two key constraints. First, exactly M MAs will be placed, which is enforced by ∑n=1Nbn=M.Σ _n=1^Nb_n=M. (12) Second, to mitigate mutual coupling between any pair of MAs, the distance between their positions must be at least the minimum separation dmind_ [30], i.e., ‖n−n′‖≥bnbn′dmin,∀n,n′∈,n≠n′,\|p_n-p_n \|≥ b_nb_n d_min,~∀ n,n ,~n≠ n , (13) where ≜1,2,…,NN \1,2,…,N\. Let k≜[hk1,hk2,…,hkN]T∈ℂNh_k [h_k1,h_k2,…,h_kN]^T ^N denote the channel between all N CPs and the k-th UE. The received signal of the k-th UE is given by yk=k()Hksk+∑l=1,l≠kKk()Hlsl+nk,y_k=h_k(b)^Hw_ks_k+ _l=1,l≠ k^Kh_k(b)^Hw_ls_l+n_k, (14) where k()∈ℂMh_k(b) ^M denotes the channel between the selected CPs and the k-th UE, k∈ℂMw_k ^M represents the corresponding transmit beamformer, and nkn_k represents the AWGN with power σk2 _k^2. The sum rate of all K UEs is then expressed as RMA(,;)=∑k=1Klog2(1+|k()Hk|2∑l=1,l≠kK|kH()Hl|2+σk2),R^MA(b,w;h)= _k=1^K _2 (1+ |h_k(b)^Hw_k|^2Σ _l=1,l≠ k^K|h_k^H(b)^Hw_l|^2+ _k^2 ), (15) where =[1T,2T,…,KT]T∈ℂKMw=[w_1^T,w_2^T,…,w_K^T]^T ^KM, and =[1T,2T,…,KT,1T,2T,…,NT]T∈ℂKN+2Nh=[h_1^T,h_2^T,…,h_K^T,p_1^T,p_2^T,…,p_N^T]^T ^KN+2N. We aim to maximize RMAR^MA with respect to b and w, which is expressed as (P7): max∈0,1N,∈ℂKM _b∈\0,1\^N,w ^KM RMA(,;) R^MA(b,w;h) (16a) s.t. (12), (13), ( mbinb), ( mbinc), ∑k=1K‖k‖2≤Pmax, _k=1^K \|w_k \|^2≤ P_max, (16b) where PmaxP_max is the maximum transmit power of the BS. The problem (P7) also shares the same form with (P1), with Nb=N_b=N, Nw=KMN_w=KM, Nh=KN+2NN_h=KN+2N, Ω=M2−M+1 =M^2-M+1, and Ψ=1 =1. Using Property 1, the problem (P7) can be reformulated as (P8): max∈2,∈ℂKM _A∈ 2^N,w ^KM RMA(,;) R_MA(b,w;h) (17a) s.t. (12), (13), (16b), ( mbinb), ( mbinc), ( mbind), bn=1,∀n∈,0,∀n∉. b_n= \ array[]l1,~~∀ n ,\\ 0,~~∀ n . array . (17d) Let ℱMA(⋅):ℂKN+2N→2×ℂKMF^MA(·):C^KN+2N→ 2^N×C^KM represent the to-be-designed joint antenna positioning and beamforming scheme, (P8) can be further represented in the form of DL of (P3) as (P9): maxℱMA(⋅) _F^MA(·) [RMA(,;)] _h[R^MA(b,w;h)] (18a) s.t. (12), (13), (16b), (17d), ( mbinb), ( mbinc), ( mbind), ( mdisb), ,=ℱMA(). \A,w\=F^MA(h). (18b) I Proposed General DL Framework In this section, we propose a general DL framework to learn the desired mapping ℱ(⋅):ℂNh→2b×ℂNwF(·):C^N_h→ 2^N_b×C^N_w in (P3). The proposed framework consists of two distinct models: a DVLN for discrete outputs and a CVLN for continuous outputs. These models are cascaded and trained jointly to provide near-optimal solutions while satisfying the constraints at the same time, as shown in Fig. 2. Specifically, the DVLN ℱA(⋅):ℂNh→2bF_A(·):C^N_h→ 2^N_b produces the support set A of the discrete variable b by =ℱA()A=F_A( h). Subsequently, the continuous variable w is produced by =ℱw(,)w=F_w(A,h), where ℱw(⋅,⋅):2b×ℂNh→ℂNwF_w(·,·):2^N_b×C^N_h ^N_w denotes the CVLN. Figure 2: The overall architecture of the proposed DL framework. To tackle the zero-gradient problem of discrete outputs, we first model the entries in A as random variables and aim to learn their conditional probability distribution given the system parameter h, i.e., p(|)p(A|h). Noticing that the constraints in (1b) are coupled, which means that each element in A depends on other elements, we express p(|)p (A|h ) in its factorized form p(|)=∏t=1Tp(at∣t−1,),p(A|h)= _t=1^Tp(a_t _t-1,h), (19) where at∈a_t denotes the selected element of A at the t-th step, t−1≜a1,a2,…,at−1A_t-1 \a_1,a_2,…,a_t-1\ represents the elements selected up to the (t−1t-1)-th step with 0≜∅A_0 , and T denotes the cardinality of A. In some cases, the value of T is exactly determined from the constraints in (1b). For example, in Example 2 of Section I-C, T=MT=M, which is governed by (12). On the other hand, in Example 1 of Section I-C, the constraints (5) and (6) only provide an upper bound on the number of elements in A. In this case, we need a mechanism that allows us to stop adding element to the set A before reaching the upper bound. To this end, we employ an end token in deep learning, which will be elaborated later. To facilitate understanding, we focus on the case where we know the exact number of elements in A at this moment. It is observed from (19) that this factorization enables a sequential construction of A over T steps. At each step t, we calculate the probability of each possible value of ata_t, i.e., p(at=n∣t−1,)n∈b∖t−1\p(a_t=n _t-1,h)\_n∈N_b _t-1 where b∖t−1N_b _t-1 denotes remaining candidates. The next element ata_t is then selected by probabilistic sampling or greedy selection based on p(at∣t−1,)p(a_t _t-1,h). In the following subsections, we present the architectures of ℱA(⋅)F_A(·) and ℱw(⋅,⋅)F_w(·,·), and demonstrate that the proposed ℱA(⋅)F_A(·) effectively captures the non-SPSD property inherent in the discrete optimization problem. I-A Design of ℱA(⋅)F_A(·) To learn the conditional probability p(|)p(A|h) while exploiting (19), we propose a DVLN with an encoder-decoder structure, where the encoder generates the embeddings of all NbN_b elements in bN_b, and the decoder outputs the conditional probabilities in T steps. The whole process of the DVLN is summarized in Fig. 3, and the design details are introduced as follows. 1) Design of the encoder: The encoder takes the system parameter h as the input, and outputs a dh×Nbd_h× N_b embedding: ≜[1,2,…,Nb]=E(),R [r_1,r_2,…,r_N_b]=G_E (h ), (20) where n∈ℝdhr_n ^d_h is a dhd_h-dimensional vector representing the embedding of the n-th element of b, and E(⋅):ℂNh→ℝdh×NbG_E(·):C^N_h ^d_h× N_b denotes the encoding scheme. Figure 3: Overview of the DVLN. The encoder network can be implemented using various architectures, such as multilayer perceptron (MLPs), GNNs, or transformers. In Sections V and VI, we will design custom architectures of E(⋅)G_E(·) for the two examples presented in Section I-C. 2) Design of the decoder: As described earlier, the decoder incrementally constructs the support set A over T steps. At the t-step, it selects an element ata_t based on p(at=n|t−1,)n∈b∖t−1\p(a_t=n|A_t-1,h)\_n∈N_b _t-1. These probabilities are predicted via an attention mechanism that computes the relevance between the embeddings in R and a context representation of the current system state (t−1,) (A_t-1,h ). In particular, we introduce a context embedding network to capture the current system state: t⋆=ℱctx(t−1,,),c _t=F_ctx(A_t-1,h,R), (21) where t⋆∈ℝdhc _t ^d_h denotes a dhd_h-dimensional context vector of the system state at step t, and ℱctx(⋅,⋅,⋅)F_ctx(·,·,·) is a function parameterized by a N. Then, given t⋆c_t , a compatibility score for each candidate is computed as [31, 32]: un,t=ATT(n,t⋆),∀n∈b∖t−1,u_n,t=ATT(r_n,c _t),~~∀ n∈N_b _t-1, (22) where ATT(⋅,⋅)ATT(·,·) denotes a general attention-based N [31, 32] that quantifies the relevance between nr_n and t⋆c _t. Finally, the conditional probability is then obtained via a softmax normalization over the compatibility scores: p(at=n∣t−1,)=eun,t∑n′=1Nbeun′,t,∀n∈b∖t−1.p(a_t=n _t-1,h)= e^u_n,t _n =1^N_be^u_n ,t, ∀ n∈N_b _t-1. (23) However, the attention mechanism described above does not inherently respect the discrete constraints in (1b), as it may assign non-zero probabilities to candidates that would render the solution infeasible. To guarantee feasibility at every decoding step, we explicitly mask out any candidate that violates the constraints if added to the current support set. Specifically, we modify the attention scores in (22) by: un,t=ATT(n,t⋆),∀n∈b∖t−1∖~b,t,−∞,∀n∈~b,t, splitu_n,t&= casesATT(r_n,c _t)&,~∀ n∈N_b _t-1 N_b,t,\\ -∞&,~∀ n∈ N_b,t, cases\\ split (24) where ~b,t⊆b N_b,t _b is the set of infeasible candidates at step t, which is formally defined as ~b,t≜ N_b,t n∈b∖t−1| \n _b .A_t-1| (25) ∃m∈1,…,Ω,fm((t−1∪n))>0, .∃ m∈\1,..., \,~f_m (b(A_t-1∪\n\) )>0 \, with (At−1∪n)b(A_t-1∪\n\) representing the discrete variable b with support set t−1∪nA_t-1∪\n\. It can be seen from (23), (24) and (25) that for all elements violating the constraints in (1b), the corresponding conditional probability p(at=n|t−1,)p(a_t=n|A_t-1,h) will approach zero, which ensures the fulfillment of the discrete constraints in (1b). In cases where the constraints (1b) only provide an upper bound to T (in the worst case, we can always set the upper bound as NbN_b), we need the capability to stop adding element to A before reaching the upper bound. In such cases, we introduce an end token β. Specifically, at each decoding step t, we augment the candidate set b∖t−1N_b _t-1 as b∖t−1∪βN_b _t-1∪\β\. The attention score associated with β is defined in the form of (24) but with uβ,t=ATT(β,t⋆)u_β,t=ATT (r_β,c_t ), where β∈ℝdhr_β ^d_h is a trainable vector. If β is selected at step t, the decoding process will stop and the final support set is fixed as =t−1A=A_t-1. I-B Design of ℱw(⋅,⋅)F_w(·,·) The CVLN ℱw(⋅,⋅)F_w(·,·) takes A and h as input, and output the continuous variable w. While numerous methods have been developed to learn such continuous mappings [14, 15, 16, 17, 18, 19], the specific architecture of ℱw(⋅,⋅)F_w(·,·) usually depends on the problem structure, and the case-specific implementation will be given in Sections V and VI, respectively. I-C Resolving the Non-SPSD Challenge in Discrete Optimization In this subsection, we formalize the non-SPSD property of (P3) and demonstrate that the proposed framework is explicitly designed to capture this property. The non-SPSD property characterizes scenarios in which identical environmental conditions may correspond to significantly different solutions, which is a phenomenon commonly observed in mixed-discrete wireless resource allocation problems. Property 2 (Non-SPSD Property of (P3)): Given that the system parameter of (P3) consists of NbN_b blocks, i.e., =[1T,2T,…,NbT]Th=[h_1^T,h_2^T,…,h_N_b^T]^T, proximity of parameter blocks does not guarantee identical membership in A. More precisely, for any given tolerance ε≥0 ≥ 0, there exist instances of (P3) (i.e., choices of h) and distinct indices i and j such that the corresponding blocks are ε -close, yet their membership status in A differs. Formally, non-SPSD property is defined as ∃i,j: ∃ i,j: ‖i−j‖≤ε and ||h_i-h_j||≤ and ((i∈ and j∉) or (i∉ and j∈)). ((i and j ) or (i and j ) ). Many mixed-discrete wireless resource allocation problems possess this non-SPSD property, including the two examples elaborated in Section I-C. For instance, in (P6), when the UE-AP pair (k,l)(k,l) and (k′,l′)(k ,l ) have identical channel conditions, i.e., kl=k′l′h_kl=h_k l , their association solutions can be (k,l)∈(k,l) and (k′,l′)∉(k ,l ) , due to the strong interference between them as modeled in the rate function (LABEL:equ9). Similarly, in (P9), when CP n and CP n′n have near-identical channel conditions and positions, i.e., |hkn−hkn′|<ϵ|h_kn-h_kn |<ε and ‖n−n′‖<ϵ\|p_n-p_n \|<ε, their positioning solutions can be n∈n and n′∉n , due to the coupled discrete constraints (13). Crucially, thanks to our proposed attention-based encoder-decoder structure with the dynamic context embedding, the proposed DVLN is capable of capturing the non-SPSD property on the discrete solutions, as stated below. Proposition 1 (Non-SPSD Property of ℱA(⋅)F_A(·)): Let =[1T,2T,…,NbT]Th=[h_1^T,h_2^T,...,h_N_b^T]^T. The DVLN ℱA(⋅)F_A(·) is capable of producing the output A such that (i∈ and j∉) or (i∉ and j∈) (i and j ) or (i and j ) even if i=jh_i=h_j. Proof. When i=jh_i=h_j, the encoder E(⋅)G_E(·) in (20) outputs identical embeddings i=jr_i=r_j, since the encoder cannot distinguish between identical inputs without a judicious design [24]. We can see from (24) that their attention scores are also the same, i.e., ui,t=uj,tu_i,t=u_j,t if both i and j belong to the candidate set ~b,t N_b,t at step t. However, since at most only one of them (say i) can be selected to be included in tA_t, this makes the context vector t+1∗c_t+1^* at step t+1t+1 different from the previous t∗c_t^* at step t. Consequently, the attention score of the remaining j will also change in the subsequent decoding steps. This enables the capability of ℱA(⋅)F_A(·) to produce A such that i∈i while j∉j (or vice versa). ∎ IV Proposed Training Algorithm During training, each element ata_t is obtained via probabilistic sampling from its conditional distribution, i.e., at∼p(at|t−1,)a_t p(a_t|A_t-1,h), to expand the scope of exploration. Then, based on the probabilistic modeling p(|)p(A|h), the joint training of (P3) can be formulated as (P10): maxA,w _ θ_A, θ_w J(A,w)=,∼p(|)[U(,;)], J( θ_A, θ_w)=E_h,A p(A|h)[U(b,w;h)], (26) where A θ_A and w θ_w denote the trainable parameters of ℱA(⋅)F_A(·) and ℱw(⋅,⋅)F_w(·,·), respectively, and the original constraints in (1b) and (1c) are enforced through the design of ℱA(⋅)F_A(·) and ℱw(⋅,⋅)F_w(·,·), respectively. Next, we show how to update A θ_A and w θ_w. 1) Update of A θ_A: Since A is sampled according to p(|)p(A|h) during training, U(,;)U(b,w;h) is non-differentiable with respect to A θ_A. However, since p(|)p(A|h) itself is differentiable with respect to A θ_A, we equivalently express the gradient of J(A,w)J( θ_A, θ_w) with respect to A θ_A as ∇AJ(A,w)=[∑∼p(∣)U(,;)∇Ap(∣)] _ θ_AJ( θ_A, θ_w)=E_h [ _A p(A )U(b,w;h) _ θ_Ap(A ) ] (27) =[∑∼p(∣)U(,;)p(|)∇Alogp(∣)] =E_h [ _A p(A )U(b,w;h)p(A|h) _ θ_A p(A ) ] =,∼p(∣)[U(,;)∇Alogp(∣)]. =E_h,A p(A ) [U(b,w;h) _ θ_A p(A ) ]. In practice, the expectation in (27) is approximated via Monte Carlo sampling over mini-batches, which corresponds to the well-known policy gradient method. To reduce the variance of the gradient estimates, we replace U(,;)U(b,w;h) in (27) with U(,;)−U^()U(b,w;h)- U(h), where U^(⋅):ℂNh→ℝ U(·):C^N_h serves as an estimate of the sum rate, also known as the critic-network [33], whose N-architecture requires a case-specific design. The parameters A θ_A are then updated by a mini-batch stochastic gradient ascent algorithm, which can be implemented with the Adam optimizer [34]. Correspondingly, the trainable parameter of U^(⋅) U(·), denoted by C θ_C, is updated by a mini-batch stochastic gradient descent algorithm with the following mean-square-error loss function: LMSE(C)=,∼p(|)[(U^()−U(,;))2].L_MSE( θ_C)=E_h,A p(A|h) [ ( U(h)-U(b,w;h) )^2 ]. (28) Algorithm 1 Joint Training of ℱA(⋅)F_A(·) and ℱw(⋅,⋅)F_w(·,·) 1: Input number of epochs NeN^e, steps per epoch NsN^s, and batch size B 2: Initialize A θ_A, w θ_w, and C θ_C 3: for epoch=1,2,…,Ne=1,2,…,N^e do 4: for step=1,2,…,Ns=1,2,…,N^s do 5: Select a batch of input \h\ from the training dataset 6: Output p(|)p(A|h), A, and w corresponding to each training sample h 7: w←Adam(w,−∇wJ(A,w)) θ_w ( θ_w,- _ θ_wJ( θ_A, θ_w)) 8: A←Adam(A,−∇AJ(A,w)) θ_A ( θ_A,- _ θ_AJ( θ_A, θ_w)) 9: C←Adam(C,∇CLMSE(C)) θ_C ( θ_C, _ θ_CL_MSE( θ_C)) 10: end for 11: end for 2) Update of w θ_w: We assume that U(,;)U(b,w;h) is differentiable with respect to w and consequently with respect to w θ_w [14, 15, 16, 17, 18, 19, 21, 24, 25]. Thus, we can compute the gradient of J(A,w)J( θ_A, θ_w) with respect to w θ_w by ∇wJ(A,w)=,∼p(∣)[∇wU(,;)]. _ θ_wJ( θ_A, θ_w)=E_h,A p(A ) [ _ θ_wU (b,w;h ) ]. (29) Correspondingly, w θ_w is updated by a mini-batch stochastic gradient ascent algorithm using the Adam optimizer. The joint training algorithm is summarized in Algorithm 1, where the negative gradient is adopted in line 7 and line 8 to set a gradient ascent update in the Adam optimizer. After training, ℱA(⋅)F_A(·) and ℱw(⋅,⋅)F_w(·,·) with parameters A θ_A and w θ_w are used to predict b and w given any system parameter h. Remark 1: To enhance the stability of the joint training, we can alternatively pretrain ℱw(⋅,⋅)F_w(·,·) according to line 7 of Algorithm 1, where the input A corresponding to a given h can be randomly generated. V Case Study 1: Joint UE-AP Association and Beamforming in CF Systems Figure 4: The network architecture in Example 1. In this section, we apply our proposed DL framework to solve a widely encountered wireless resource allocation problem: joint UE-AP association and beamforming in CF systems, as described in Example 1 of Section I. We first detail the design of ℱA(⋅)F_A(·) and ℱw(⋅,⋅)F_w(·,·), and then present simulations to demonstrate the superiority of the proposed DL framework compared with existing approaches. The whole N architecture of our proposed framework for this case study is illustrated in Fig. 4. V-A ℱA(⋅)F_A(·) Design Details 1) Design of the encoder: We need to design the encoder E(⋅)G_E(·) as shown in (20), which takes =[11T,12T,…,1LT,21T,…,KLT]Th=[h^T_11,h^T_12,…,h^T_1L,h^T_21,…,h^T_KL]^T as the input, and outputs the embeddings =[11,12,…,1L,21,…,KL]R=[r_11,r_12,…,r_1L,r_21,…,r_KL]. To this end, we model the CF system as a graph, which includes two types of nodes, i.e., K UE-nodes and L AP-nodes, and there exists an edge between each UE-node and each AP-node. Thus, E(⋅)G_E(·) can be designed as a GNN. Specifically, we adopt the edge-node GNN (ENGNN) [13] for its ability to jointly process edge and node features, offering greater expressive power. In the ENGNN implementation, we denote the k-th UE-node feature and l-th AP-node feature at the lEl_E-th (lE=0,1,…,LEl_E=0,1,…,L_E) hidden layer as UE,k[lE]f^[l_E]_UE,k and AP,l[lE]f^[l_E]_AP,l, and the feature of their corresponding edge as kl[lE]e^[l_E]_kl. The initial edge features are obtained by kl[0]=MLP1E(kl),∀k∈,l∈ℒ,e_kl^[0]=MLP_1^E(h_kl),~~∀ k ,l , (30) where MLP1EMLP^E_1 is an MLP, and the node features AP,l[0]f_AP,l^[0] and UE,k[0]f_UE,k^[0] are initialized as zero. Then, the node and edge features are updated by the node and edge update strategy, which is detailed in Appendix A. After LEL_E update layers, the final edge features serves as the required embedding, i.e., kl=kl[LE]r_kl=e_kl^[L_E]. 2) Design of the decoder: We design the context embedding network ℱctx(⋅,⋅,⋅)F_ctx(·,·,·) as shown in (21), which takes t−1A_t-1, R, and h as the input, and outputs a context vector t⋆c_t . To extract the CF network-level features, it is natural to employ a GNN as the backbone of ℱctx(⋅,⋅,⋅)F_ctx(·,·,·). Moreover, to incorporate the structural information t−1A_t-1 into the N, unlike in the encoder E(⋅)G_E(·) where all edges are treated homogeneously, we distinguish between two types of edges. Specifically, we categorize edges as associated edges, i.e., ∀(k,l)∈t−1∀(k,l) _t-1, and candidate edges, i.e., ∀(k,l)∉t−1∀(k,l) _t-1, and employ distinct learnable parameters to extract their features. The detailed update equations are provided in Appendix B. After computing the context vector t⋆c _t, the compatibility score for each candidate edge is computed via (24), where the function ATT(⋅,⋅)ATT(·,·) is achieved via a cross-attention mechanism, which is expressed as ATT(kl,t⋆)=C⋅tanh(tTkldh),ATT(r_kl,c _t)=C· ( q_t^Tk_kl d_h ), (31) where t=Qt⋆q_t=W_Qc _t and kl=Kklk_kl=W_Kr_kl with trainable matrices Q,K∈ℝdh×dhW_Q,W_K ^d_h× d_h, and the compatibilities are clipped within [−C,C] [-C,C ] with a hyper-parameter C. In this case study, the total cardinality T of the support set A is upper bounded by minKLmax,LKmax \KL_ ,LK_ \, according to constraints (5) and (6). To allow the possibility of the model to terminate before reaching this upper bound, we introduce the end token β described at the end of Section I-A. 3) Design of the critic-network: In training period, the critic-network U^(⋅) U(·) is introduced to assist the training of ℱA(⋅)F_A(·), which takes the channel h as the input, and outputs the estimated sum rate. Its architecture closely mirrors that of E(⋅)G_E(·), with the only difference in the output layer. The output layer of U^(⋅) U(·) is expressed as U^()= U(h)= ReLU[1K∑k=1KFC1C(~UE,k[LC])+1L∑l=1LFC2C(~AP,l[LC]) [ 1K _k=1^KFC_1^C( f_UE,k^[L_C])+ 1L _l=1^LFC_2^C( f_AP,l^[L_C]) . (32) +1KL∑k=1K∑l=1LFC3C(~kl[LC])], .+ 1KL _k=1^K _l=1^LFC_3^C( e_kl^[L_C]) ], where FC1C(⋅)FC_1^C(·)-FC3C(⋅)FC_3^C(·) are single-layer fully connected networks, LCL_C denotes the number of hidden layers of the critic-network, ~UE,k[LC] f_UE,k^[L_C], ~AP,l[LC] f_AP,l^[L_C], and ~kl[LC] e_kl^[L_C] correspond to the features of the k-th UE-node, the l-th AP-node, and their associated edge features at the final hidden layer, respectively, and the ReLU activation is applied to ensure the non-negative output. V-B ℱw(⋅,⋅)F_w(·,·) Design Details We need to design ℱw(⋅,⋅)F_w(·,·), which takes h and A as the input, and outputs the beamforming variable w. In particular, we adopt an architecture nearly identical to that of ℱctx(⋅,⋅,⋅)F_ctx(·,·,·), differing only in the first and last layers. Specifically, the first layer of ℱw(⋅,⋅)F_w(·,·) can be adjusted from (45), by removing klr_kl from the input and replacing t−1A_t-1 with A. The node features are initialized as zero. After LWL_W layers of updates (similar to (46)-(50)), the beamformer w is obtained by [ℜklT,ℑklT]T=FCW(^kl[LW]), [ \w_kl^T\, \w^T_kl\]^T=FC^W( e_kl^[L_W]), (33a) kl←Pmaxbklklmax∑k′=1Kbk′l‖k′l‖2,Pmax,∀k∈,l∈ℒ. _kl← P_maxb_klw_kl \ _k =1^Kb_k l\|w_k l\|^2,P_ \,~∀ k ,l . (33b) where ^kl[LW] e_kl^[L_W] denotes the edge feature between the k-th UE and the l-th AP at the final hidden layer, FCW(⋅)FC^W(·) denotes a single-layer fully connected network, and (33b) implements a projection layer that enforces the per-AP power constraints in (9b). Remark 2: Although an optimal beamforming structure of CF systems can be derived [35], which avoids the learning of the mapping by a N, it entails an a matrix inversion of dimension LMLM. This incurs prohibitively high computational complexity when LMLM is large. Consequently, we opt to learn the mapping directly using a N without leveraging this solution structure. V-C Numerical Results In this subsection, we provide numerical results to demonstrate the benefits of the proposed DL framework for joint UE-AP association and beamforming in CF systems. 1) Simulation setting, choices of hyper-parameters, and benchmarks: The considered CF system consists of L=8L=8 APs and K=20K=20 UEs. Each AP is equipped with M=4M=4 antennas and each UE has a single antenna. The maximum number of serving UEs of each AP is set as Kmax=6K_ =6, and each user is served with at most Lmax=2L_ =2 APs. The APs and UEs are randomly distributed at a 500 m×500 m500 m× 500 m square area. The large-scale fading coefficients are generated according to the path-loss model 30.5+36.7log10Dkl+Fkl30.5+36.7 _10D_kl+F_kl in dB, where DklD_kl is the distance in meters between the k-th UE and the l-th AP, and Fkl∼(0,4)F_kl (0,4) denotes the shadow fading component. The small-scale channel is generated from uncorrelated Rayleigh fading. The noise power of each UE is set as σk2=−100 dBmσ^2_k=-100 dBm. The hyper-parameters of the proposed DL framework are summarized as follows. All MLPs in our designed NNs have 22 hidden layers, each with a fully connected layer, a batch normalization layer, and a ReLU activation. The encoder E(⋅)G_E(·) has LE=2L_E=2 update layers, and the dimension of the hidden representation is set as dh=128d_h=128. The context embedding network ℱctx(⋅,⋅,⋅)F_ctx(·,·,·) has LX=2L_X=2 update layers. The N-based ℱw(⋅,⋅)F_w(·,·) has LW=2L_W=2 hidden layers with dimension of hidden representation 128128. The critic-network has LC=6L_C=6 update layers. The hyper-parameter C in (31) is set as 88. In joint training, the number of training samples and validation samples are 5120051200 and 10241024, respectively. The number of training epochs is set as 200200 with a batch size of 10241024. The learning rate is set as 10−410^-4. After training, the performance is evaluated on 10241024 test samples, and in the testing stage, ata_t is determined by a greedy selection, i.e., at=argmax(k,l)∈×ℒp(at=(k,l)|t−1,)a_t=argmax_(k,l)∈K×L~p(a_t=(k,l)|A_t-1,h). The implementation of all simulations (including the compared methods) is under the PyTorch version 2.1.2+cu11.8, operating on an NVIDIA GeForce GTX 4090 GPU. (a) Sum rate performance comparison under different power budgets (versus DL-based methods). (b) Sum rate performance comparison under different power budgets (versus model-based methods). (c) The average computation time comparison. Figure 5: Sum rate and computation time comparison. Figure 6: The sum rate performance with different context embedding network designs. For comparison, we provide the simulation results of the following benchmarks, including DL-based approaches and model-based approaches. The DL-based approaches include: • Straight-Through-Estimator (STE): This method employs an ENGNN to output discrete UE–AP association variables and continuous beamforming variables in parallel [21]. The discrete association variables are handled by STE. To satisfy the association constraints in (5), we deactivate extra links of those overloaded APs during the inference stage. • Gumbel-Softmax: This approach also utilizes an ENGNN to generate association and beamforming variables in parallel [25]. It further incorporates the Gumbel-Softmax reparameterization technique to approximate discrete sampling with differentiable operations [23]. Extra links to overloaded APs are deactivated during inference to satisfy the constraints in (5). We also consider the following model-based approaches: • Greedy+WMMSE: This approach first determines the UE-AP association via a greedy method [6]. When a UE is accessed, it is served by LmaxL_ APs with the largest channel gains. If the selected AP already has KmaxK_ serving users, then the UE will be associated with the AP with next strongest channel gain. Given the association variable, the beamforming is designed by the the WMMSE algorithm [35], with the maximum number of iterations set to 5050 to ensure convergence. • Greedy+P-RZF: This method includes greedy-based UE-AP association and the partial regularized zero-forcing [36] beamforming. 2) Performance Evaluation: We first evaluate the sum rate performance of our proposed DL framework against DL-based baseline methods. In Fig. 5(a), we show the sum rate performance against other DL-based approaches under different power budgets. It can be seen that the proposed DL framework consistently outperforms all baselines across all power levels. In particular, unlike STE, which treats discrete variables as continuous and relies on the surrogate gradient, our approach directly handles discrete variables, thereby avoiding the gradient mismatch inherent in STE and achieving superior performance. A similar performance degradation occurs in Gumbel-Softmax, due to the approximation error of discrete output by the Gumble-softmax function. These approximation-based strategies struggle when the association variables are highly coupled and exhibit strong combinatorial structure, leading to severe performance degradation, e.g., over 30%30\% loss in sum rate in high-SNR regimes. Furthermore, in contrast to existing DL-based methods that can only handle the SPSD mappings [24], our framework explicitly possesses the non-SPSD property, enabling more expressive modeling and thus delivering superior sum rate performance. We next evaluate the sum rate performance against model-based baselines, as shown in Fig. 5(b). The proposed DL framework significantly outperforms the Greedy heuristic, as the scheduling strategy becomes too complex to be handled by heuristic approaches. Moreover, as transmit power increases, interference becomes dominant over noise, making effective interference management essential. We can see that the proposed approach excels in such high-interference environments. Furthermore, we compare the computational complexity of different methods, with results shown in Fig. 5(c). We can see that our proposed DL framework achieves significantly faster inference speed than iterative optimization–based baselines Greedy+WMMSE due to its computationally efficient feed-forward architecture. Next, we investigate the effect of changing the architecture design. In particular, in one implementation, we replace the proposed GNN-based context embedding network by an attention–based network, while all other components of the framework remain unchanged. Furthermore, we also experimented with replacing the ℱA(⋅)F_A(·) with the greedy algorithm, but keep the ENGNN-based beamforming as described in Subsection B. As shown in Fig. 6, the proposed GNN-based design achieves higher sum rates than the attention-based approach, owing to its ability to effectively model edge features and thus offer greater expressive power [13]. Nevertheless, both designs significantly outperform the Greedy baseline, particularly at high-SNR regimes where interference management becomes critical. Figure 7: Average sum rate and association rate on the validation set versus training epoch when Pmax=10dBmP_ =10dBm. Finally, we examine the role of the end token in enabling the framework to learn an appropriate stopping point during the sequential UE-AP association process. In Fig. 7, we set Pmax=10dBmP_ =10dBm, and plot the average sum rate and the average association rate on the validation set over the number of training epochs. The average association rate is defined as the obtained cardinality T divided by its upper bound, i.e., T/minLKmax,KLmaxT/ \LK_ ,KL_ \. As shown Fig. 7, the association rate is low initially, indicating that the end token is triggered early in the sequence. As training progresses and the agent explores more effective association strategies, both the association rate and the sum rate increase significantly. This shows that the end token learns to determine an effective stopping point dynamically, rather than filling up the set A until the upper bound determined by the constraints. VI Case Study 2: Joint Antenna Positioning and Beamforming in MA-aided Systems In this section, we employ our DL framework to address another wireless resource allocation problem: joint antenna positioning and beamforming for MA-aided systems, as described in Example 2 of Section I. VI-A ℱA(⋅)F_A(·) Design Details 1) Design of the encoder: We need to design the encoder E(⋅)G_E(·), which takes =[1T,2T,…,KT,1T,2T,…,NT]Th=[h_1^T,h_2^T,…,h_K^T,p_1^T,p_2^T,…,p_N^T]^T as the input, and outputs the embeddings =[1,2,…,N]R=[r_1,r_2,…,r_N]. To this end, we model the MA-aided system as a graph, which includes two types of nodes, i.e., K UE-nodes and N CP-nodes, and there exists an edge between each UE-node and each CP-node. We also adopt the ENGNN [13] as the foundation architecture of E(⋅)G_E(·), where the k-th UE-node feature and n-th CP-node feature of the lEl_E-th hidden layer are expressed as UE,k[lE]f^[l_E]_UE,k and CP,n[lE]f^[l_E]_CP,n, and the corresponding edge feature is represented by kn[lE]e^[l_E]_kn. The edge features and CP-node features are initialized as kn[0]=MLP1E(hkn),∀k∈,n∈, _kn^[0]=MLP^E_1 (h_kn ),~∀ k ,n , (34a) CP,n[0]=MLP2E(n),∀n∈, ^[0]_CP,n=MLP^E_2 (p_n ),~~∀ n , (34b) where MLP1EMLP_1^E and MLP2EMLP_2^E are 2 MLPs, and the UE-node feature UE,k[0]f^[0]_UE,k is initialized as zero. Using the ENGNN update equations (42)-(44), after LEL_E hidden layers, the updated CP-node feature CP,n[LE]f^[L_E]_CP,n serves as the required embedding nr_n. 2) Design of the decoder: We design the context embedding function ℱctx(⋅,⋅,⋅)F_ctx(·,·,·) which takes t−1A_t-1, h, and R as the input, and outputs the context vector t⋆c _t. First, when t>2t>2, we perform information aggregation to extract a preliminary graph-level feature, expressed as t= _t= MLP2X(1t−1∑t′=1t−1MLP1X(at′), _2^X ( 1t-1Σ _t =1^t-1MLP_1^X(r_a_t ), . (35) 1N∑n=1NMLP4X(n,1K∑k=1KMLP3X(hkn))), . 1N _n=1^NMLP_4^X (p_n, 1K _k=1^KMLP^X_3 (h_kn ) ) ), where MLP1XMLP^X_1-MLP4XMLP^X_4 are 4 MLPs. When t=1t=1, as t−1=∅A_t-1= , we use a trainable parameter ⋆∈ℝdhr ^d_h to replace the first input term of MLP2XMLP^X_2. Next, we employ a multi-head attention (MHA) module to extract the deeper contextual features and refine the context vector. The MHA module calculates relevance scores between tc_t and the embeddings of CPs, and integrates these score to enhance the representation of tc_t. The specific update equations are omitted due to space limitations. Given t⋆c _t, the compatibility score for each CP is computed via (24), where the function ATT(⋅,⋅)ATT(·,·) is achieved similar to (31). In this case study, the total number of decoding steps can be determined as T=MT=M because of the constraint (12). 3) Design of critic-network: The critic-network’s architecture closely mirrors that of E(⋅)G_E(·), with the only difference in the output layer. The output layer of U^(⋅) U(·) is expressed as U^() U(h) =ReLU[1K∑k=1KFC1C(~UE,k[LC])+1N∑n=1NFC2C(~CP,n[LC])] =ReLU [ 1K _k=1^KFC_1^C( f_UE,k^[L_C])+ 1N _n=1^NFC_2^C( f_CP,n^[L_C]) ] (36) +1KN∑k=1K∑n=1NFC3C(~kn[LC])], .+ 1KN _k=1^K _n=1^NFC_3^C( e_kn^[L_C]) ], where FC1C(⋅)FC^C_1(·)-FC3C(⋅)FC^C_3(·) are single-layer fully-connected networks, LCL_C denotes the number of hidden layers of the critic-network, and ~UE,k[LC] f_UE,k^[L_C], ~CP,n[LC] f_CP,n^[L_C] , and ~kn[LC] e_kn^[L_C] correspond to the features of the k-th UE-node, the l-th CP-node, and their associated edge features at the final hidden layer, respectively. VI-B ℱw(⋅,⋅)F_w(·,·) Design Details In this subsection, we introduce the design of ℱw(⋅,⋅)F_w(·,·). First, we note the optimal solution of the beamformer in this case study [37] k= w_k= pk(M+∑i=1Kμiσk2i()i()H)−1k()‖(M+∑i=1Kμiσk2i()i()H)−1k()‖, p_k (I_M+ _i=1^K _i _k^2h_i(b)h_i(b)^H )^-1h_k(b) \| (I_M+ _i=1^K _i _k^2h_i(b)h_i(b)^H )^-1h_k(b) \|, (37) ∀k∈, ~∀ k , where ≜[μ1,μ2,…,μK]T∈ℝ+K μ [ _1, _2,…, _K]^T _+^K and ≜[p1,p2,…,pK]T∈ℝ+K p [p_1,p_2,…,p_K]^T _+^K are unknown parameters and should satisfy ∑k=1Kμk=∑k=1Kpk=Pmax _k=1^K _k= _k=1^Kp_k=P_max such that (16b) can be satisfied. Exploiting this solution structure can simplify the mapping to be learned by the N, by decreasing the number of unknown parameters from 2KM2KM to 2K2K. The remaining task is to predict these low-dimensional parameters. For this purpose, we adopt the ENGNN, which consists of two types of nodes, i.e., M MA-nodes and K UE-nodes, and there exists an edge between each MA-node and each UE-node. The edge features are initialized as ^km[0]=MLP1W([k()]m),∀k∈,m∈ℳ, e^[0]_km=MLP^W_1([h_k(b)]_m),~∀ k ,~m , (38) and the node features ^MA,m[0] f^[0]_MA,m and ^UE,k[0] f^[0]_UE,k are initialized as zero. After LWL_W layers node and edge updating (similar to (42)-(44)), we obtain μ and p by [μk,pk]T=FCW(^UE,k[LW]),∀k∈, [μ_k,p_k]^T=FC^W( f^[L_W]_UE,k),~~∀ k , (39a) ←Pmax⋅softmax(), μ← P_ ·softmax( μ), (39b) ←Pmax⋅softmax(), p← P_ ·softmax( p), (39c) where FCW(⋅)FC^W(·) is a single-layer fully-connected network, and the softmax activation is employed to normalize μ and p. Finally, μ and p are substituted to (37) to get w. VI-C Numerical Results 1) Simulation setting and choices of hyper-parameters: We consider an MA-aided system where the BS is equipped with [4,9][4,9] MAs to serve 44 single-antenna UEs. The size of the 2D rectangular transmit area is 2λ×2λ2λ× 2λ, where each side is uniformly sampled with [5,8][5,8] points, resulting in 25,36,49,64\25,36,49,64\ CPs, and the wavelength λ=60 mλ=60 m. The minimum distance between every two MAs is set as dmin=30 mmd_min=30 m. The distance between the k-th UE and the BS, denoted as DkD_k, is uniformly distributed within [100,200][100,200] in meters. The noise power is set as −100-100 dBm. We consider the field-response channel model[38], where the channel between the n-th CP and the k-th UE is expressed as hkn=LpTkkn,∀k∈,n∈, h_kn=1_L_p^T _kg_kn,~~∀ k ,~n , (40) where Lp∈ℝLp1_L_p ^L_p is the all-one field response vector (FRV), and Lp=16L_p=16 denotes the path number. The diagonal matrix k≜diag[ηk,1,ηk,2,…,ηk,Lp]T _k \ [ _k,1, _k,2,…, _k,L_p ]^T \, where ηk,lp _k,l_p denotes the response coefficient of the lpl_p-th path to the k-th UE, and it is independently and identically distributed as (0,L0Dk−α)CN (0,L_0D_k^-α ) with L0=34.5L_0=34.5 dB and α=3.67α=3.67, respectively. Moreover, kng_kn denotes the transmit FRV between the n-th CP and the k-th UE, which is given by kn=[ejρk,1n,ejρk,2n,⋯,ejρk,Lpn]T, _kn= [e^j _k,1^n,e^j _k,2^n,·s,e^j _k,L_p^n ]^T, (41) where ρk,lpn=2π/λ[(xn−x1)cosθk,lpsinϕk,lp+(yn−y1)sinθk,lp] _k,l_p^n=2π/λ[ (x_n-x_1 ) _k,l_p _k,l_p+(y_n-y_1) _k,l_p] represents the phase difference between the n-th CP and the first CP at the lpl_p-th path, and θk,lp _k,l_p and ϕk,lp _k,l_p represent the elevation angle of departure (AoD) and the azimuth AoD of the lpl_p-th path. The probability density function of AoDs is fAoD(θk,lp,ϕk,lp)=cosθk,lp/2π,θk,lp∈[−π/2,π/2],ϕk,lp∈[−π/2,π/2]f_AoD (θ_k,l_p, _k,l_p )= θ_k,l_p/2π,~θ_k,l_p∈ [-π/2,π/2 ],~ _k,l_p∈ [-π/2,π/2 ]. For the proposed N model, all the MLPs are implemented by 2 linear layers, each followed by a ReLU activation function. The encoder E(⋅)G_E(·) has LE=3L_E=3 layers, with dh=128d_h=128. The hyper-parameter is set to C=8C=8. The critic-network has LC=6L_C=6 update layers. Moreover, ℱw(⋅,⋅)F_w(·,·) has LW=3L_W=3 hidden layers, each with a hidden dimension of 6464. In the training procedure, the number of epochs is set to 100100, where each epoch consists of 5050 mini-batches with a batch size of 10241024. A learning rate 10−410^-4 is adopted to update the trainable parameters through Algorithm 1 using the Adam optimizer. The GPU and environment for training and testing is the same as in Section V. 2) Benchmarks: We consider four baseline methods for comparison: • Random+WMMSE: This method involves randomly selecting M CPs. If the selected CPs violate the constraints in (13), the selection is repeated until the constraints are satisfied. Subsequently, with the MA positions determined, the beamformer is computed using the WMMSE algorithm[35]. • Greedy+WMMSE: For this approach, the average channel gain of each CP with respect to all K UEs is first calculated and treated as the equivalent channel gain for that CP. The CPs exhibiting the strongest channel gains are then selected iteratively over M steps, with CPs violating (13) masked in each step. Following the MA positioning, the beamformer is obtained via the WMMSE algorithm. • Greedy+ZF: This method combines greedy algorithm for determining the MA position, and the zero-forcing beamforming [39]. • FP-C: We modify an approach that jointly optimizes continuous antenna positioning and beamforming for MA-aided systems [9]. To obtain discrete antenna positions that satisfy the discrete constraints in (13), in each iteration of FP-C, we project the position of each MA to its nearest discrete CP that satisfies (13) sequentially. To the best of our knowledge, existing DL approaches cannot effectively handle the distance constraints in (13); therefore, we do not include DL-based benchmarks. (a) The sum rate performance comparison under different power budgets. (b) The sum rate performance comparison under different M’s. (c) The sum rate performance comparison under different N’s. Figure 8: The sum rate performance comparison under different power budgets, M’s, and N’s. 3) Performance Evaluation: We first compare the sum rate performance under different power budgets. In Fig. 8(a), with M=6M=6 and N=49N=49, it can be observed that Random+WMMSE and Greedy+ZF perform badly due to their heuristic antenna positioning and beamforming strategies, respectively. Moreover, the proposed DL framework outperforms all baselines particularly at higher transmit power. This demonstrates the great capability of the proposed framework in mitigating interference, particularly when inter-user interference becomes stronger. Next, we evaluate the performance of the proposed DL framework under different system settings. We first set Pmax=20P_max=20 dBm and N=49N=49, and plot the sum rate against M in Fig. 8(b). As observed, the proposed DL framework achieves highest sum rates across different values of M. Next, in Fig. 8(c), we plot the sum rate against N when Pmax=20P_max=20 dBm and M=6M=6. We can see that the proposed DL framework achieves superior performance across different values of N. In contrast to heuristic positioning methods such as Greedy and Random that perform poorly, our proposed method achieves the best performance no matter when the number of CPs is limited, or when the number of CPs is sufficient but with more challenging constraints. Finally, we compare the average computation time of various approaches. It can be observed from Fig. 9 that the computation time of the proposed DL framework is substantially shorter than the iterative optimization-based methods, including WMMSE and FP-C. Figure 9: The average computation time comparison when M=6M=6 and N=49N=49. VII Conclusion In this paper, we have proposed a DL framework for solving a general class of mixed-discrete wireless resource allocation problems with discrete constraints. A general problem formulation has been established and a DVLN with probabilistic formulation has been designed to handle discrete variables. It employs an encoder-decoder architecture with a sequential decoding, enabling it to rigorously enforce the coupled discrete constraints while satisfying the non-SPSD property on discrete solutions. With a CVLN developed to predict the solutions of continuous variables, the DVLN and CVLN are jointly trained in an unsupervised manner. Simulation results on two representative case studies have demonstrated the superior performance with significantly less computation time compared to existing model-based and DL-based approaches. Appendix A Update Mechanism of ENGNN The ENGNN comprises two types of nodes: transmitter (TX) nodes and receiver (RX) nodes, with cardinalities NTN_T and NRN_R, respectively. Define the TX-node feature and the RX-node feature in the l-th layer as TX,nt[l]f_TX,n_t^[l] and RX,nr[l]f_RX,n_r^[l], and there corresponding edge features as ntnr[l]e_n_tn_r^[l]. At the l-th layer, the node and edge features are updated by TX,nt[l]= _TX,n_t^[l]= MLP2[l](TX,nt[l−1],1NR∑nr=1NRMLP1[l](RX,nr[l−1],ntnr[l−1])), _2^[l] (f_TX,n_t^[l-1], 1N_R _n_r=1^N_RMLP_1^[l] (f_RX,n_r^[l-1],e_n_tn_r^[l-1] ) ), (42) ∀n=1,…,NT, ∀ n=1,…,N_T, RX,nr[l]= _RX,n_r^[l]= MLP4[l](RX,nr[l−1],1NT∑nt=1NTMLP3[lE](TX,nt[l−1],ntnr[l−1])), _4^[l] (f_RX,n_r^[l-1], 1N_T _n_t=1^N_TMLP_3^[l_E] (f_TX,n_t^[l-1],e_n_tn_r^[l-1] ) ), (43) ∀nr=1,…,NR, ∀ n_r=1,…,N_R, ntnr[l]= _n_tn_r^[l]= MLP7[l](ntnr[l−1],1NT∑n′=1NTMLP5[l](n′nr[l−1],RX,nr[l−1]), _7^[l] (e_n_tn_r^[l-1], 1N_T _n =1^N_TMLP_5^[l] (e^[l-1]_n n_r,f_RX,n_r^[l-1] ), . (44) 1NR∑n′=1NRMLP6[l](ntn′[l−1],TX,nt[l−1])), . 1N_R _n =1^N_RMLP_6^[l] (e_n_tn ^[l-1],f_TX,n_t^[l-1] ) ), ∀nt=1,…,NT,nr=1,…,NR. ~∀ n_t=1,…,N_T,~n_r=1,…,N_R. where MLP1[l]MLP_1^[l] - MLP7[l]MLP_7^[l] are 7 MLPs. Appendix B Design of ℱctxF_ctx for Example 1 In the t-th decoding step, the edge features of ℱctx(⋅,⋅,⋅)F_ctx(·,·,·) are initialized as ¯kl,t[0]=MLP1X(kl,kl),∀(k,l)∈t−1,MLP2X(kl,kl),∀(k,l)∉t−1, split e_kl,t^[0]&= casesMLP_1^X (h_kl,r_kl ),~∀(k,l) _t-1,\\ MLP_2^X (h_kl,r_kl ),~∀(k,l) _t-1, cases\\ split (45) where MLP1XMLP_1^X and MLP2XMLP_2^X are two distinct MLPs, and the node features ¯UE,k,t[0] f_UE,k,t^[0] and ¯AP,l,t[0] f_AP,l,t^[0] are initialized as zero. Furthermore, the node and edge features are updated as ¯UE,k,t[lX]=MLP3X,[lX](¯UE,k,t[lX−1],1L∑l=1Lkl,t[lX]),∀k∈, f_UE,k,t^[l_X]=MLP_3^X,[l_X] ( f_UE,k,t^[l_X-1], 1L _l=1^Lg_kl,t^[l_X] ),~∀ k , (46) ¯AP,l,t[lX]=MLP4X,[lX](¯AP,l,t[lX−1],1K∑k=1Kkl,t[lX]),∀l∈ℒ, f_AP,l,t^[l_X]=MLP_4^X,[l_X] ( f_AP,l,t^[l_X-1], 1K _k=1^Kn_kl,t^[l_X] ),~∀ l , (47) ¯kl,t[lX]=MLP5X,[lX](¯kl,t[lX],¯UE,k,t[lX],¯AP,l,t[lX]),∀(k,l)∈t−1,MLP6X,[lX](¯kl,t[lX],¯UE,k,t[lX],¯AP,l,t[lX]),∀(k,l)∉t−1, split e_kl,t^[l_X]&= casesMLP_5^X,[l_X] ( e_kl,t^[l_X], f^[l_X]_UE,k,t, f^[l_X]_AP,l,t ),~∀(k,l) _t-1,\\ MLP_6^X,[l_X] ( e_kl,t^[l_X], f^[l_X]_UE,k,t, f^[l_X]_AP,l,t ),~∀(k,l) _t-1, cases\\ split (48) where MLP3X,[lX]MLP^X,[l_X]_3-MLP6X,[lX]MLP^X,[l_X]_6 are 4 MLPs, and kl,t[lX]g_kl,t^[l_X] and kl,t[lX]n_kl,t^[l_X] denote the messages aggregated from edges to nodes, which are computed as kl,t[lX]=MLP7X,[lX](¯AP,l,t[lX−1],¯kl,t[lX−1]),∀(k,l)∈t−1,MLP8X,[lX](¯AP,l,t[lX−1],¯kl,t[lX−1]),∀(k,l)∉t−1, splitg_kl,t^[l_X]= casesMLP_7^X,[l_X] ( f^[l_X-1]_AP,l,t, e_kl,t^[l_X-1] ),∀(k,l) _t-1,\\ MLP_8^X,[l_X] ( f^[l_X-1]_AP,l,t, e_kl,t^[l_X-1] ),∀(k,l) _t-1, cases\\ split (49) kl,t[lX]=MLP9X,[lX](¯UE,k,t[lX−1],¯kl,t[lX−1]),∀(k,l)∈t−1,MLP10X,[lX](¯UE,k,t[lX−1],¯kl,t[lX−1]),∀(k,l)∉t−1, splitn_kl,t^[l_X]= casesMLP_9^X,[l_X] ( f^[l_X-1]_UE,k,t, e_kl,t^[l_X-1] ),~∀(k,l) _t-1,\\ MLP_10^X,[l_X] ( f^[l_X-1]_UE,k,t, e_kl,t^[l_X-1] ),~∀(k,l) _t-1, cases\\ split (50) where MLP7X,[lX]MLP^X,[l_X]_7-MLP10X,[lX]MLP^X,[l_X]_10 are 4 additional MLPs. The associated and candidate edges are modeled as distinct edge types, each processed by dedicated MLPs to capture their unique characteristics. Finally, after LXL_X layers, the graph-level context vector is obtained by aggregating node and edge representations: t⋆= _t= 1K∑k=1KFC1X(¯UE,k,t[LX])+1L∑l=1LFC2X(¯AP,l,t[LX]) 1K _k=1^KFC_1^X( f_UE,k,t^[L_X])+ 1L _l=1^LFC_2^X( f_AP,l,t^[L_X]) (51) +1KL∑k=1K∑l=1LFC3X(¯kl,t[LX]), + 1KL _k=1^K _l=1^LFC_3^X( e_kl,t^[L_X]), where FC1X(⋅)−FC3X(⋅)FC_1^X(·)-FC_3^X(·) are single-layer fully connected networks. References [1] M. Guenach, A. A. Gorji, and A. Bourdoux, “Joint power control and access point scheduling in fronthaul-constrained uplink cell-free massive MIMO systems,” IEEE Trans. on Commun., vol. 69, no. 4, p. 2709–2722, 2021. [2] C. Huang, S. Hu, G. C. Alexandropoulos, A. Zappone, C. Yuen, R. Zhang, M. D. Renzo, and M. Debbah, “Holographic MIMO surfaces for 6G wireless networks: Opportunities, challenges, and trends,” IEEE Wireless Commun., vol. 27, no. 5, p. 118–125, 2020. [3] K.-K. Wong, A. Shojaeifard, K.-F. Tong, and Y. Zhang, “Fluid antenna systems,” IEEE Trans. on Wireless Commun., vol. 20, no. 3, p. 1950–1962, 2021. [4] L. Zhu, W. Ma, B. Ning, and R. Zhang, “Movable-antenna enhanced multiuser communication via antenna position optimization,” IEEE Trans. on Wireless Commun., vol. 23, no. 7, p. 7214–7229, 2024. [5] Y.-F. Liu, T.-H. Chang, M. Hong, Z. Wu, A. Man-Cho So, E. A. Jorswieck, and W. Yu, “A survey of recent advances in optimization methods for wireless communications,” IEEE J. Sel. Areas Commun., vol. 42, no. 11, p. 2992–3031, 2024. [6] S. Buzzi, C. D’Andrea, L. Wang, A. Hasim Gokceoglu, and G. Peters, “Co-existing/cooperating multicell massive MIMO and cell-free massive mimo deployments: Heuristic designs and performance analysis,” IEEE Open J. of the Commun. Soc., vol. 5, p. 6180–6200, 2024. [7] M. Sanjabi, M. Razaviyayn, and Z.-Q. Luo, “Optimal joint base station assignment and beamforming for heterogeneous networks,” IEEE Trans. on Signal Process., vol. 62, no. 8, p. 1950–1961, 2014. [8] K. Shen and W. Yu, “Fractional programming for communication systems—part I: Uplink scheduling via matching,” IEEE Trans. on Signal Processing, vol. 66, no. 10, p. 2631–2644, 2018. [9] Z. Cheng, N. Li, J. Zhu, X. She, C. Ouyang, and P. Chen, “Sum-rate maximization for fluid antenna enabled multiuser communications,” IEEE Commun. Lett., vol. 28, no. 5, p. 1206–1210, 2024. [10] Y. Wu, D. Xu, D. Wing Kwan Ng, W. Gerstacker, and R. Schober, “Globally optimal movable antenna-enabled multiuser communication: Discrete antenna positioning, power consumption, and imperfect CSI,” IEEE Trans. on Commun., vol. 73, no. 10, p. 9903–9923, 2025. [11] Y. Yang, Y. Gong, and Y.-C. Wu, “Intelligent-reflecting-surface-aided mobile edge computing with binary offloading: Energy minimization for iot devices,” IEEE Internet of Things J., vol. 9, no. 15, p. 12 973–12 983, 2022. [12] —, “Energy-sensitive binary offloading for reconfigurable-intelligent-surface-assisted wireless-powered mobile-edge computing,” IEEE Internet of Things J., vol. 11, no. 7, p. 11 593–11 605, 2024. [13] Y. Wang, Y. Li, Q. Shi, and Y.-C. Wu, “ENGNN: A general edge-update empowered GNN architecture for radio resource management in wireless networks,” IEEE Trans. on Wireless Commun., vol. 23, no. 6, p. 5330–5344, 2024. [14] J. Guo and C. Yang, “Learning power allocation for multi-cell-multi-user systems with heterogeneous graph neural networks,” IEEE Trans. on Wireless Commun., vol. 21, no. 2, p. 884–897, 2022. [15] S. Liu, J. Guo, and C. Yang, “Multidimensional graph neural networks for wireless communications,” IEEE Trans. on Wireless Commun., vol. 23, no. 4, p. 3057–3073, 2024. [16] Z. Liu, Y. Li, Y.-C. Wu, and Y. Gong, “Learning to optimize resource allocation in dynamic wireless environments: Embracing the new while engaging the old,” IEEE Trans. on Wireless Commun., vol. 24, no. 9, p. 7346–7359, 2025. [17] Y. Li and Y.-F. Liu, “HPE transformer: Learning to optimize multi-group multicast beamforming under nonconvex QoS constraints,” IEEE Trans. on Commun., vol. 72, no. 9, p. 5581–5594, 2024. [18] Y. Li, Z. Chen, Y. Wang, C. Yang, B. Ai, and Y.-C. Wu, “Heterogeneous transformer: A scale adaptable neural network architecture for device activity detection,” IEEE Trans. on Wireless Commun., vol. 22, no. 5, p. 3432–3446, 2023. [19] Y. Leng, Q. Lin, L.-Y. Yung, J. Lei, Y. Li, and Y.-C. Wu, “Unveiling the power of complex-valued transformers in wireless communications,” IEEE Trans. on Commun., vol. 74, p. 612–627, 2026. [20] A. Van Den Oord, O. Vinyals, and K. Kavukcuoglu, “Neural discrete representation learning,” in Advances in Neural Inf. Process. Syst. (NeurIPS), 2017. [21] Y. Li, Y. Liu, and W. Yu, “Multimodal visual image based user association and beamforming using graph neural networks,” IEEE Trans. on Wireless Commun., 2025, doi: 10.1109/TWC.2025.3582887. [22] J. Park, F. Sohrabi, J. Du, and J. G. Andrews, “Self-nomination: Deep learning for decentralized csi feedback reduction in MU-MIMO systems,” IEEE Trans. on Wireless Commun., vol. 25, p. 10 321–10 336, 2026. [23] E. Jang, S. Gu, and B. Poole, “Categorical reparameterization with gumbel-softmax,” in Int. Conf. on Learn. Representations (ICLR), 2017. [24] S. Liu, C. Yang, and S. Han, “Learning wideband user scheduling and hybrid precoding with graph neural networks,” IEEE Trans. on Wireless Commun., 2025, doi: 10.1109/TWC.2025.3590424. [25] Q. Lyu and M. Vu, “Joint beamforming and integer user association using a GNN with gumbel-softmax reparameterizations,” IEEE Trans. on Veh. Technol., 2025, doi: 10.1109/TVT.2025.3576051. [26] F. Liang, C. Shen, W. Yu, and F. Wu, “Towards optimal power control via ensembling deep neural networks,” IEEE Trans. on Commun., vol. 68, no. 3, p. 1760–1776, 2020. [27] Q. Chen, L. Li, Q. Li, J. Wu, A. Wang, R. Sun, X. Luo, T.-H. Chang, and Q. Shi, “When GNNs meet symmetry in ILPs: an orbit-based feature augmentation approach,” in Int. Conf. on Learn. Representations (ICLR), 2025. [28] Y. Li, Y.-F. Liu, F. Xu, Q. Shi, and T.-H. Chang, “Learning to optimize qos-constrained beamforming in multi-user systems: A penalty-dual framework,” IEEE Trans. on Wireless Commun., vol. 23, no. 11, p. 16 123–16 138, 2024. [29] C. D’Andrea and E. G. Larsson, “User association in scalable cell-free massive MIMO systems,” in Asilomar Conf. on Signals, Syst., and Comput., 2020. [30] Y. Jin, Q. Lin, Y. Li, H. Zhu, B. Cheng, Y.-C. Wu, and R. Zhang, “A general optimization framework for tackling distance constraints in movable antenna-aided systems,” IEEE Trans. on Wireless Commun., vol. 25, p. 10 869–10 885, 2026. [31] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin, “Attention is all you need,” in Advances in Neural Inf. Process. Syst. (NeurIPS), 2017. [32] W. Kool, H. Van Hoof, and M. Welling, “Attention, learn to solve routing problems!” in Int. Conf. on Learn. Representations (ICLR), 2019. [33] V. R. Konda and J. N. Tsitsiklis, “Actor-critic algorithms,” in Advances in Neural Inf. Process. Syst. (NeurIPS), 1999. [34] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” arXiv:1412.6980, 2014. [35] S. S. Christensen, R. Agarwal, E. De Carvalho, and J. M. Cioffi, “Weighted sum-rate maximization using weighted MMSE for MIMO-BC beamforming design,” IEEE Trans. on Wireless Commun., vol. 7, no. 12, p. 4792–4799, 2008. [36] G. Interdonato, M. Karlsson, E. Björnson, and E. G. Larsson, “Local partial zero-forcing precoding for cell-free massive MIMO,” IEEE Trans. on Wireless Commun., vol. 19, no. 7, p. 4758–4774, 2020. [37] E. Björnson, M. Bengtsson, and B. Ottersten, “Optimal multiuser transmit beamforming: A difficult problem with a simple solution structure,” IEEE Signal Process. Mag., vol. 31, no. 4, p. 142–148, 2014. [38] L. Zhu, W. Ma, and R. Zhang, “Movable antennas for wireless communication: Opportunities and challenges,” IEEE Commun. Mag., vol. 62, no. 6, p. 114–120, 2024. [39] Q. Spencer, A. Swindlehurst, and M. Haardt, “Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels,” IEEE Trans. on Signal Process., vol. 52, no. 2, p. 461–471, 2004.