Joint Clustering and Prediction of the Quality of Service in Vehicular Cellular Networks Oscar Stenhammar⋆†, Gábor Fodor⋆† , Carlo Fischione⋆ ⋆
KTH Royal Institute of Technology, Sweden. E-mail: ostenh|sraz|gaborf|[email protected] †
Ericsson Research, Sweden. E-mail: oscar.stenhammar|[email protected]
arXiv:2604.12903v1 [cs.NI] 14 Apr 2026
Abstract Machine learning models are increasingly deployed in wireless networks with stringent performance requirements. However, dynamic propagation environments and fluctuating traffic densities introduce concept drift, which complicates the ability to maintain accurate predictive machine learning models. We propose a distributed optimization framework that jointly clusters cells and trains cluster-level predictive models, enabling nodes to cooperatively predict quality of service (QoS) distributions under communication constraints. The proposed method models QoS as a multivariate Gaussian/lognormal distribution and uses a novel clustering mechanism that groups cells with similar network conditions, allowing each cell to select the most appropriate predictor without retraining new models for each cell. By leveraging block coordinate descent, our solution efficiently clusters the cells and updates the predictive models to mitigate concept drift, while maintaining a compact model set to minimize computation overhead. Evaluation using data from realistic simulations with the Sionna ray-tracer and the ns-3 simulator shows that the method converges and yields cluster constellations that adapt to changes in the network that cause concept drift. The experimental evaluation focuses on providing a prediction of the distribution latency, jitter, and RSRP over a one-hour prediction horizon. The proposed method significantly outperforms the traditional single global predictive model approach and reduces the mean absolute error by 9–27% compared to local cell-level predictors. This demonstrates that the proposed method effectively captures local variability using far fewer models through scalable distributed clustering.
I. I NTRODUCTION HE transition toward the 6th generation (6G) in wireless communication systems is partly driven by emerging applications that demand quality of service (QoS) improvements compared to 5G [1], [2], [3]. For example, automated connected users with high mobility impose requirements on low latency, sufficient throughput, and extremely high reliability. In vehicular applications such as teleoperated driving, collision avoidance, or platooning, excessive communication interruptions may directly compromise safety. Network operators therefore offer service level agreements (SLAs) that guarantee predefined QoS levels. In this context, predicting future QoS enables proactive and risk-aware scheduling in the radio access network (RAN). By anticipating SLA violations due to congestion or channel variations, the network can allocate resources in advance rather than reacting after a breach has occurred [4], [5]. This motivates new architectural and algorithmic solutions across the RAN [6]. Wireless applications with the need for proactive service management, and the rapid advancements of machine learning (ML), have generated great research interest in the development of predictive QoS (pQoS) for next-generation networks [7], [8], [9]. The pQoS algorithms utilize ML to predict the QoS, which forms the foundation of proactive service management. With the correct setup, ML has the ability to make accurate and reliable QoS predictions. Non-independent and identically distributed (IID) data, which frequently arise in wireless communication settings, can substantially degrade the prediction accuracy for ML algorithms [10]. In vehicular communication networks, dynamic environmental factors such as propagation environments, urban infrastructure, and traffic density lead to concept drift in the received QoS [10], [11]. The term concept drift has various definitions in the literature. In this paper, we adopt the definition of concept drift from [12], which states that ”Concept drift is a phenomenon in which the statistical properties of a target domain change over time in an arbitrary way.” In wireless networks, there can be various types of concept drift, including sudden, gradual, incremental, and recurring drift. These drifts cause models trained on past data to become misaligned with current conditions. In the context of data-driven pQoS ML models, concept drift can significantly deteriorate predictive accuracy, since ML relies on training from actual data [12], [13]. For example, consider a highway road segment where the pQoS model was trained under moderate traffic load and stable channel conditions. Over time, traffic density increases due to seasonal commuting patterns, leading to higher interference levels, more frequent handovers, and stronger fluctuations in signal-to-interference-plus-noise ratio (SINR). These spatiotemporal shifts alter the underlying mapping from features (e.g., reference signal received power (RSRP), channel quality, user speed) to QoS outcomes (e.g., reliability, latency, jitter), meaning that the previously learned model no longer reflects the current network dynamics. This exemplifies concept drift in a pQoS context and highlights the need for adaptive learning mechanisms in pQoS systems [12]. An important aspect of pQoS mechanisms is their prediction horizon, the time interval into the future for which the QoS is predicted [14]. Short-term predictions, ranging from milliseconds to a few seconds, enable the network to react to rapid channel fluctuations by proactive handover decisions and adaptive modulation and coding. Medium-term prediction horizons, on the order of tens of seconds to minutes, enable operators to anticipate traffic volume peaks and optimize resources across cells.
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Long-term predictions, ranging from hours to days, can support strategic network management functions and infrastructure planning. By selecting an appropriate prediction horizon, or by combining multiple horizons, pQoS can serve both real-time control and broader operational planning. This strengthens pQoS as a key enabler of proactive management in future 6G networks. Needless to say, the prediction task becomes more challenging as the prediction horizon increases. A. Literature Review The majority of prior work on pQoS for wireless networks relies on a single global model [10], [15], which is defined as a predictive model trained centrally using aggregated data collected from all devices across a wide geographical region. Such a model is typically trained offline using historical data aggregated at a central server to update model weights periodically, rather than in real-time. The model is then deployed at the base stations (BSs) to predict future QoS outcomes from locally observed network features. This centralized training paradigm assumes that a single global model can adequately capture the heterogeneous wireless conditions experienced throughout the network and performs well in relatively stable environments where stationarity assumptions approximately hold. However, a large single-model paradigm is inherently vulnerable to concept drift because of spatial and temporal dynamic conditions in smaller regions [13]. To address these limitations, more adaptive pQoS algorithms are necessary to mitigate concept drift and improve robustness. Strategies range from continual learning schemes that update models with fresh data, to domain adaptation approaches that detect and react to distributional shifts [14], [16]. These methods aim both to preserve predictive accuracy and to limit the operational cost of retraining in large-scale networks. Among the distributed solutions, a clustered version of federated learning (FL) has emerged as a promising direction [17], [18]. By grouping clients with similar data distributions and performing federated updates within clusters, these approaches reduce negative transfer across heterogeneous devices or locations while retaining the privacy and scalability benefits of FL. Clustered federated schemes can dynamically reassign clients as their local distributions evolve, which helps to contain concept drift across user populations [19], [20]. However, many of the clustered approaches focus QoS predictions on the user side instead of predicting the QoS on the network side [21], [22], which is where the network management algorithms run. Notably, much of the prior literature focuses on short-term QoS forecasting, where the QoS is treated as a single random variable [10]. Considerably less work has addressed longer prediction horizons or the prediction of full QoS distributions. Mid-term prediction efforts exist in previous work. For instance, some studies forecast QoS over horizons on the order of one minute [23]. As the prediction horizon increases, temporal autocorrelation weakens, reducing the usefulness of conventional time-series forecasting models. This motivates the need for probabilistic approaches that explicitly model the Gaussian distribution of future QoS rather than a single expected value [24], [25]. B. Contributions This paper takes the next step in pQoS by forecasting QoS Gaussian distributions over long-term prediction horizons and by introducing an adaptive, joint clustering and training framework tailored for cellular networks serving SLA users. Our main contributions are: • We propose a pQoS framework that groups cells with similar network conditions and learns a dedicated, cluster-specific probabilistic model for each group. The framework predicts the mean vector and covariance matrix of a d-dimensional Gaussian distribution over a prediction horizon in the magnitude of hours. • We formulate this problem as a joint clustering and prediction optimization over cluster assignments, neural network parameters, and the number of clusters. The problem is relaxed via a nuclear-norm regularization. The resulting problem is optimized using block coordinate descent (BCD), alternating between model parameter training with FL and cluster assignment updates based on a matrix of distance metrics, singular value thresholding (SVT), and simplex projection. • Under standard assumptions, such as L-smoothness and bounded loss, we prove convergence of the proposed BCD procedure to a stationary point. We demonstrate through simulations in heterogeneous deployments that our method converges and outperforms single-model and non-clustered baselines in both distributional accuracy and robustness. Together, these contributions provide a reliable approach for pQoS over long-term prediction horizons for cellular networks that must support SLAs. The remainder of the paper is organized as follows: The system model is defined in Section II. The original joint optimization problem to be solved is formulated in Section III. Section IV-A formulates a relaxed problem that is possible to solve, Section IV-B provides the algorithm of the proposed solution, and Section IV-C develops the theoretical analysis to prove convergence for the proposed algorithm. Numerical results to assess the proposed method’s performance are provided in Section V. The results are discussed in VI, and finally, Section VII summarizes and concludes the paper. II. S YSTEM M ODEL We consider a cellular network deployed over a geographical region comprising a set of N BSs, each serving individual geographical cells. The system considers user density that varies across the area, for instance, between urban and rural regions,
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Cluster c1
Cluster c2
Cluster c3
Fig. 1. An illustration of the system model. Due to mobile users moving, the scenario evolves between times t1 and t2 , resulting in changes in the load and interference in the network. This changes the distribution pq for a certain KPI Q in the different cells. Rather than performing expensive re-training of the existing pQoS models, we re-cluster the cells to ensure that each cell is associated with a cluster whose pQoS model is well aligned with the scenario of the cell and QoS distribution pq .
leading to spatially heterogeneous interference levels and network load. The region contains a road grid with diverse propagation environments, along which a set of vehicles travel. Each vehicle runs applications that rely on the cellular network, making reliable QoS predictions essential for ensuring the SLAs. To support these applications, the network operator seeks to manage resources proactively to ensure that SLAs can be met by relying on accurate QoS forecasts over a long-term prediction horizon. The goal of our approach is to efficiently capture spatial heterogeneity in the network by partitioning N cells from the network into a smaller number C ≤ N of clusters. The clusters are formed based on the similarity of the data distributions across cells. Each cluster will be assigned a predictive model. By optimizing the number of clusters, we can balance model complexity and predictive accuracy: too few clusters may fail to capture local variations in QoS, while too many clusters increase training and maintenance overhead. For each cluster, we train a predictive model that outputs the parameters of a multivariate distribution of the QoS metrics. We target the Gaussian distribution instead of an exact value since the QoS over a long-term prediction horizon is influenced by highly variable and stochastic factors, making point predictions difficult to obtain with high confidence. We target prediction horizons in the magnitude of hours. By producing distributional forecasts over such time horizons, the system supports proactive network management and more reliable decision-making. The vector ŷi representing the QoS distribution parameters in cell i can be predicted by ŷi (t + h) = ω(xi ; θc ),
(1)
using the predictive function ω with weights θc , specific for each cluster c. The cell-specific input data xi include network-, mobility-, packet-, and radio-level measurements collected during the most recent sampling period. It also contains the expectation vector and covariance matrix at time t in cell i. Note that yi (t + h) denotes the true distribution parameters at time t + h, while ŷi (t + h) reflects the predicted parameters. Observed QoS metrics typically arise from the aggregation of many independent random effects [26], which, by the central limit theorem, lead to either approximately a Gaussian distribution or a lognormal distribution [27]. If a QoS metric follows a lognormal distribution, which is often the case for latency as an example [26], applying a logarithmic transformation maps
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the metric to a Gaussian distribution, allowing both cases to be handled within a unified Gaussian modeling framework. This motivates our choice to consider the QoS to follow a Gaussian distribution. The predictive framework updates associated models with each cell locally at the corresponding BS using its own data. The cluster-level models are then updated through an FL procedure [28], in which BSs periodically share model gradients with a central server to produce an aggregated update without exchanging raw data. The update period τ is a design parameter that can be chosen based on system requirements such as network latency and available communication resources. In practice, it is typically defined by the network operator or system designer to balance convergence speed and communication overhead. Shorter periods allow faster adaptation to changing conditions, while longer periods reduce signaling and computational load at the BSs. In the literature of QoS and network performance prediction, neural network architectures with three or more hidden layers are frequently used to achieve sufficient model capacity [10]. While such depth improves modeling accuracy, it also substantially increases the number of trainable parameters and the associated training overhead. In the predictive function (1) of our system M×F model, each cluster c has a neural network ω parameterized by the weights θF , that is shared among all cells in c ∈ R that cluster. To reduce model complexity and training overhead, we restrict the cluster-specific parameters θc ∈ RM×1 to correspond only to the final layer of the neural network with weights θF c = {θbase , θ c }, which are pre-trained on a global dataset. Here, θ base ∈ RM×F −1 represents the first F − 1 layers. The pre-training dataset is global in the sense that the data is from all the BSs covering the entire considered geographical area. This design enables transfer learning, whereby knowledge learned from the global predictor is reused and adapted locally with a small number of trainable parameters, allowing efficient specialization to cluster-specific conditions while preserving the generalization capabilities of the global model [29]. While the full parameter set is given by {θbase , θc }, the weights θbase remain fixed during the cluster-specific updates. Therefore, for the sake of notational brevity, we denote the predictive model with θc as in (1), where the dependence on the constant base parameters θbase is implicit. In the following, we provide an illustrative example of how this system model is used. Example 1. A high-level illustration of the system model is provided in Fig. 1. We model the QoS Qi (t) ∈ Rd as a Gaussian random vector with d QoS metrics of interest for each cell i ∈ I at the current time t. To provide a concrete example in this section, Qi (t) is a vector consisting of latency Li and packet loss Pi that follows a bivariate Gaussian distribution: L Qi (t) = i ∼ N (µi , Σi ) ∀i = 1, ..., N. Pi
Here, Li ≥ 0, Pi ≥ 0. Note that we view the elements Li (t) and Pi (t) of vector Qi (t), and the parameters of the Gaussian 2 distribution µi (t) ∈ Rd and Σi (t) ∈ Rd , to be time dependent. However, we suppress the notation of time t in the variables for readability. The goal is to train the predictive model in (1) for each cluster that outputs the Gaussian distribution parameters µ(i,h) and Σ(i,h) for the future multivariate distribution Qi (t + h) ∼ N (µ(i,h) , Σ(i,h) ) over a time horizon h. By predicting the Gaussian ⊺ distribution, we can calculate the probability that a user in cell i receives at least the desired future QoS Qd = Ld Pd , where Ld is the allowed latency and Pd is the allowed packet loss rate. This is done by the following procedure: Pr(Li ≤ Ld , Pi ≤ Pd ) = Φ Ld , Pd ; µ̂(i, h) , Σ̂(i, h) ,
where Φ is a two-dimensional real vector’s cumulative distribution function (CDF) with expectation µ̂(i,h) and covariance Σ̂(i,h) . This can be calculated by the probability density function (PDF) of the predicted bivariate Gaussian distribution. We assume that µ̂(i, h) and Σ̂(i, h) are such that the probability Pr(Li < 0, Pi < 0) is negligible. This assumption is justified by the fact that the Gaussian approximation is applied in operating regimes where the predicted mean values are strictly positive and the estimation variance is small relative to the mean. Hence, the probability mass assigned to the non-physical negative region is insignificant. Building upon the system model, the next section defines the formal problem formulation addressed in this study. III. P ROBLEM F ORMULATION To construct C clusters from N cells, optimize the number of clusters C, and train a predictive model for each cluster that predicts the set of Gaussian distribution parameters, we formulate an optimization problem in this section. First, we introduce binary assignment variables aic : ( 1, if cell i is assigned to cluster c, aic = 0, otherwise. Here, each cell i must be assigned to exactly one cluster c: C X c=1
aic = 1,
∀i = 1, . . . , N.
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Depending on latency, scalability, and coordination requirements, this neural network can be deployed on edge servers based on BSs, centralized RAN controllers, cloud-based network management platforms, or even dedicated accelerator hardware integrated into open RAN (O-RAN). To formulate this problem, we define three loss terms to optimize in the following three subsections. A. Prediction Loss The prediction loss between the predicted ŷi (t + h) from (1) and the ground truth yi (t + h) is modeled by: C X N X c=1 i=1
aic ℓ ω(xi ; θc ), yi (t + h) .
We sum the loss functions ℓ(·, ·) over the number of clusters C and cells N to retrieve the total loss. For practical purposes, we choose the loss function to be the mean squared error (MSE). The binary cluster assignment variable aic ∈ {0, 1} is 1 if cell i is assigned to cluster c. Otherwise, aic = 0. In other words, each cell is counted only once in the sum of the prediction loss. B. Clustering Consistency Loss To encourage grouping of cells with similar data distributions, we penalize a high separation of data distributions among cells within the same cluster, using a distance measure D: λ
C X N X
c=1 i,j=1
aic ajc D Ni , Nj ,
where λ ≥ 0 is a weighting factor. For simplicity, we denote N (µi , Σi ) as Ni . If the distributions Ni and Nj are dissimilar, the outcome of the distance metric D Ni , Nj will be high. On the contrary, the distance metric is low if Ni = Nj . We will discuss suitable choices of the distance metric in Section IV-C and choose a distance metric in Section V-A. From the distance metric, we want to find the binary weights aic to cluster cells with similar data distributions. The weights ∗ aic are utilized to find the cluster sets {Sc }C c=1 from Sc = {i ∈ I | ci = arg maxc aic } that contains the cells i that belongs to cluster c. C. Cluster Number Penalty To optimize the number of clusters, we introduce the regularization term βC, where β > 0. Optimizing the number of clusters implies training fewer predictive models. By minimizing the number of predictive models, training and communication become more efficient. D. Optimization Problem Now we are in the position to formulate a structured training problem by adding all three terms defined in the Sections III-A– III-C above. We propose the following optimization problem: min
C, {aic }, {θc }
C X N X
aic ℓ ω(xi ; θc ), yi (t + h)
c=1 i=1 C X N X
+λ
c=1 i,j=1
s. t.
C X
aic = 1,
aic ajc D Ni , Nj + βC
(2)
∀i = 1, . . . , N,
c=1
aic ∈ {0, 1},
∀i = 1, . . . , N, c = 1, . . . , C.
Because this problem is non-convex, both due to the neural network ω with non-linear activation functions on weights θ c and the combinatorial nature of the clustering assignments, we propose a BCD approach to arrive at an approximate solution. One of the BCD blocks is solved at each BS separately, and the other BCD block is solved on a central server. In the following section, we present our solution to this problem and show how to perform such a challenging approach.
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IV. S OLUTION A PPROACH In this section, we address the computational challenges of the original optimization problem (2) by introducing a suitable relaxation that renders the formulation tractable while preserving its essential structure. Building on a relaxed problem, we then propose a BCD–based algorithm that decomposes the optimization into two solvable blocks. Finally, we analyze the convergence properties of the proposed method and establish conditions under which the algorithm converges. A. Problem Relaxation In order to obtain a formulation that allows us to derive an approximate solution, we relax optimization problem (2). But first, we need to make some definitions. We set Θ = [θ1 , . . . , θC ] ∈ RM×C where M is the length of the vectors θc with c = 1, ..., C, containing the weights for the predictive models. We define the matrix A ∈ RN ×C as A = [aic ]i=1,...,N, c=1,...,C . With these definitions, we can now relax (2). We do that by the following two steps: • The elements aic ∈ {0, 1} in A indicates whether cell i is in cluster c from our definition in Section III-A. Since aic is binary, it means that if two cells are in the same cluster, their corresponding rows are identical. Intuitively, this means that the rank of A, which counts the number of independent rows, is equal to the number of clusters C. For example, if we only have two sets of unique rows in matrix A, the cells will be placed in only 2 different clusters. Such an observation suggests a relaxation of 2 by replacing the number of clusters C in the last term of (2) with the rank of A. However, this would make the problem combinatorial and NP-hard [30], because the rank function is non-convex and discontinuous. To circumvent this difficulty, we replace the number of clusters C in the last term of (2) with the nuclear norm kAk∗ , defined as the sum of the singular values of A. This convex relaxation is a common relaxation of the rank in the literature, because the number of non-zero singular values is equal to the rank [31]. • The original assignment constraints in (2) enforce a hard clustering, requiring aic ∈ {0, 1}. This discrete constraint makes the optimization problem combinatorial, non-convex, and generally intractable for large N and C. To address this, we relax the binary constraints by allowing the assignment variables aic to lie in the probability simplex. In this relaxed formulation, aic can be interpreted as the likelihood that cell i belongs to cluster c. The final discrete cluster assignment is obtained by selecting the cluster with the largest probability for each cell i by finding c∗i = arg maxc aic . For each ∗ max cluster, we find the cluster sets {Sc }C c=1 from Sc = {i ∈ I | ci = arg maxc aic } that contains the cells i that belongs to cluster c. The number of clusters are then found by computing C = |{c ∈ {1, . . . , Cmax } : Sc 6= ∅}|. With the relaxations defined above, we now modify (2). The expression for the relaxed optimization problem becomes: min
A, Θ
F (A, Θ) =
C X N X c=1 i=1
+λ
C X N X
c=1 i,j=1
s. t. aic ≥ 0,
C X
aic ℓ ω(xi ; θc ), yi (t + h)
aic ajc D Ni , Nj + βkAk∗ ,
aic = 1,
(3)
∀i = 1, . . . , N,
c=1
Optimization problem (3) is a continuous but still nonconvex optimization problem because the assignments aic multiply the per-cluster regression losses. Furthermore, computing a global minimizer of the relaxed formulation remains computationally intractable because the formulation can be classified as k-means and similar NP-hard clustering problems [19], [32]. However, (3) is block-multiconvex. The Θ-subproblem is convex for fixed A, and the A-subproblem is convex for fixed θ (when choosing D properly, which we will show in Section IV-C). This justifies the use of BCD, as we show next. In order to map optimization problem (3) to the BCD framework in [33], we first define the functions N C X X aic ℓ ω(xi ; θc ), yi (t + h) + g(A), (4) f (A, Θ) = c=1 i=1
g(A) = λ
C X N X
c=1 i,j=1
aic ajc D Ni , Nj ,
rA (A) = βkAk∗ . M×C
(5) (6)
N ×C
Again, for clarification, Θ ∈ R and A ∈ R . Now, using vector notation, we can find an alternative way to write F (A, Θ) in (3). Let D ∈ RN ×N with [D]ij = D(Ni , Nj ), and define vectors ℓc , ac ∈ RN by [ℓc ]i = ℓ(ω(xi ; θc ), yi (t + h)) and [ac ]i = aic . Using D, ℓc , and ac , we can simplify F (A, Θ) in (3) as F (A, Θ) =
C X c=1
a⊤ c ℓc + λ
C X c=1
a⊤ c D ac + βkAk∗ .
(7)
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By the definitions (4)-(7), we can implement the non-trivial BCD formulation from [33], as presented in the following subsection. B. Implementation of Block Coordinate Descent To approach the objective function F (A, Θ) in (7), we use BCD. In the following subsections, we define the two blocks to jointly optimize A and Θ in (3). 1) Block 1: We define the problem of block 1 as o n (8) Θk = arg min f (Ak−1 , Θ) + αΘ kΘ − Θk−1 k2 . Θ
k−1 Here, 2αΘ = LΘ denotes the step size for each update of Θk−1 . By solving (8), we update the weights Θk . We now calculate the gradient for each predictive model with weights θ c ∈ Θ. The first-order optimality condition leads to
(θc − θck−1 ) = 0. ∇θc f (Ak−1 , θc ) + Lθk−1 c By solving for θ kc , we get θkc = θck−1 −
1 Lθk−1 c
∇θc f (Ak−1 , θck−1 ).
(9)
. The update in (9) is performed locally at each BS This corresponds to a gradient descent step with a step size of 1/Lθk−1 c to obtain the model weights θkc,i , using only the data associated with cell i. To leverage information across the cells in a cluster, a cluster model is aggregated centrally via FL, where the updated parameters from all participating cells in a cluster are combined to form the new cluster model parameters 1 X k θc,i . (10) θkc = |Sc | i∈Sc
where Sc is the set of cells contributing to cluster c. 2) Block 2: Initially, we set C = Cmax = N . Furthermore, we define the nuclear norm as the regularization function rA (A) = kAk∗ . We aim to optimize the following expression: o n (11) Ak = arg min f (A, Θk ) + αA kA − Ak−1 k2 + kAk∗ , A
k−1 LA
As in block 1, we define 2αA = as the step size. The first and third terms in (11) represent solving the relaxed optimization problem (3), while the second term controls how fast we update A. To solve this, we compute the gradient ∇aic f (Ak−1 , Θk ) = ℓ(ω(xi ; θkc ), yi (t + h)) + 2λ
N X j=1
and perform a gradient step descent: k−1 k = aic − zic
k−1 ajc D Ni , Nj ,
1 ∇a f (Ak−1 , θ kc ). LkA ic
k The values zic are computed locally at each BS i. They are then transmitted to the central server to form the temporary matrix k Z as k . Zk = zic i=1,...,N,c=1,...,C
After the gradient step, we apply singular value thresholding (SVT) [34] to optimize the nuclear norm of A. We update the BCD step as Ak = Sτ Zk . The function Sτ (X) is the SVT operator:
Sτ (X) = Sτ (UΣVT ) = UΣτ VT . Here, X = UΣVT is the singular value decomposition of X, and Sτ applies soft-thresholding to the singular values σ, which are located on the diagonal of Σ : στ = max(σ − τ, 0). After we have retrieved Ak , we employ Dykstra’s projection algorithm to map each row of Ak onto the probability simplex [35], thereby maintaining the probabilistic validity of the cluster assignment. The algorithm iteratively projects the
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Algorithm 1 BCD for joint clustering and prediction. 0
k C 1: Input: λ, τ , Θ , A0 , step sizes Lk Θ , LA , {Sc }c=1 , maximum iterations K.
2: for k = 1, 2, . . . , K do 3: Block 1: Update Neural Network Weights Θ 4: 5:
6:
for c = 1, . . . , C at cell i if i ∈ Sc do Compute gradient at BS i:
∇θ c fi (Ak−1 , θck−1 ).
Update θkc,i at each BS i: θkc,i = θck−1 −
7: 8: 9:
end for Transmit all θkc,i to the central server. Aggregate the distributed model updates: θkc =
1 ∇θ fi (Ak−1 , θck−1 ). LkΘ c
1 X k θc,i . |Sc | i∈Sc
10: 11:
Block 2: Update Cluster Assignment Matrix A Compute gradient step at each BS i: ∇aic f (Ak−1 , Θk ) = ℓ(ω(xi ; θkc ), yi (t + h)) + 2λ
N X j=1
12:
k Update zic locally at BSs i, ∀c: k−1 k zic = aic −
13:
14: 15:
k−1 ajc D Ni , Nj .
1 ∇a f (Ak−1 , θkc ). LkA ic
k Transmit zic from BSs i to the central server and form k . Zk = zic i=1,...,N,c=1,...,C
Compute Zk = UΣVT . Apply thresholding to singular values:
Στ = max(Σ − τ, 0). 16: 17: 18:
Reconstruct Ak = UΣτ VT . Project Ak onto probability simplex by Dykstra’s [35]. Compute the new cluster set Sc for each cluster c: Sc = {i ∈ I | c∗i = arg max aic }. c
19: end for K
20: Output: Θ
, AK , {Sc }C c=1 .
unconstrained updates onto the valid range in (3), ensuring that aic stays between 0 and 1 and the rows of Ak sum to unity. This projection effectively finds the closest valid Ak to the BCD updates in the Euclidean sense. max Finally, the cluster assignments in Ak are used to set c∗i = arg maxc aic . We find the cluster sets {Sc }C c=1 from Sc = {i ∈ ∗ I | ci = arg maxc aic } that contains the cells i that belongs to cluster c. The number of clusters are then found by computing C = |{c ∈ {1, . . . , Cmax } : Sc 6= ∅}|. That concludes the BCD algorithm. The complete algorithm from both blocks is provided in Algorithm 1, with update period τ to keep the model parameters and cluster assignments updated to the dynamic environment. In the next subsection, we study under which assumptions Algorithm 1 converges.
C. Convergence Analysis To prove the convergence of Algorithm 1, we first have to make some standard assumptions in the literature [33], [36].
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Assumption 1. (L-smoothness) The function f (A, Θ) in (4) is L-smooth if for every bounded set B, there exists a constant LB > 0 for all (A, Θ), (A′ , Θ′ ) ∈ B such that: k∇f (A, Θ) − ∇f (A′ , Θ′ )k ≤ LB k(A, Θ) − (A′ , Θ′ )k.
(12) Assumption 2. (Bounded Loss Function) There exists a constant ℓ∗ such that ℓ ω(xi ; θc ), yi (t + h) ≥ ℓ∗ for all θ c and xi .
Assumption 3. (Convexity) The function g : RN ×C → R ∪ {+∞} defined in (5) is convex, lower semi-continuous, and proper in the sense of convex analysis. Assumption 4. (Block update parameter bounds) The parameters LkA used in the block steps satisfy uniform bounds 0 < LA ≤ LkA ≤ LA < ∞ for all k. The same goes for the parameters LkΘ . Assumption 5. (Lower bounded) The objective function F (A, Θ) is bounded below on its effective domain: inf F (A, Θ) > −∞.
(A,Θ)
Moreover, the initial point (A0 , Θ0 ) satisfies F (A0 , Θ0 ) < +∞. From Assumption 3, we need g(A) in (5) to be convex, lower semi-continuous, and proper. Hence, we need to make sure that block 2 is convex. Proposition 1. Let Assumptions 1 and 5 hold. Then, Assumption 3 holds if the matrix D with entries Dij = D Ni , Nj is positive semidefinite (PSD). Proof. The proof of Proposition 1 is provided in Appendix. With Assumptions 1–5 and Proposition 1, we are ready to present the main analytical result. In the following proposition, we prove that Algorithm 1 converges to a critical point of the function F (A, Θ). Proposition 2. Let (Ak , Θk )∞ k=1 be the sequence generated by Algorithm 1. Under Assumptions 1-5, the sequence asymptotically converges to a critical point of the function F (A, Θ), i.e., for à = limk→∞ Ak and Θ̃ = limk→∞ Θk , we have ∂L(Ã, Θ̃) = 0.
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Proof. The proof of Proposition 2 is provided in Section A. We note that Algorithm 1 returns a sub-optimal solution to (3). Since the optimization problem is an approximation of the original problem (2), the solution produced by Algorithm 1 can be interpreted as an approximate, feasible solution to (2). D. Computational–Communication Complexity In the proposed clustered framework, we use FL [28] as follows: each BS computes local model updates using its own dataset. These updates are periodically transmitted to a central server, which aggregates them to update the cluster-level models. The update period τ is a design parameter that trades off convergence speed against communication overhead. Once updated, the cluster-level models are redistributed to the BSs belonging to each cluster. To contextualize the proposed approach, we compare it with a global approach in which a single model is trained on a central server, and a local approach in which each cell maintains and trains its own predictive model. In the local approach, all computation is performed on-site and no model parameters or gradients are exchanged, resulting in zero communication overhead. In contrast, both the global and our proposed clustered approaches require backhaul communication from the BSs to the server. Assuming the same update frequency of 15 minutes and number of local training epochs, the bandwidth consumption up in the backhaul communication for the global model is Bglobal = N F b = 12 · 108172 · 4 ≈ 5.2MB, where F is the number of model parameters and b is the number of bytes that represent each scalar parameter. For the cluster approach, the same up communication cost is Bcluster = N Fℓ b = 12 · 1548 · 4 ≈ 74KB, where Fℓ is the number of model parameters for the last layer. The numbers of model parameters from the MLP are calculated by simple arithmetic from the basics of neural networks. The ratio F/Fℓ ≈ 70 reveals that the upstream traffic is reduced by a factor of 70. In the downstream communication of the global approach, a single updated model is broadcast to all BSs with a down communication cost of Bglobal = F b = ·108172 · 4 ≈ 430KB. In the clustered approach, C distinct cluster models are multicast from the central server to the BSs. Consequently, the downlink communication cost scales linearly with the number down of clusters C, leading to the communication cost Bclustered = CFℓ b = 6 · 1548 · 4 ≈ 37KB if we assume a constellation of 6
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clusters. This motivates keeping C as small as possible while still capturing spatial heterogeneity in the network. We see that the ratio of the communication cost between the clustered approach and the global is F/(CFℓ ) ≈ 12. When it comes to the computational complexity, the global approach updates all model parameters, resulting in a per-round training complexity O(EF Strain ) where the total number of parameters F , the number of epochs E, and the total number of local training samples Strain . In the clustered approach, the computational complexity is O(EFℓ Strain + N C 2 ) assuming that the number of epochs and total number of training samples are identical to the global approach. The N C 2 term comes from the singular value decomposition (SVD) in the clustered approach. Assuming the same update frequency, this leads to a reduction in arithmetic complexity in our proposed clustered approach, the global training. This reduction is particularly significant for deep models where the last layer constitutes a small fraction of the total number of parameters. V. N UMERICAL E VALUATION To evaluate the proposed framework, we generate a dataset using a combination of the ns-3 and Sionna simulators to obtain realistic data. The two upcoming subsections describe how the dataset is generated and present the numerical results of our predictive framework based on the simulated dataset. A. Experimental Setup To evaluate the proposed pQoS framework, we conduct extensive wireless network simulations featuring high-mobility vehicular users in a realistic deployment scenario. For the purpose of reproducibility, the simulation environment is available in our public repository1. The simulations capture both spatial and temporal variations in network conditions and build on a previous integration of the ns-3 network simulator [37] with the Sionna [38] link-level ray-tracing engine [39]. The Sionna environment enables the incorporation of realistic propagation characteristics into the ns-3 discrete network simulator by replacing the standard channel models with ray-tracing. We define vehicular mobility traces representative of typical city traffic patterns in the predefined Munich scene from Sionna. We deploy 30 connected vehicles along the defined mobility traces. Each vehicle acts as a user equipment (UE), connecting to the cell with the strongest signal. To reflect realistic variations in network load over a day, the traffic is modeled as a timedependent function over 24 hours. For example, a low network traffic background intensity is applied during early morning hours, a moderate background load during daytime, and peak demand during rush hours with high background. We utilize a channel quality-aware scheduler that supplies throughput fairness. To further capture the effect of network heterogeneity, user mobility speeds are varied between 30 km/h and 100 km/h. This dynamic mobility profile allows us to analyze the robustness of the pQoS models under varying conditions. The 12 cells are placed at inter-site distances to cover the entire map with reasonable connectivity. The carrier frequency is set to 3.5 GHz with a bandwidth of 20 MHz, and the maximum transmit power of each cell is fixed at 30 dBm. To collect a dataset with network data from different layers, we integrate a FlowMonitor logging system with a sampling period of 1 second. The FlowMonitor is a network monitoring package within ns-3 that can retrieve samples from the simulation stack at regular intervals across all active flows. We record a set of network, mobility, and radio-level measurements that characterize the communication downlink conditions and the user context. The logs include the current timestamp, identifiers for the active data flow and serving cell, and the instantaneous load of that cell. User mobility is captured through the UE’s geographical position, speed, and movement direction. Packet-level statistics comprise the packet size, the number of packets transmitted from the cell, and the number of packets received at the UE during the sampling interval. The packet-level statistics also include average end-to-end latency, throughput, packet loss rate, and jitter over the sampling period. Radio conditions are described by the measured SINR, RSRP, and an indicator of line-of-sight or non-line-of-sight propagation. The predictive model is implemented as a multilayer perceptron (MLP) designed to predict the future multidimensional distribution of key QoS indicators within a given cell over the prediction horizon of 1 hour. For each prediction cycle, the neural network receives as input all network-, mobility-, packet-, and radio-level measurements collected during the most recent sampling period. These features provide a detailed description of the instantaneous operating conditions, reflecting traffic load, user movement, packet performance, and radio propagation quality. To incorporate a long-term temporal context, the input also includes the parameters of a fitted multidimensional Gaussian distribution that summarizes the joint behavior of latency, jitter, and RSRP in the same cell over the preceding 15-minute period. Together, these inputs represent both short-term dynamics and historical trends that are relevant for a long-term prediction horizon. The output of the neural network is the predicted joint distribution of latency, jitter, and RSRP 1 hour into the future for the same cell. The dataset for each cell and hour, containing around 10,000 samples, is split into 70% training, 10% validation, and 20% test sets. The predictive MLP is implemented with three hidden layers of 256, 256, and 128 neurons using ReLU activations and 20% dropout. The model is trained using the Adam optimizer with learning rate 10 −3, batch size 32, MSE loss, and a fixed random seed for reproducibility. Furthermore, we set the update period τ = 15 minutes. 1 https://github.com/osst3224/ns3-rt-mobility
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