Federated Spectrum Learning for Reconfigurable Intelligent Surfaces-Aided Wireless Edge Networks

In the depicted context, several new research challenges emerge for the conventional FL process:

• •

  Challenge-1: Before the start of local training at each iteration of the conventional FL process, the edge server first needs to perform resource allocation (such as users selection and bandwidth allocation) by solving the corresponding optimization problem. Due to the existence of shadowing and multi-path fading in radio environments, some users may not be able to reliably send their requests to the base station (BS) via the direct links, which may be severely degraded and then unreliable. In this case, the BS cannot perform the resource allocation well since some of the requests may be missed.

• •

  Challenge-2: In conventional FL, the users who finish the local training send via wireless links their local models to the BS for global model aggregation. Due to the possible unreliability of the wireless direct links and the limited wireless bandwidth, the convergence of FL may be degraded because of the low efficiency of uploading the local models. In this case, the accuracy of FL prediction may be degraded since some of the local models may be received unsuccessfully at the BS.

To address the aforementioned challenges in conventional FL implementations, emerging and advanced technologies such as terahertz (THz) communications and ultra-massive multiple-input-multiple-output (MIMO), can be exploited to improve the wireless transmission. However, excessive hardware costs and energy consumption pose design challenges due to the relatively high operating frequency of THz communications and the large number of RF chains required. In recent years, reconfigurable intelligent surfaces (RISs), benefited from breakthroughs on the fabrication of programmable metasurfaces, have emerged as a promising technology for improving the quality of wireless links and for configuring the wireless environment by, e.g., appropriately reflecting the incident signals with the aid of a large number of nearly-passive reflecting elements with low power consumption [12, 13, 14, 15, 16]. Recently, innovative RIS technologies improving wireless communications with low cost and energy consumption have been proposed [17, 18]. For example, holographic MIMO surfaces (HMIMOS) attracted great research attention as a possible solution for realizing massive MIMO systems at a lower cost and power consumption [19, 20]. In addition, to collect users’ training requests subject to deep fading conditions, cooperative RF spectrum sensing constitutes a promising approach for improving the sensing performance by capitalizing on the spatial sensing diversity of distributed sensors [21, 22, 23]. Motivated by these considerations, it is desirable to address the two mentioned challenges of FL by integrating FL with RISs, thereby achieving cooperative RF spectrum sensing (to improve the performance of optimization) and enhancing the quality of the wireless links (to improve the accuracy of FL prediction) [24].

Recently, some researchers have investigated the potential benefits of using RISs for improving the performance of FL [25, 26]. In these papers, however, the considered system scenarios are limited to a single-RIS setup, and the user-RIS association problem was not relevant. A more complex and general FL system has been investigated in [27], where multiple RISs are deployed for performance enhancement. However, none of the above existing works explored the interplay between FL and RISs. Thus, this motivates us to design a novel federated spectrum learning (FSL) framework, thereby achieving efficient spectrum learning for wireless edge networks.

In wireless networks, the potential opportunities offered by the distributed FL paradigm have not been fully exploited, mainly because of the straggler effect and the unreliability of wireless channels under limited wireless resources [28]. Due to the unreliability of wireless channels, in particular, it may be difficult to receive correct local updates at the edge, especially when the wireless channel undergoes deep fading. As a result, the convergence performance of FL may be degraded. In this context, it is imperative to improve the FL performance from the communications perspective [28, 29, 30, 31]. Specifically, Chen et al. [28] investigated a joint learning, wireless resource allocation, and user selection problem in FL by taking into account the wireless link packet errors and the availability of wireless resources. Samarakoon et al. [29] introduced FL to help with the joint design of power control and resource allocation for application to vehicular communication networks. To mitigate the straggler effect, Yang et al. [30] proposed a fast global model aggregation approach to improve the performance and the convergence rate of FL via over-the-air computation (AirComp). Since uploading of the local parameters at every iteration is often inefficient in FL, Wang et al. [31] presented a control algorithm that determines the best tradeoff between local update and global aggregation so as to minimize the loss function of FL.

Unlike other transmission technologies such as active relay, RISs can help to reflect the incident signals through adjusting the phase shifts of their scattering elements smartly, thereby improving the wireless transmissions [32]. In order to explore the benefits of RISs in wireless networks, designing RIS-aided wireless communication systems based on deep learning techniques has recently received major attention [33]. In particular, Hu et al. [34] considered a metasurface assisted RF sensing method which can sense the locations of objects in a 3D space, and proposed a deep reinforcement learning (DRL) algorithm to solve the formulated optimization problem. Huang et al. [35] analyzed the joint design of transmit beamforming at the base station and phase shifts at the RIS to maximize the sum rate utilizing DRL. Cao et al. [36] investigated an RIS-assisted multi-user downlink aerial-terrestrial communication system via multi-task learning. Yang et al. [37] proposed an RIS-assisted anti-jamming solution for securing wireless communications via RL. Although some works focused on deep learning for RIS systems, e.g., reflecting beamforming matrix optimization [34, 37, 35, 36] and channel estimation [38, 39, 40], endowing RISs with active learning capabilities for data-driven wireless networks still presents many open issues to be explored and addressed.

In this paper, our goal is to propose a novel FSL framework to improve the accuracy of FL with the aid of RISs, as well as to endow conventional RISs with spectrum sensing capabilities, thereby improving the system performance in a cooperative way, especially in radio environments subject to deep fading channels. In this context, our major contributions are summarized as follows:

1. 1.

   We develop a novel FSL framework for RISs-aided wireless edge networks by investigating the interplay between FL and RISs. Based on collected RF data traces, the mobile users train their local models and then upload them to an edge server via RIS-assisted links, thereby achieving a fast yet reliable model aggregation. At the beginning of each iteration, in addition, the aggregated model is deployed at each RIS controller to help the BS collect the requests of users by cooperatively sensing the spectrum, thereby improving the performance, especially in complex shadowing and fading radio environments.

2. 2.

   We jointly optimize the user-RIS association, the phase shifts configuration and the wireless bandwidth allocation. In the considered FSL framework subject to limited wireless resources and a limited number of RISs, specifically, we formulate a joint user-RIS association, phase shifts design, and wireless bandwidth allocation problem, whose goal is to maximize the FL system utility. To solve the user-RIS association problem, we propose a matching game-based association scheme in which the users who achieve larger gains in terms of achievable SNR have higher probabilities to be associated with an RIS. The optimized phase shifts guarantee that the achieved gain of each user served by the RISs is maximized. Also, we employ the bisection search method to handle the wireless bandwidth allocation problem.

3. 3.

   We demonstrate the advantages of the proposed FSL framework in terms of the training and inference performance. The proposed FSL framework is tested by using real RF traces, and numerical simulations demonstrate that the proposed FSL framework outperforms other benchmark schemes regarding spectrum sensing accuracy and system utility. Based on the conducted study, the following considerations can be made:

   • •

     The proposed FSL framework assisted by a larger number of RISs can achieve a better prediction accuracy and lower training loss.

   • •

     As the number of reflecting elements of each RIS increases, the FL system utility can be significantly improved.

   • •

     The performance of the proposed FSL framework is determined by the learning rate in the presence of multiple RISs. Therefore, an appropriate learning rate needs to be chosen as a function of the number of available RISs.

To our best knowledge, this is an early work that investigates the interplay between FL and RISs for efficient spectrum learning over wireless edge networks. The rest of this paper is organized as follows. We describe the proposed FSL framework in Section II. We introduce the system model and problem formulation in Section III. We present the algorithm solution to the considered optimization problem in Section IV. We illustrate the simulation results in Section V. Finally, conclusions are drawn in Section VI.

Notations: A bold letter indicates a vector or matrix. An upper case letter indicates a random variable or random parameter and a lower case letter indicates a realization of a random variable or a random parameter. $\max\{\cdot\}$ and $\min\{\cdot\}$ represent the maximum value and the minimum value, respectively. The amplitude of a complex number $a$ is denoted by $\left|a\right|$. The symbol $\leftarrow$ denotes the ‘assignment’ relation.

As shown in Fig. 1, we consider a wireless network that consists of one base station (BS) co-located with an edge server, which serves a set $\cal M$ of $M$ users via $K$ RISs, where ${\cal M}=\{1,2,\ldots,M\}$. There may exist obstacles (e.g., buildings) between the users and the BS, and the RISs can help improve the quality of wireless transmission between the users and the BS. We denote the $m$th user as $U_{m},\ m\in{\cal{M}}$, the local training dataset by ${\cal D}_{m}$ with $D_{m}=|{\cal D}_{m}|$ being the number of data samples. In the dataset ${\cal D}_{m}=\{\bm{x}_{m,s},{y}_{m,s}\}^{D_{m}}_{s=1}$, each data sample is constituted by an input vector $\bm{x}_{m,s}\in\mathbb{R}^{N_{d}\times 1}$ and its corresponding output value ${y}_{m,s}\in\mathbb{R}$. As far as the data locality is concerned, we assume that the collected datasets are non-overlapping with each other, i.e., $\bm{x}_{i}\neq\bm{x}_{j}$, $i\neq j,\ \forall i,j\in\cal M$.

In this paper, a novel FSL framework is proposed to exploit the interplay between FL and RISs. In the considered FL wireless network, specifically, only some users may send requests (called requested users) to participate in the training procedure of FL. Therefore, it is necessary to identify them and to appropriately allocate the available resources (e.g., wireless bandwidth) for maximizing the FL performance. In the proposed FSL framework, the requested users are inferred with the aid of neural networks deployed at the controller of the RISs. The latter neural networks are in turn trained by leveraging an FL framework that relies on the availability of RISs for reliable data transmission. These two phases are intertwined and, therefore, each training iteration of the proposed FSL framework consists of two stages: 1) the RIS-aided resource allocation stage, and 2) the RIS-aided FL training stage.

Specifically, in the RIS-aided resource allocation stage shown in Fig. 1(a), an initial convolutional neural network (CNN) model is deployed at each RIS controller to collaboratively infer the requested users from the impinging RF signals¹¹1We assume that the RIS controller has sufficient computation resources, e.g., field programmable gate array (FPGA) [41], to run the trained model for inference., and then the obtained estimates are reported to the BS for resource allocation. In the RIS-aided federated training stage shown in Fig. 1(b), given to the allocated resources, the CNN model available at the RISs’ controllers are further trained with the aid of FL, where the requested users upload their local model parameters via RISs-aided wireless links. It is worth noting that FL training and resource allocation (or optimization) are two intertwined problems, as illustrated in Fig. 2. In particular, if the resource allocation cannot yield an optimal solution, i.e., the wireless bandwidth is not appropriately allocated to the requested users, then the accuracy of the CNN model trained via FL worsens. If the accuracy of the trained CNN model deteriorates, in turn, the performance of resource allocation worsens. Indeed, this is a typical ‘chicken-and-egg’ issue. To address this issue, we propose the FSL framework by leveraging a fully-trained CNN model at each RIS controller and by jointly optimizing the user-RIS association, the phase shifts configuration and the wireless bandwidth allocation.

As illustrated in Fig. 1(a), the RIS-aided resource allocation stage involves three steps: i) Users send requests: the users who aim to participate in the current iteration of FL send requests to the BS; ii) RIS controllers infer the set of requested users: the CNN model deployed at each RIS controller estimates the ID of the requested users, which are then reported to the BS via a dedicated channel; iii) Edge server performs configurations: Based on the inferred results, the edge server performs the resource allocation by solving the optimization problem and broadcasts the result to the users and the RIS controllers. Specifically, when arriving at the RISs, the incident RF signals first undergo analog-to-digital conversion (ADC) and frequency down-conversion. Then the baseband In-phase (I) and Quadrature (Q) sequences are fed into the trained CNN model to perform online inference at the RIS controller. To achieve wireless spectrum learning at the RIS controllers, we introduce Assumption 1.

By performing feed-forward calculation via the trained CNN model, the set of inferred requested users obtained by the $k$th RIS controller is represented by

where $\widetilde{n}_{k}$ indicates the inferred total number of requested users in the incident signal.

In the considered FSL framework, for a data sample with input $\bm{x}_{m}$, the task is to find a model parameter vector $\bm{\omega}_{m}$ that characterizes the output $y_{m}$ by minimizing the loss function $f(\bm{\omega}_{m},\bm{x}_{m,s},{y}_{m,s})$. Since the dataset of $U_{m}$ is ${\cal D}_{m}$, the loss function of $U_{m}$ can be obtained as

We denote $D=\sum_{m\in\cal M}D_{m}$ as the total data samples of all users, and the FL training process is performed to solve the following optimization problem:

The FL training process is illustrated in Fig. 1(b), which includes three steps: i) Users train local models: each user $U_{m}$ trains the local model $\bm{\omega}_{m}$ based on the local dataset ${\cal D}_{m}$; ii) Users upload local models: the users upload their trained local models (e.g., $\bm{\omega}_{1},...,\bm{\omega}_{m}$) via the RISs-aided wireless links to the edge server; iii) Model aggregation and broadcasting: upon receiving all the uploaded gradients, the edge server aggregates the local updates and generates an updated model, $\bm{\omega}_{G}$, which is sent back to the requested users and the RIS controllers²²2Note that, different learning algorithms can be used to update the local FL model, e.g., gradient descent, to update the local FL model. The update of the global model $\bm{\omega}_{G}$ is given by $\bm{\omega}_{G}=\frac{\sum_{m=1}^{M}D_{m}\bm{\omega}_{m}}{D}$.. This training procedure is repeated several times until the global CNN model converges.

For each communication round, the uploading of the user’s local FL model ($\bm{\omega}_{m}$) can be considered as a transmission that occurs in one time slot, which may not be recovered correctly at the edge server due to the impact of the wireless channel between the users and the edge server, especially when the wireless channel undergoes deep fading due to the presence of obstacles. In this case, the local FL model parameters cannot be used for the aggregation at the edge server. To improve the quality of the global model that is aggregated at the edge server, the users can upload their local model parameters with the aid of RISs³³3Note that the user-RIS association may need to be updated at the beginning of each FL round by using the proposed user-RIS association algorithm..

Therefore, the aggregated FL model at a considered iteration in the proposed FSL framework is

We suppose that the cooperative prediction accuracy of the FL model at a considered iteration is $\eta$. We introduce a binary variable ${\alpha}_{m}$ for indicating whether the $m$th user sends a request or not and let $\widetilde{\alpha}_{m}$ be the inferred value of ${\alpha}_{m}$, so we have

Moreover, the binary variable $Q(\bm{\omega}_{m})$ in (4) indicates that the local model received at the edge server can be recovered correctly or not, which is given by

In (6), $p={\rm{Pr}}\left\{\gamma_{m}\geq\gamma_{T}\right\}$ indicates the probability that $\bm{\omega}_{m}$ is received correctly at the BS, and $1-p$ otherwise. $\gamma_{m}$ denotes the received SNR of $U_{m}$ and $\gamma_{T}$ is an SNR threshold. For instance, $Q(\bm{\omega}_{m})=0$ indicates that the received local FL model parameter cannot be correctly received, otherwise, $Q(\bm{\omega}_{m})=1$.

Different from the conventional FL procedure that is usually subject to the wireless bandwidth for model parameters uploading, the proposed RIS-aided FSL framework aims at improving FL performance by achieving wireless spectrum learning at the RIS controllers. Here, we consider an intuitive application scenario - the coexistence of multiple wireless systems, in which the wireless spectrum learning problem must be appropriately designed and addressed. Specifically, we suppose that two different wireless systems, such as WiFi and LTE-U systems, share the same unlicensed band and there exists interference between the two systems. To coordinate a fair share of the spectrum without causing undesired interference, improved wireless spectrum sensing and signal identification to detect spectrum users are needed [6]. However, traditional power estimation and spectrum sensing can only detect whether the spectrum is occupied or not as in energy detection. To address this issue, the proposed FSL framework achieves appropriate wireless resources coordination of the two systems via RISs-enabled cooperative spectrum sensing, where the CNN model deployed at the RIS controllers for RF spectrum learning is trained via FL.

An intuitive example with two system users (denoted by ${\bf U}=\{U_{1},U_{2}\}$) and two RISs is considered, where $U_{1}$ is a WiFi user and $U_{2}$ is a LTE-U user. A CNN model is trained via the proposed FSL framework and then deployed at each RIS controller to learn the features of RF traces. In particular, upon receiving the superimposed incident signals at each RIS, the well-trained CNN infers the set of requested users by performing feed-forward calculation. Since each user has a binary state, i.e., ‘active’ and ‘inactive’, in this case, there exist four combinations (i.e., classes) of signals from the perspective of each RIS:

• •

  Class-1: Idle. This indicates that both $U_{1}$ and $U_{2}$ are inactive, so the collected RF traces include only the noise.

• •

  Class-2: Only ${U_{1}}$. This indicates that only $U_{1}$ is active, so the collected RF traces include only $U_{1}$.

• •

  Class-3: Only ${U_{2}}$. This indicates that only $U_{2}$ is active, so the collected RF traces include only $U_{2}$.

• •

  Class-4: ${U_{1}\!+\!U_{2}}$. This indicates that both $U_{1}$ and $U_{2}$ are active, so the collected RF traces include $U_{1}$ and $U_{2}$.

Via inference, each RIS controller identifies the composition of the superimposed incident RF signals and sends the inferred result to the BS via a dedicated channel for data fusion. Note that the spectrum identification in the considered example boils down to a four-class classification problem. According to the inferred result, the BS can coordinate spectrum resources between the two systems appropriately. Also, according to the allocated resources, the requested users upload their local trained models to the BS via RISs-aided wireless links.

By leveraging the presence of RISs, the users can upload the local gradients via the RIS-assisted wireless uplinks more reliably. For the local uploading of the model parameters, we consider an orthogonal-access schemes such as orthogonal frequency division multiple access (OFDMA) in an synchronous manner. We suppose that each RIS, denoted as $R_{k}$, $k\in{\cal K}$, is equipped with $N_{k}$ reflecting elements, which can be appropriately configured by the RIS controller to reflect the associated user’s signal effectively towards the BS through non-overlapping frequency band. In general, the RISs can be appropriately deployed so that line-of-sight (LoS) links can be established between them and the BS. Moreover, we assume that the channel state information (CSI) can be perfectly estimated at the BS by sending training symbols at the user and adjusting the reflection states of the RIS according to a predesigned training reflection pattern [42, 43]. Then, the estimated CSI is fed back to the RIS controller via a dedicated channel⁴⁴4Since channel estimation in RISs-aided wireless communications has been widely investigated, it is not explicitly considered in this paper., such as [16, 44]. Note that due to the movement of the mobile users, the received signals are sometimes subject to a multipath fading and Doppler effect, which can be effectively eliminated and/or mitigated by real-time tuneable RISs [45].

As for $R_{k}$, the amplitude reflection coefficient is assumed to be equal to one for all the $N_{k}$ reflecting elements, so the phase reflection matrix is denoted by

where ${\bm{\theta}}_{k}=\left(\theta_{1}^{k},\theta_{2}^{k},...,\theta_{N_{k}}^{k}\right)$ denotes the vector of phase shifts that need to be optimized.

In the considered FSL framework, the channels from the user $U_{m}$ to $R_{k}$ and from $R_{k}$ to the BS are given by $\textit{{h}}_{m,k}\in\mathbb{C}^{N_{k}\times 1}$ and $\textit{{g}}_{k}\in\mathbb{C}^{1\times N_{k}}$, respectively. The channel gain of the direct link from $U_{m}$ to the BS is denoted by $h_{d,m}$. Besides, the channels are assumed to be quasi-static and remain nearly-constant during the transmission [15]. Accordingly, the received local model parameters that are uploaded by the total $M$ users at the BS include the signals via the direct links and the links reflected by the RIS and the white Gaussian noise, i.e.,

where $p_{m}$ denotes the transmit power of $U_{m}$, $z_{m}$ is the unit-power information signals sent from $U_{m}$, and $\bm{n}=[n_{1},n_{2},...,n_{M}]^{\rm T}$ denotes the white Gaussian noise vector.

Besides, the user-RIS association matrix ${\bm{R}}$ with dimension of ${K\times M}$ is given by

where $r_{m,k}\in\{0,1\}$, $r_{m,k}=1$ indicates that $R_{k}$ is allocated to $U_{m}$, and $r_{m,k}=0$, otherwise. We assume that only one RIS can be allocated to one user at most, so we have $\sum_{m\in{\cal M}}r_{m,k}\leq 1,\ \forall k\in{\cal K}$.

In conventional RIS-aided communication systems, it is known that each RIS reflects all the incident electromagnetic waves, and will steer them towards reflecting directions that depend on the direction of incidence and the phase shifts applied by the RIS elements. To achieve user-RIS association in the proposed FSL framework, we introduce Assumption 2.

Therefore, the SNR at the BS for user $U_{m}$ is given by

where $B_{m}$ denotes the bandwidth allocated to $U_{m}$, and ${\cal N}_{0}$ indicates the noise power spectral density.

Suppose that the total bandwidth between the users and the BS is $B$, and let ${\bm{\beta}}\!=\!\{\beta_{1},\beta_{2},...,\beta_{M}\}$ denote the wireless bandwidth allocation vector, so we have $B_{m}=\beta_{m}B$ and $\sum_{m=1}^{M}\beta_{m}\!\leq\!1$. Based on (9), the achievable data rate of $U_{m}$ can be obtained as

Having defined the latency model in the previous section, the problem is formulated to maximize the number of correctly received parameters of the local FL model per unit time at each iteration. Here, we define a metric for evaluating the FL performance at each iteration, called FL system utility, which is calculated as the ratio between the total number of correctly received local FL model parameters at the BS and the total delay at the considered iteration, i.e.,

where the binary variable $Q(\bm{\omega}_{m})=1$ indicate that the local FL model parameter is correctly received at the BS, otherwise, $Q(\bm{\omega}_{m})=0$.

In this paper, the objective is to maximize the FL system utility (${\bm{\Theta}}=\{{\bm{\theta}}_{1},{\bm{\theta}}_{2},...,{\bm{\theta}}_{K}\}^{\rm T}$) by jointly optimizing the RIS allocation matrix (${\bm{R}}$), the wireless bandwidth activation vector (${\bm{\beta}}$), and the phase shifts matrix of the RISs whose size is ${K\times N}$⁶⁶6For simplicity, we assume that each RIS has the same number of reflecting elements, i.e., $N=N_{k_{1}}=N_{k_{2}}$, where $k_{1}\neq k_{2}$.. To this end, a joint RIS phase shifts, user-RIS association and bandwidth allocation problem can be formulated as follows:

In (17a), the binary value $r_{m,k}=1$ indicates that $R_{k}$ is allocated to $U_{m}$, and $r_{m,k}=0$, otherwise. (17b) indicates that at most one RIS can be allocated to a user at a time. (17c) indicates that the achievable SNR needs to be larger than a threshold. (17d) indicates that each RIS reflecting element can only provide a phase shift $\theta_{n}^{k}\in[0,2\pi)$ without amplifying the signals. Finally, (17e) indicates that the sum of the wireless bandwidths that are allocated to the users cannot exceed the total bandwidth.

By exploiting the structure of the objective function and its constraints, we observe that the original problem in (17) has a high complexity. By using the Tammer decomposition method, we decompose the original problem into two subproblems with separated objective and constraints without changing the optimality of the solutions [49].

First, we rewrite the original problem as an equivalent problem:

To solve the equivalent problem $\widetilde{\mathbb{P}}$, we further decompose it into two subproblems, as illustrated in Remark 1.

Since ${\rm Q}^{*}$ in $\mathbb{P}_{2}$ can be obtained by solving $\mathbb{P}_{1}$, $\mathbb{P}_{2}$ can be rewritten as the bandwidth allocation subproblem ($\widetilde{\mathbb{P}}_{2}$) to minimize the FL training latency:

Before solving the subproblem $\mathbb{P}_{1}$ in (19), Observation 1 is presented as follows.

We first optimize the phase shift vector ${\bm{\Theta}}$ of problem $\mathbb{P}_{1}$ in (19), where $r_{m,k}=1$. Accordingly, the optimal value of ${\bm{\Theta}}$ can be calculated by solving the following problem:

According to the triangle inequality, we have

where the equality holds when ${\rm arg}\left(h_{d,m}\right)={\rm arg}\left(\textit{{g}}_{k}\mathbf{\Phi}_{k}\textit{{h}}_{m,k}\right)$ is fulfilled⁷⁷7This suggests that the phase of the signal reflected through the user-RIS-BS links is aligned with that of the user-BS direct link..

We let ${\bm{u}}_{k}{\bm{v}}_{k}\triangleq\textit{{g}}_{k}\mathbf{\Phi}_{k}\textit{{h}}_{m,k}$, where,

Then the problem in (22) is equivalent to:

We observe that the optimal solution to the problem in (24) is ${\bm{u}}_{k}^{*}=e^{j({\rm arg}\left(h_{d,m}\right)-{\rm arg}\left({\rm diag}(\textit{{g}}_{k})\textit{{h}}_{m,k}\right))}$. Therefore, the $n$th phase shift of the $k$th RIS is calculated as

where $\textit{g}^{n}_{k}$ and $\textit{h}_{m,k}^{n}$ are the $n$th element of $\textit{{g}}_{k}$ and $\textit{{h}}_{m,k}$, respectively.

As a result, we have ${\bm{\theta}}_{k}^{*}=\left(\theta_{1}^{k*},\theta_{2}^{k*},...,\theta_{N_{k}}^{k*}\right)$. The optimal phase shifts matrix of the RISs is obtained as ${\bm{\Theta}^{*}}=\{{\bm{\theta}}_{1}^{*},{\bm{\theta}}_{2}^{*},...,{\bm{\theta}}_{K}^{*}\}^{\rm T}$. Having the optimal RISs phase shifts, we move to the second-step to compute the optimal user-RIS association via matching theory.

Via the aggregated global FL model, each RIS controller infers the set of requested users at each iteration and then sends the estimated result to the BS. Based on the fusion result, the user-RIS association can be optimized. To tackle the combinatorial problem, we apply matching theory to map the user-RIS association subproblem into a matching game.

Specifically, we denote the set of the requested users whose signals over the direct links cannot be correctly received as ${\cal M}^{\prime}$, where ${\cal M}^{\prime}\subseteq\cal M$. Denote the total number of the requested users within ${\cal M}^{\prime}$ as $M^{\prime}$. We solve the user-RIS association subproblem in two cases: $M^{\prime}\leq K$ (case-1) and $M^{\prime}>K$ (case-2). For these two cases, we first formulate the user-RIS association as a two-sided matching game, followed by the definition of the utility. Then, we present a multi-granularity based matching algorithm to achieve a stable matching.

In Case-1, the number of RISs is no smaller than the number of users in ${\cal M}^{\prime}$, i.e., $K\geq{M}^{\prime}$. In this case, each user can be served by one RIS at a time.

We solve the subproblem $\widetilde{\mathbb{P}}_{2}$ to minimize the FL latency, as described in Theorem 1.

In the proposed FSL framework, the deployment of RISs has a twofold contribution: 1) the RIS controllers help detecting the set of requested users in the resource allocation stage, and 2) the RISs can improve the data transmissions in the FL training stage. Based on this, the performance improvement given by deployment of RISs is presented in Theorem 2.

From Theorem 2, we observe that the performance improvement brought by RISs in the proposed FSL framework is proportional to the number of RISs.

In this section, we demonstrate the feasibility of the proposed scheme by clarifying the RIS elements design, analyzing the signaling overhead, storage, as well as quantifying the computation complexity at the RIS controller.

In this section, we present the testing results of the considered four-class inference example, by using real RF samples. In the considered example with two users and two RISs, the CNN model is trained through the proposed FSL framework to achieve RF signal classification. The testing results are illustrated in Table I.

For each user in the considered scenario, historical RF traces are collected using a universal software radio peripheral (USRP2) testbed, which is wired connected via Gigabit Ethernet to a host server with an implementation of the GNU Radio. To be specific, each user is emulated through a laptop that is responsible for baseband processing while a USRP2 platform is used for the up/down-conversion, the digital-to-analog/analog-to-digital conversion, and wireless transmission of the signals. Considering that each USRP2 testbed usually has one transmuting antenna, in order to collect the signals with USRP2, we let multiple USRP2 testbeds transmit RF signals simultaneously to a receiver. Then the RF signals can be received and stored as I/Q sequences for training the CNN model, by including the wireless channel, for a wide range of SNR values (from $0$ to $20$ dB with interval of $5$ dB) in order to account for different channel conditions. Besides, the window size $w$ (i.e., the number of time steps of the collected RF data) in Table I is $32$, $128$, and $512$, respectively.

In the considered example, we assume that the RIS₁ and RIS₂ assist the wireless transmissions of $U_{1}$ and $U_{2}$, respectively, and the total wireless bandwidth is shared equally between the two users. For the local training, the users train the CNN with $80\%$ of RF dataset (i.e., I and Q samples), validate it by using $10\%$ of the dataset, and test it by using the remaining $10\%$ of the dataset. The trained CNN model consists of two convolutional (Conv) layers with ReLU activation functions, followed by two dense fully connected (FC) layers. The trained CNN model includes $256$ filters (1$\times$3) in the first Conv layer, $128$ filters (1$\times$3) in the second Conv layer, $256$ neurons in the first FC layer, and $9$ neurons in the second FC layer (output). Within each FL round, the two users upload their local models once the local training ends. It is observed from Table I that the inference accuracy of the converged CNN model with two RISs is, in general, greater than $95\%$. Compared to other classes, the ‘Idle’ class has the main characteristic that no user transmits and only background noise exists. Due to the distinguishable pattern compared to the other three classes, the CNN model predicts the ‘Idle’ class (i.e., the Class-1 in Table I) perfectly.