## 2 System Model and Problem Formulation
### System Model  
We consider a downlink multi-RIS-aided multi-user system consisting of \(N\) single-antenna users, each of which would benefit from the collaboration of \(M\) single-antenna BSs and \(N\) RISs, as illustrated in Figure 2. Each RIS \(r\) equipped with elements \(\mathcal{L}=\{1,\cdots,L\}\) passively reflects the signals from BSs to assist a dedicated user \(n\), such that the system performance can be enhanced. For simplicity, the sets of users/RISs and BSs are, respectively, denoted by \(\mathcal{N}=\{1,\cdots,N\}\) and \(\mathcal{M}=\{1,\cdots,M\}\). \begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Description** \\ \hline \(M\) & Number of BSs \\ \(N\) & Number of RISs/users \\ \(L\) & Number of RIS-reflecting elements \\ \(\theta_{r,l}\) & Phase shift of the \(l\)-th element of RIS \(r\) \\ \(\mathbf{\Phi}_{r}\) & Reflection coefficients matrix of RIS \(r\) \\ \(h_{m,n}\) & Direct channel from BS \(m\) to user \(n\) \\ \(g_{m,r}\) & Channel from BS \(m\) to RIS \(r\) \\ \(f_{r,n}\) & Channel from RIS \(r\) to user \(n\) \\ \(\alpha_{m,n}^{2}\) & Average power gain of \(h_{m,n}\) \\ \(\beta_{m,r}^{2}\) & Average power gain of \(g_{m,r}\) \\ \(\eta_{r,n}^{2}\) & Average power gain of \(f_{r,n}\) \\ \(\gamma_{m,n}\) & User \(n\)’s selection of BS \(m\) \\ \(s_{n}\) & Transmitted symbol of user \(n\) \\ \(w_{m,n}\) & Precoding weight used by BS \(m\) for user \(n\) \\ \(y_{n}\) & Desired signal received by user \(n\) \\ \(I_{n}\) & Interference received by user \(n\) \\ \(\tilde{\lambda}_{n}\) & Average SINR received at the user \(n\) \\ \(\lambda_{n}\) & Lower bound of \(\tilde{\lambda}_{n}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Variable definitions. In such a system, we assume that a CPU exists for network central processing, and carries out transmission scheduling according to the network information. To ensure accurate channel state information (CSI) at the CPU, various channel acquisition techniques are employed [35; 36; 37]. The RIS comprises numerous cost-effective passive reflecting elements, each of which can be individually adjusted to manipulate the phase of incident electromagnetic signals using a programmable PIN diode. Thus, RISs can enable signals to be transmitted simultaneously over two links, i.e., direct and reflected links, resulting in more diverse transmissions. As shown in Figure 2, the RISs receive the signals sent by the BSs, and then reflect these signals to the users. To simplify the problem, we assume that the reflection amplitude of each RIS element is maximized at 1 [38]. Additionally, we assume that the signal is reflected ideally without any hardware defects. The phase shift of the \(l\)-th RIS element of the RIS \(r\) is denoted as \(\theta_{r,l}\) and belongs to the interval \([0,2\pi]\), while \(\mathbf{\Phi}_{r}\) represents the frequency response of RIS \(r\) and is defined as a diagonal matrix with diagonal elements \(\{e^{|\theta_{r,1}|},e^{|\theta_{r,2}|},\cdots,e^{|\theta_{r,l}|}\}\). The channel between BS \(m\) and user \(n\) in the direct link is denoted as \(h_{m,n}\in\mathbb{C},m\in\mathcal{M},n\in\mathcal{N}\). The fading channels \(g_{m,r}=\left[g_{m,r,1},g_{m,r,2},\cdots,g_{m,r,L}\right]^{H}\in\mathbb{C}^{L \times 1},m\in\mathcal{M},r\in\mathcal{N}\) and \(f_{r,n}^{H}=\left[f_{r,n,1},f_{r,n,2},\cdots,f_{r,n,L}\right]\in\mathbb{C}^{1 \times L},r,n\in\mathcal{N}\) are, respectively, the channel between the single-antenna BS \(b\) and RIS \(r\), and between RIS \(r\) and the single-antenna user \(n\). Under the hypothesis of independent Rayleigh fading channels, we have \(h_{m,n}\sim\mathcal{CN}(0,\alpha_{m,n}^{2})\), \(g_{m,r}\sim\mathcal{CN}(0,\beta_{m,n}^{2},I_{L})\) and \(f_{r,n}\sim\mathcal{CN}(0,\eta_{r,n}^{2}I_{L}),m\in\mathcal{M},r,n\in\mathcal{N}\). The average power gains \(\alpha_{m,n}^{2}\), \(\beta_{m,r}^{2}\) and \(\eta_{r,n}^{2}\) depend on the distances between the BSs, RISs and users.