Decentralized Active & Passive Beamforming

• The paper introduces a decentralized framework that jointly optimizes active beamforming at BSs and passive beamforming at RIS/BDRIS using consensus and convex optimization techniques.
• It addresses sum-rate maximization under imperfect CSI, limited hardware scalability, and stringent communication overhead through distributed learning and iterative algorithms.
• It demonstrates that decentralized approaches can achieve near-optimal performance compared to centralized schemes while significantly reducing complexity and backhaul requirements.

A decentralized active and passive beamforming framework refers to algorithms and protocol designs enabling geographically distributed transmitters (such as Base Stations, Access Points) and passive electromagnetic devices (Reconfigurable Intelligent Surfaces, RISs; and its generalizations such as Beyond Diagonal RIS, BDRIS) to jointly optimize the transmission of multi-stream wireless signals without reliance on a central processor. These frameworks address sum-rate maximization and robustness under imperfect channel state information (CSI), scalable hardware, and stringent communication overhead constraints. Recent advances leverage consensus optimization, successive convex/concave approximation, assignment algorithms, and distributed learning to architect efficient cooperative beamforming in cell-free multi-user MIMO and OFDM systems (Katsanos et al., 19 Jan 2026, Katsanos et al., 13 Jan 2026, Huang et al., 2020, Ni et al., 2022, Le et al., 2024).

1. System Architecture and Signal Model

Decentralized frameworks operate within cell-free wideband systems comprised of $B$ multi-antenna BSs/APs and $R$ multi-group RIS/BDRIS panels, serving $U$ multi-antenna or single-antenna users across $K$ OFDM subcarriers (Katsanos et al., 19 Jan 2026). Each transmitter's output on subcarrier $k$ is

$x_{b,k}=\sum_{u=1}^U W_{b,u,k}\,s_{u,k}$

with precoding matrix $W_{b,u,k}\in\mathbb{C}^{N_t\times N_s}$ obeying per-node power constraint

$\sum_{u,k}\|W_{b,u,k}\|_F^2\le P_b^{\max}.$

BDRIS/STAR-RIS elements are partitioned into groups, each with a frequency-selective response modeled by capacitive matrices and circuit primitives ($R_0,L_1,L_2$), encapsulated in the reciprocal scattering matrix. Assignment/permutation matrices $Q_{p,r}$ define flexible reordering of elements for dynamic group connectivity (Katsanos et al., 19 Jan 2026).

User channels comprise direct BS–UE and cascaded multi-hop BS-RIS-UE components. In STAR-RIS, elements split incident power (energy splitting) for simultaneous transmission/reflection, with

$(\beta_{lm}^T)^2 + (\beta_{lm}^R)^2 \le 1$

where amplitude coefficient $\beta_{lm}^\chi$ and phase-shift $\theta_{lm}^\chi$ are adapted for user region $\chi$ (Le et al., 2024).

Imperfect CSI is modeled as

$$\hat h = h + e,\quad e\sim\mathcal{CN}(0,\delta|h|^2)$$

and robust frameworks optimize the expected rate under such uncertainty.

2. Optimization Problem Formulation

The key objective is expected sum-rate maximization, jointly optimizing

• Precoding matrices $\{W_{b,u,k}\}$ (active)
• Tunable capacitances and permutation matrices $\{ C_{r,g}, Q_{p,r} \}$ (passive)

Under constraints including power budgets, capacitor symmetry/box bounds, and permutation feasibility:

$$\max_{\{W,C,Q_p\}}\,\, \mathbb{E}\left[ \sum_{u=1}^U \sum_{k=1}^K \log_2 \left| I_{N_s} + S_{u,u,k}^H P_{u,k}^{-1} S_{u,u,k}\right| \right]$$

with permissible configurations and consensus requirements over the network (Katsanos et al., 19 Jan 2026).

STAR-RIS and distributed multi-RIS settings extend the formulation to reflect energy-splitting, multi-region coverage, and multiple passive surfaces, incorporating unit-modulus/phasing and amplitude coupling constraints (Le et al., 2024, Ni et al., 2022, Huang et al., 2020).

3. Algorithmic Frameworks

Decentralized beamforming relies on decomposed optimization with minimal inter-node exchange. Typical components include:

• Active Beamforming: Per-node precoder optimization via Successive Concave Approximation (SCA), WMMSE, or Majorization-Minimization (MM), yielding closed-form updates of the form

$$W_{b,u,k}^{\rm opt}(\lambda) = (E_{b,u,k} + \lambda I )^{-1} J_{b,u,k}$$

where Lagrange parameters enforce power (Katsanos et al., 19 Jan 2026, Ni et al., 2022, Katsanos et al., 13 Jan 2026, Huang et al., 2020).

• Passive Beamforming: Distributed or consensual tuning of RIS/BDRIS capacitive matrices using convex projections (Dykstra’s alternating projection), Linear Assignment Problems (Hungarian algorithm for permutation matrices), and gradient-tracking (Katsanos et al., 19 Jan 2026, Katsanos et al., 13 Jan 2026, Huang et al., 2020).
• Consensus and Message-Passing: Dynamic average consensus for passive variables ensures network-wide agreement. Adaptive, doubly-stochastic weight matrices $V_{\rm net}$ are utilized for fast error reduction (Katsanos et al., 19 Jan 2026, Katsanos et al., 13 Jan 2026, Huang et al., 2020). Robustness to estimation error is maintained through stochastic linearization and proximal regularization.
• Graph Neural Network Approaches: In distributed STAR-RIS configurations, Heterogeneous Graph Neural Networks (HGNN) with parameter tying are used to output both active and passive coefficients, enabling permutation-equivariant, scalable solutions across arbitrary system sizes while achieving competitive sum-rate performance (Le et al., 2024).

4. Architectural Models and Practical Trade-offs

The BDRIS Dynamic Group-Connected (DGC) model partitions $M$ elements into $G$ flexible groups, reducing hardware complexity and CSI sharing overhead compared to fully-connected architectures, with only a slight reduction in achievable sum-rate (<5%) (Katsanos et al., 19 Jan 2026). The consensus-based distributed approach enables near-global performance without the need for a central processing unit (CPU) or heavy backhaul signaling.

Partially distributed implementations typically locate active beamforming at local APs/BSs, while passive beamforming, such as phase-shift vector updates, remains centralized or is managed with minimal iterative exchanges (Ni et al., 2022). Full decentralization (e.g., incrementally-ordered ADMM ring) is feasible with consensus constraints on replicated phase vectors for multi-IRS configurations (Huang et al., 2020).

STAR-RIS energy splitting and phase control introduce additional constraints due to simultaneous transmission and reflection, which are tractably handled in distributed or GNN-based frameworks (Le et al., 2024).

5. Convergence, Complexity, and Communication Overhead

The proposed CSD-SCA algorithms and distributed MM/ADMM methods converge to stationary points under standard diminishing-step update conditions, typically within $15$–$30$ iterations depending on cooperative/non-cooperative mode and system scale (Katsanos et al., 19 Jan 2026, Huang et al., 2020, Katsanos et al., 13 Jan 2026, Ni et al., 2022). Adaptive network weights yield fast consensus errors ($\leq 5$ iterations) versus static weights (Katsanos et al., 19 Jan 2026).

Complexity per iteration is dominated by matrix inversions (active updates, $O(N_t^3)$) and convex projections/gradient calculation for passive variables (up to $O(R^2 M^2 K)$ for large RIS arrays) (Katsanos et al., 19 Jan 2026, Katsanos et al., 13 Jan 2026, Huang et al., 2020). Communication overhead is proportional to exchanged grader pricing and capacitor/phase blocks ($\approx 2M$ real scalars/neigh per AP for RIS, significant reduction vs centralized forwarding of $B U K N$ channel coefficients) (Katsanos et al., 13 Jan 2026, Katsanos et al., 19 Jan 2026). GNN-based approaches reduce runtime substantially, scaling efficiently to large networks (Le et al., 2024).

6. Performance Evaluation and Comparative Insights

Decentralized cooperative schemes with DGC BDRIS show near-centralized sum-rate performance, outperforming non-cooperative and conventional diagonal RIS benchmarks, with sum-rate gains up to $3$ b/s/Hz at $35$ dBm, and within $2$ b/s/Hz of fully-connected BDRIS (Katsanos et al., 19 Jan 2026). Robustness to CSI errors is evident, with decentralized cooperation matching centralized perfect CSI up to $25$ dBm.

Tables below organize architectural schemes and their comparative performance profiles.

Architecture	Complexity	Sum-Rate Loss vs FC	Consensus Overhead
Fully-connected BDRIS	Highest	Baseline	Largest
DGC BDRIS	Reduced	$\sim$2 b/s/Hz	Minimal
Diagonal RIS	Lowest	Largest	Minimal

Algorithmic Mode	Communication Load	Performance	CPU Dependency
Centralized	Largest	Optimal/Upper Bound	Strong
Decentralized (CSD)	Small	Near-optimal (<6%)	None
Partially Distributed	Medium/O(B)	Close-to-optimal	Moderate
GNN-based (HGNN)	Small	1–2% gap to AO–SCA	None

A plausible implication is that architectures such as DGC BDRIS and consensus-based distributed optimization provide a scalable pathway for next-generation cell-free MIMO/OFDM networks under realistic hardware and CSI conditions. Hardware complexity and backhaul signaling can be dramatically reduced without significant sacrifice in throughput, especially when robust design principles and fast consensus are applied.

7. Technical Challenges and Open Research Directions

Key ongoing challenges include the accommodating of highly imperfect or quantized CSI in fully decentralized settings, scaling algorithms to massive user/elements regimes, real-time adaptation to nonstationary environments, and rigorous convergence analysis in non-convex multi-constraint formulations. Continued innovation in decentralized algorithmic design—including graph neural architectures and dynamic group-connected architectures—remains critical for practical deployment of distributed active–passive beamforming in smart wireless environments (Katsanos et al., 19 Jan 2026, Le et al., 2024).