Over-the-Air Consensus in Wireless Networks

• Over-the-air consensus is a set of protocols that leverages the analog superposition property of wireless channels to perform in-air aggregation for decentralized consensus and optimization.
• These methods bypass traditional digital exchanges by using techniques such as non-coherent aggregation and ratio consensus to counteract fading and synchronization challenges.
• The approach significantly reduces communication overhead and scales well in dense wireless networks, with applications in multi-agent control, federated learning, and blockchain.

Over-the-air consensus is a class of distributed algorithms that exploit the physical-layer superposition property of wireless multiple-access channels (MACs) to implement consensus or decentralized optimization primitives directly in the analog domain. Rather than circumventing interference or enforcing strict orthogonalization, these protocols leverage the fact that simultaneous transmissions inherently compute a linear (or nonlinear) function of all participating signals at the receiver. This paradigm has significant implications for communication efficiency, scalability, and robustness in large-scale wireless multi-agent systems, particularly where spectrum is scarce and strict coordination is costly or infeasible.

1. Fundamental Principles and Problem Formulation

The canonical consensus problem concerns a network of $N$ nodes, each with an initial value $x_i(0)$. The goal is for all nodes to iteratively update their states $\{x_i(k)\}$ such that $$\lim_{k\to\infty} x_i(k) = x^* = \frac{1}{N}\sum_{j=1}^N x_j(0)$$. Traditionally, this is achieved by repeated local exchanges and linear-weighted averaging over a logical network graph. In over-the-air consensus, the analog superposition over the wireless channel is directly exploited for aggregation, so that the consensus step is performed by physical-layer mixing (Yang et al., 11 Mar 2024, Charalambous et al., 30 Jul 2025, Lee et al., 2022).

A general system model involves:

• Nodes equipped with analog/RF transceivers; often under half-duplex constraints
• Channel models incorporating flat or frequency-selective fading, potentially with time variation and reciprocal path-gains
• Transmission of analog signals encoding the node’s local state or update, possibly with power pre-scaling
• Receivers obtain a superposed (linear, possibly noisy and non-coherent) sum of all transmitting neighbors’ signals

The consensus step thus becomes an implicit analog summation—rather than explicit digital message exchange—subject to the idiosyncrasies of fading, noise, and lack of instantaneous channel state information (CSI).

2. Over-the-Air Consensus Variants and Algorithmic Structures

Several core classes of OTA consensus protocols have been developed:

2.1 Non-Coherent Aggregation Schemes

Non-coherent OTA consensus algorithms (e.g. (Yang et al., 11 Mar 2024, Deng et al., 8 Apr 2025, Michelusi, 2022)) employ amplitude- or energy-based signal encoding, with no requirement for carrier-phase synchronization or channel inversion. Each node encodes its local state as a transmit energy or amplitude, resulting in a received signal

$y_i = \sum_j h_{ij} s_j + n_i$

where $h_{ij}$ is the complex fading coefficient. Receivers typically use amplitude-squared or energy measurements $|y_i|^2$ to form locally unbiased estimates of weighted sums, with appropriate noise subtraction. The update step often includes a diminishing step-size $\alpha(k)$ for stochastic approximation. The key attribute is that only long-term channel statistics (e.g., average pathloss) and local neighborhood knowledge are needed, facilitating operation in networks with time-varying topologies and no detailed CSI.

2.2 Ratio Consensus and Normalization

Protocols based on ratio consensus (e.g. (Charalambous et al., 30 Jul 2025)) generalize average consensus to arbitrary (possibly directed) communication graphs and unknown MAC weights. Each node maintains two states (numerator, denominator), both updated via the OTA MAC and normalized by local measurements of summed channel gains: $$y_j[k+1] = \frac{\sum_{i\in\mathcal{N}_j^-} h_{ji}[k] y_i[k]}{\sigma_j[k]}$$

$$x_j[k+1] = \frac{\sum_{i\in\mathcal{N}_j^-} h_{ji}[k] x_i[k]}{\sigma_j[k]}$$

with $\mu_j[k] = y_j[k]/x_j[k]$. This approach compensates for arbitrary and time-varying channel coefficients, enabling provable convergence under mild connectivity and fading assumptions.

2.3 Distributed Optimization and Federated Learning

In decentralized stochastic or gradient-based optimization, over-the-air consensus is used to implement the “mixing” step of decentralized SGD (or DGD) directly over the MAC (e.g. (Ozfatura et al., 2020, Shi et al., 2021, Zhai et al., 2023, Madhan-Sohini et al., 2022, Michelusi, 2022)). Here, model updates for federated learning or distributed estimation are encoded as analog waveforms, averaged “in the air” through superposition, and processed at each node to update local state:

$$w_{i, k+1} = \Pi_\mathcal{M}\big[w_{i,k} + \gamma_k d_{i,k} - \eta_k g_{i,k}\big]$$

with $d_{i,k}$ the non-coherently extracted consensus estimate, and $\eta_k$, $\gamma_k$ stepsizes. These protocols have been shown to converge with rates matching or exceeding classical digital approaches, with the key advantage of dramatic reductions in required communication resources as $N$ grows (Michelusi, 2022, Michelusi, 2022).

3. Theoretical Guarantees and Convergence Analysis

OTA consensus protocols have been rigorously analyzed under a range of noise, fading, and topology models:

• Mean-square and almost-sure convergence: Under appropriate diminishing stepsizes, unbiasedness of energy aggregation, and joint network connectivity, these protocols are proven to reach consensus in mean square and almost surely, even in the presence of AWGN and non-coherent fading (Yang et al., 11 Mar 2024, Deng et al., 8 Apr 2025, Michelusi, 2022).
• Convergence rates: In distributed optimization, NCOTA-DGD achieves $\mathcal{O}(K^{-1/4})$ for the expected error after $K$ steps under fixed consensus/learning stepsizes (Michelusi, 2022). When using diminishing stepsizes, rates of $\mathcal{O}(1/\sqrt{k})$ are established for strongly convex objectives (Michelusi, 2022).
• Spectral properties: The spectral gap of the implicit mixing matrix (induced by average pathloss or dynamically by channel statistics) directly governs the speed of consensus (Zhai et al., 2023, Deng et al., 8 Apr 2025).
• Bias under non-coherent aggregation: Without careful design, non-coherent interference induces bias in vanilla protocols; reformulating as decentralized projected gradient descent (D-PGD) and implementing transmit power control and receive scaling can eliminate this bias without sacrificing speed (Deng et al., 8 Apr 2025).

4. Performance, Resource Efficiency, and Scalability

Over-the-air consensus is fundamentally motivated by the need for scalable, spectrally efficient coordination in dense wireless networks. Key empirical findings include:

• Resource consumption: OTA consensus reduces per-round communication resource blocks by up to $N$-fold versus digital or orthogonal analog schemes, as a single MAC slot or resource block achieves network-wide aggregation (Yang et al., 11 Mar 2024, Ozfatura et al., 2020, Michelusi, 2022). In multi-agent control and platooning scenarios, the total transmission count required for consensus scales far more favorably with network size (Lee et al., 2022, Epp et al., 21 Mar 2024).
• Application-layer runtime: In direct comparisons, OTA-DGD performs orders of magnitude more consensus iterations within the same wall-clock time budget compared to orthogonal-channel implementations, with faster convergence to optimality and test error (Michelusi, 2022, Michelusi, 2022).
• Adaptivity to topology and fading: Many schemes operate without explicit knowledge of the network graph or instantaneous channel states, supporting dynamic, mobile, and ultra-dense deployments (Agrawal et al., 2023, Michelusi, 2022).
• Robustness: OTA consensus protocols demonstrate resilience to physical-layer noise, topology changes, and interference. For example, in blockchain, the AirCon protocol achieves byzantine fault tolerance up to $0.39K$ adversarial nodes with low consensus-error ratios (CER) even at moderate SNR (Xie et al., 2022).

5. Practical Implementation Details and Protocol Design

Correct OTA consensus protocol design entails several critical considerations:

• Slot-level synchronization: Simultaneous or slotted transmissions by all nodes are required to guarantee superposition, but only up to symbol boundaries—carrier-phase alignment is unnecessary for non-coherent schemes (Yang et al., 11 Mar 2024).
• Power control and scaling: Nodes may employ local power scaling (pre-coding) and normalization based on either locally known statistics or online estimation to ensure fair mixing weights and to compensate for pathloss differences (Yang et al., 11 Mar 2024, Deng et al., 8 Apr 2025).
• Channel requirements: While protocols can demand per-link CSI and synchronization (coherent schemes or AirComp with beamforming (Zhai et al., 2023)), robust non-coherent solutions only require loose knowledge of average pathloss or approximately reciprocal gains, and can operate even in the presence of significant time variation or user mobility (Michelusi, 2022, Agrawal et al., 2023).

In large networks, scalability can be further enhanced by hierarchical or clustered aggregation, as exemplified in clustered wireless federated learning (CWFL) (Madhan-Sohini et al., 2022), where local OTA aggregation within clusters combines with digital or analog inter-cluster consensus for improved performance-complexity tradeoff.

6. Application Domains

Over-the-air consensus has been validated and deployed in several key networked systems:

• Multi-agent control and vehicle platooning: OTA consensus enables efficient computation of neighbor-averaged states for distributed controllers, leading to improved string stability and control performance (e.g., $14.22\%$ gain in positioning error in AirCons) (Lee et al., 2022), with order-of-magnitude reductions in channel usage for platoon-scale coordination (Epp et al., 21 Mar 2024).
• Federated learning and distributed optimization: OTA consensus underlies decentralized SGD, gradient tracking with variance reduction, and fully decentralized federated learning algorithms, achieving fast convergence with low communication overhead for convex and smooth objectives (Shi et al., 2021, Zhai et al., 2023, Madhan-Sohini et al., 2022).
• Blockchain and distributed ledgers: AirCon implements byzantine-fault-tolerant consensus via physical-layer hash aggregation, scaling from $O(K^2)$ to $O(K)$ channel uses and tolerating adverse wireless conditions (Xie et al., 2022).
• Random field estimation and sensor fusion: OTA consensus protocols for distributed convex optimization and field reconstruction can be coupled with “superiorization” for properties such as sparsity, achieving best-in-class spectral efficiency (Agrawal et al., 2023).

7. Limitations, Open Challenges, and Future Directions

Despite demonstrated robustness and scalability, several limitations persist:

• Scalability in group sizes: In some designs, the required resource block size grows exponentially with the number of concurrently-aggregated nodes or subcarrier groupings (e.g., $2^{|S_n|-1}$ subcarriers in AirCons), making hierarchical or cluster-based aggregation advisable for very large $N$ (Lee et al., 2022).
• Bias and residual error: In naive primal-averaging under non-coherent aggregation, consensus values can be biased by interference; careful formulation as a decentralized optimization problem is necessary to achieve unbiased consensus (Deng et al., 8 Apr 2025).
• Channel and synchronization robustness: While non-coherent schemes relax many physical constraints, severe fading, lack of reciprocity, and imperfect synchronization can still degrade performance, especially at low SNRs or with multipath (Xie et al., 2022, Michelusi, 2022).
• Generalization to nonlinear functions: Most OTA consensus methods focus on linear function computation; extension to more general nomographic or nonlinear forms remains an active area (Xie et al., 2022).
• Integration with higher-layer architectures: Joint optimization of transceiver beamformers, mixing matrices, and OTA consensus steps—for example, via alternating optimization—offers a path to maximize learning performance in federated scenarios under realistic wireless impairments (Zhai et al., 2023).

A plausible implication is that OTA consensus constitutes a foundational communication primitive for ultra-dense, rapidly time-varying cyber-physical systems. Its spectral efficiency and protocol scalability make it a key enabler for edge intelligence, massive Internet-of-Things deployments, and distributed autonomy in wireless environments. Continued development is anticipated on robustness to asynchrony, support for nonlinear aggregation, and rigorous analysis under non-convex objective landscapes.