⚡ electrical engineering

Transmit Coefficients and Receive Combining Vector Design for OTA-FL with Imperfect CSI

This paper proposes a joint optimization framework for transmit coefficients and receive combining vectors in over-the-air federated learning under imperfect channel state information, utilizing Lyapunov-based optimization to minimize long-term mean squared error and preserve model accuracy across multiple datasets.

Original authors: Xiaoyan Ma, Shahryar Zehtabi, Yinan Zou, Taejoon Kim, Christopher G. Brinton

Published 2026-07-21

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Xiaoyan Ma, Shahryar Zehtabi, Yinan Zou, Taejoon Kim, Christopher G. Brinton

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where your smartphone, your smartwatch, and your neighbor's tablet all want to learn together to get smarter, but they can't share their private photos or messages. This is the heart of Federated Learning (FL), a clever way for many devices to train a single, shared "brain" without ever sending their personal data to a central server. Instead of mailing their homework, they just send their "ideas" (mathematical updates) back and forth.

Now, imagine trying to shout these ideas over a crowded, noisy radio wave. If everyone shouts at once, their voices mix together. This is Over-the-Air (OTA) computation, a technique that treats the mixing of radio waves like a feature, not a bug. By shouting simultaneously, the devices can merge their ideas instantly, saving time and battery. However, there's a catch: the radio waves are fickle. They bounce off buildings, fade away, and get distorted by the environment. If the devices don't know exactly how the waves are behaving (a concept called Channel State Information or CSI), their mixed-up voices might come out garbled, ruining the learning process. This paper tackles the messy reality of these imperfect radio waves to make sure the group learning stays on track.

The Great Radio Choir: Fixing the Noise

In this study, the authors, Xiaoyan Ma and her team, looked at a specific problem: what happens when the devices trying to learn together don't have a perfect map of the radio waves? In the real world, knowing the exact state of the channel is like trying to predict the wind while running a marathon—it's always changing, and you can never be 100% sure. Most previous studies assumed the devices had a perfect, crystal-clear map of the wind, which is nice in theory but rarely happens in practice.

The researchers asked a big question: If the devices are shouting into a foggy radio channel, how do we make sure their combined voice is still clear enough to teach the global model? They focused on minimizing the Mean Squared Error (MSE), which is basically a fancy way of measuring "how much did the message get garbled?" The goal was to keep this garbling as low as possible over many rounds of learning, not just for a single shout.

The Two-Step Dance: Tuning the Volume and the Ears

To solve this, the team designed a two-part strategy involving the devices (the shouters) and the central server (the listener).

The Shouters (Transmit Coefficients): Each device needs to decide how loudly to shout. If they shout too loud, they waste battery; too soft, and the server can't hear them. The authors developed a way for each device to calculate the perfect volume based on the imperfect map they have of the radio waves. They realized that if the map is blurry (imperfect CSI), the device should be more careful, perhaps shouting a bit softer to avoid making the noise worse.
The Listener (Receive Combining Vector): The server has multiple antennas (like having multiple ears). It needs to figure out how to combine the mixed-up signals from all the devices to reconstruct the original message. The team designed a special mathematical "ear" that knows how to filter out the static caused by the blurry map.

The Time Travel Problem and the "Virtual Queue"

Here is where the story gets tricky. The first solution the team came up with was incredibly smart but required time travel. To calculate the perfect volume for today, the devices needed to know the radio conditions for every future round of learning. Since we can't see the future, this "non-causal" approach was great for theory but useless for real life.

To fix this, the authors introduced a clever trick using Lyapunov optimization, which they explained using a concept called a virtual queue. Imagine each device has a "debt meter" for its battery usage. If a device shouts too loudly today, the meter goes up. The system's job is to keep the meter from overflowing over the long run. By watching this meter, the devices can make smart decisions about how loud to shout right now, using only the information they have at this exact moment (causal CSI). This allows them to plan for the long term without needing to see the future.

What They Found

The team tested their ideas using famous image datasets like Fashion-MNIST (recognizing clothes), CIFAR-10 (recognizing simple objects), and CIFAR-100 (recognizing more complex objects). They simulated a world with 20 devices and various levels of radio noise.

Their simulations showed that their new methods worked wonders. Even when the radio maps were very blurry (high channel estimation errors), their approach kept the learning accuracy high.

The "Perfect" vs. "Real" Gap: They found that their method, even with imperfect maps, performed almost as well as if the devices had perfect maps. The drop in accuracy was tiny.
Beating the Basics: When they compared their method to older, simpler strategies (like just shouting at a fixed average volume or trying to shout louder to cancel out the noise), their approach consistently won. The older methods often got confused by the noise, leading to a "garbled" global model that learned poorly.
Scaling Up: They also checked what happened when they added more devices (up to 40). The system got better as more devices joined, thanks to having more diverse data, and their method handled the extra noise gracefully.

The Bottom Line

This paper doesn't claim to have solved every problem in wireless learning, but it offers a robust, practical way to handle the messy reality of imperfect radio signals. By treating the long-term energy usage like a debt meter and designing the devices to shout and listen intelligently based on what they know right now, the authors showed that we can build Federated Learning systems that are both fast and resilient. Even when the wind is blowing hard and the maps are blurry, the choir can still sing in tune.

Technical Summary: Transmit Coefficients and Receive Combining Vector Design for OTA-FL with Imperfect CSI

Problem Statement
Over-the-air (OTA) Federated Learning (FL) offers a promising solution to the communication bottlenecks in distributed machine learning by leveraging waveform superposition for simultaneous model aggregation. However, existing designs predominantly assume perfect Channel State Information (CSI), an idealization that rarely holds in practice due to time-varying channels and the scale of participating devices. In real-world scenarios, imperfect CSI introduces aggregation distortion, degrading the final learning performance. While prior studies have addressed instantaneous Mean Squared Error (MSE) minimization under imperfect CSI, they often neglect the iterative nature of FL, where aggregation errors accumulate across multiple communication rounds to affect long-term convergence. This paper addresses the gap in optimizing OTA-FL performance over multiple global aggregation rounds under imperfect CSI conditions, specifically focusing on the long-term MSE minimization problem.

Methodology
The authors propose a system model involving $K$ single-antenna local devices and a multi-antenna Parameter Server (PS). The core of the methodology involves a joint optimization of transmit coefficients at the devices and receive combining vectors at the PS to minimize the time-averaged MSE.

Convergence Analysis: The authors first establish a theoretical upper bound for the time-averaged gradient norms. This analysis explicitly links channel estimation uncertainty to aggregation distortion and learning convergence, demonstrating that minimizing the time-averaged MSE is critical for improving long-term training performance.
Non-Causal Optimization Framework: An alternating optimization approach is developed to solve the joint design problem. The original problem is decomposed into two subproblems:
- Transmit Coefficient Design: Optimized for each device subject to maximum and average power constraints. Closed-form solutions are derived, involving a criterion check to determine if a bisection method is required to satisfy long-term average power constraints.
- Receive Combining Vector Design: Optimized at the PS to minimize the instantaneous MSE given the transmit coefficients. A closed-form solution is derived based on a modified Minimum Mean Square Error (MMSE) criterion that accounts for channel estimation errors.
Causal Optimization via Lyapunov Optimization: Recognizing that the non-causal approach (requiring future channel states) is impractical for real-time implementation, the authors introduce a Lyapunov-based optimization method. By employing virtual queues to track cumulative energy consumption, this method decouples the long-term power constraints into per-round constraints. This allows the transmit coefficients to be optimized using only causal CSI (current round information) while still satisfying long-term average power limits.

Key Contributions

Imperfect CSI Framework: The study moves beyond idealized assumptions by considering a practical setting where local devices upload parameters based on imperfect channel estimates, jointly optimizing both transmitter and receiver under these conditions.
Long-Term Performance Characterization: Unlike prior works focusing on single-round performance, this paper develops a unified analytical formulation that captures the propagation of channel estimation uncertainty through multiple aggregation rounds, explicitly linking CSI uncertainty to learning convergence.
Joint Optimization with Closed-Form Solutions: A tractable optimization framework is proposed that decomposes the joint problem into convex subproblems, yielding closed-form solutions for transmit power and receive combining vectors to reduce design complexity.
Causal Implementation: The introduction of a Lyapunov-based design scheme effectively handles temporal dependencies, enabling transmit coefficient optimization based solely on causal CSI without sacrificing long-term power constraints.

Results
Numerical evaluations were conducted using Fashion-MNIST, CIFAR-10, and CIFAR-100 datasets under both Independent and Identically Distributed (IID) and non-IID data settings.

Accuracy Recovery: The proposed algorithms significantly mitigate the degradation of test accuracy caused by imperfect OTA aggregation channels. The performance gap between the proposed designs and the ideal upper bound (perfect CSI) is shown to be minimal.
Robustness: The schemes demonstrate robustness against varying levels of channel estimation errors, maintaining stable test accuracy even as error variance increases.
Comparison with Benchmarks: The proposed non-causal and causal CSI-aided schemes outperform several benchmark methods, including average power transmission, channel inversion power control, worst-case CSI-aware power control, normalized Maximum Ratio Combining (MRC), and direct combining. Notably, benchmarks that ignore channel estimation errors or fail to account for long-term accumulation (such as standard MMSE combining) show inferior performance.
Scalability: Simulations indicate that test accuracy scales with the number of local devices up to a point, after which data redundancy leads to diminishing returns, with the proposed schemes consistently outperforming baselines across different network sizes.

Significance
The paper claims that its primary significance lies in bridging the research gap between instantaneous aggregation error minimization and long-term learning performance in OTA-FL systems under realistic channel conditions. By providing a theoretical bound that connects channel estimation errors to convergence behavior and offering a practical, causal optimization framework, the work enables more reliable and efficient OTA-FL deployments. The results verify that the proposed designs can effectively manage imperfect aggregation channels, ensuring that the scalability and low-latency benefits of OTA-FL are not compromised by channel estimation uncertainties.

The Great Radio Choir: Fixing the Noise

The Two-Step Dance: Tuning the Volume and the Ears

The Time Travel Problem and the "Virtual Queue"

What They Found

The Bottom Line

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