Constrained Over-the-Air Model Updating for Wireless Online Federated Learning with Delayed Information

We study online federated learning over a wireless network, where the central server updates an online global model sequence to minimize the time-varying loss of multiple local devices over time. The server updates the global model through over-the-air model-difference aggregation from the local devices over a noisy multiple-access fading channel. We consider the practical scenario where information on both the local loss functions and the channel states is delayed, and each local device is under a time-varying power constraint. We propose Constrained Over-the-air Model Updating with Delayed infOrmation (COMUDO), where a new lower-and-upper-bounded virtual queue is introduced to counter the delayed information and control the hard constraint violation. We show that its local model updates can be efficiently computed in closed-form expressions. Furthermore, through a new Lyapunov drift analysis, we show that COMUDO provides bounds on the dynamic regret, static regret, and hard constraint violation. Simulation results on image classification tasks under practical wireless network settings show substantial accuracy gain of COMUDO over state-of-the-art approaches, especially in the low-power region.

• We propose an effective algorithm named Constrained Over-the-air Model Updating with Delayed infOrmation (COMUDO) to solve this problem. COMUDO introduces a new lower-and-upper-bounded virtual queue, which eliminates the need for Slater's condition and enforces a minimum constraint penalty, to strictly control the power violation. The resulting closed-form local updates can adapt to unknown variations of both the local loss functions and channel states under individual power limits.

• We establish a connection between the bounds on the virtual queue and the hard constraint violation through a new Lyapunov drift analysis. We show that COMUDO provides O(T 1+maxµ,ω 2

) dynamic regret, O(T 1+ω 2 ) static regret, and O(T ν ) hard constraint violation for general convex loss functions. Here, T is the number of training rounds, and µ, ν, and ω respectively represent how fast the dynamic online benchmark, power constraints, and channel noise fluctuate over time.

• We experiment on canonical datasets over typical wireless settings to study the performance of COMUDO for both convex and non-convex loss functions. Our simulation results show that COMUDO significantly improves the learning performance over state-of-the-art benchmarks. The performance gain is more substantial in the lowpower region.