Biased Over-the-Air Federated Learning under Wireless Heterogeneity
Muhammad Faraz Ul Abrar Nicolò Michelusi
Recently, Over-the-Air (OTA) computation has emerged as a promising federated
learning (FL) paradigm that leverages the waveform superposition properties of
the wireless channel to realize fast model updates. Prior work focused on the
OTA device ``pre-scaler" design under \emph{homogeneous} wireless conditions,
in which devices experience the same average path loss, resulting in zero-bias
solutions. Yet, zero-bias designs are limited by the device with the worst
average path loss and hence may perform poorly in \emph{heterogeneous} wireless
settings. In this scenario, there may be a benefit in designing \emph{biased}
solutions, in exchange for a lower variance in the model updates. To optimize
this trade-off, we study the design of OTA device pre-scalers by focusing on
the OTA-FL convergence. We derive an upper bound on the model ``optimality
error", which explicitly captures the effect of bias and variance in terms of
the choice of the pre-scalers. Based on this bound, we identify two solutions
of interest: minimum noise variance, and minimum noise variance zero-bias
solutions. Numerical evaluations show that using OTA device pre-scalers that
minimize the variance of FL updates, while allowing a small bias, can provide
high gains over existing schemes.