Machine Learning at the Wireless Edge: Distributed Stochastic Gradient Descent Over-the-Air

We study federated machine learning (ML) at the wireless edge, where power- and bandwidth-limited wireless devices with local datasets carry out distributed stochastic gradient descent (DSGD) with the help of a parameter server (PS). Standard approaches assume separate computation and communication, where local gradient estimates are compressed and transmitted to the PS over orthogonal links. Following this digital approach, we introduce D-DSGD, in which the wireless devices employ gradient quantization and error accumulation, and transmit their gradient estimates to the PS over a multiple access channel (MAC). We then introduce a novel analog scheme, called A-DSGD, which exploits the additive nature of the wireless MAC for over-the-air gradient computation, and provide convergence analysis for this approach. In A-DSGD, the devices first sparsify their gradient estimates, and then project them to a lower dimensional space imposed by the available channel bandwidth. These projections are sent directly over the MAC without employing any digital code. Numerical results show that A-DSGD converges faster than D-DSGD thanks to its more efficient use of the limited bandwidth and the natural alignment of the gradient estimates over the channel. The improvement is particularly compelling at low power and low bandwidth regimes. We also illustrate for a classification problem that, A-DSGD is more robust to bias in data distribution across devices, while D-DSGD significantly outperforms other digital schemes in the literature. We also observe that both D-DSGD and A-DSGD perform better with the number of devices, showing their ability in harnessing the computation power of edge devices.