Non-Coherent Over-the-Air Decentralized Stochastic Gradient Descent

This paper proposes a Decentralized Stochastic Gradient Descent (DSGD) algorithm to solve distributed machine-learning tasks over wirelessly-connected systems, without the coordination of a base station. It combines local stochastic gradient descent steps with a Non-Coherent Over-The-Air (NCOTA) consensus scheme at the receivers, that enables concurrent transmissions by leveraging the waveform superposition properties of the wireless channels. With NCOTA, local optimization signals are mapped to a mixture of orthogonal preamble sequences and transmitted concurrently over the wireless channel under half-duplex constraints. Consensus is estimated by non-coherently combining the received signals with the preamble sequences and mitigating the impact of noise and fading via a consensus stepsize. NCOTA-DSGD operates without channel state information (typically used in over-the-air computation schemes for channel inversion) and leverages the channel pathloss to mix signals, without explicit knowledge of the mixing weights (typically known in consensus-based optimization). It is shown that, with a suitable tuning of decreasing consensus and learning stepsizes, the error (measured as Euclidean distance) between the local and globally optimum models vanishes with rate $\mathcal O(k^{-1/4})$ after $k$ iterations. NCOTA-DSGD is evaluated numerically by solving an image classification task on the MNIST dataset, cast as a regularized cross-entropy loss minimization. Numerical results depict faster convergence vis-\`a-vis running time than implementations of the classical DSGD algorithm over digital and analog orthogonal channels, when the number of learning devices is large, under stringent delay constraints.