Adaptive Federated Learning Over the Air

We propose a federated version of adaptive gradient methods, particularly AdaGrad and Adam, within the framework of over-the-air model training. This approach capitalizes on the inherent superposition property of wireless channels, facilitating fast and scalable parameter aggregation. Meanwhile, it enhances the robustness of the model training process by dynamically adjusting the stepsize in accordance with the global gradient update. We derive the convergence rate of the training algorithms for a broad spectrum of nonconvex loss functions, encompassing the effects of channel fading, and interference that follows a heavy-tailed distribution. Our analysis shows that the AdaGrad-based algorithm converges to a stationary point at the rate of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal{O}(\text{ln}(T)/T^{1-\frac{1}{\alpha}})$</tex-math></inline-formula>, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha$</tex-math></inline-formula> represents the tail index of the electromagnetic interference. This result indicates that the level of heavy-tailedness in interference distribution plays a crucial role in the training efficiency: the heavier the tail, the slower the algorithm converges. In contrast, an Adam-like algorithm converges at the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal{O}(1/T)$</tex-math></inline-formula> rate, demonstrating its advantage in expediting the model training process. We conduct extensive experiments that corroborate our theoretical findings and affirm the practical efficacy of our proposed federated adaptive gradient methods.