Federated Learning: A Stochastic Approximation Approach

Abstract

This paper considers the Federated learning (FL) in a stochastic approximation (SA) framework. Here, each client $i$ trains a local model using its dataset $\mathcal{D}^{(i)}$ and periodically transmits the model parameters $w^{(i)}_n$ to a central server, where they are aggregated into a global model parameter $\bar{w}_n$ and sent back. The clients continue their training by re-initializing their local models with the global model parameters. Prior works typically assumed constant (and often identical) step sizes (learning rates) across clients for model training. As a consequence the aggregated model converges only in expectation. In this work, client-specific tapering step sizes $a^{(i)}_n$ are used. The global model is shown to track an ODE with a forcing function equal to the weighted sum of the negative gradients of the individual clients. The weights being the limiting ratios $p^{(i)}=\lim_{n \to \infty} \frac{a^{(i)}_n}{a^{(1)}_n}$ of the step sizes, where $a^{(1)}_n \geq a^{(i)}_n, \forall n$. Unlike the constant step sizes, the convergence here is with probability one. In this framework, the clients with the larger $p^{(i)}$ exert a greater influence on the global model than those with smaller $p^{(i)}$, which can be used to favor clients that have rare and uncommon data. Numerical experiments were conducted to validate the convergence and demonstrate the choice of step-sizes for regulating the influence of the clients.