On the Gradient Heterogeneity Dynamics of Adversarially Robust Federated Regression

Federated learning (FL) is intrinsically heterogeneous: honest clients may have different data-generating models. On top of that, adversarial clients can make heterogeneity even more pronounced by sharing arbitrary updates. Existing analyses typically control the interaction between statistical heterogeneity and adversarial behavior through gradient-dissimilarity conditions. However, the underlying bound is imposed a priori and may yield conservative guarantees even for least-squares regression. We instead derive the gradient heterogeneity from the statistical model of linear and nonlinear regression with fresh data samples at every round. Our bounds separate heterogeneity among the honest clients' ground-truth model parameters, finite-sample label noise, and initialization. We then demonstrate that, for any $(f,κ)$-robust aggregator with coefficient $κ= O(f/n)$, where $f$ is the number of adversarial clients and $n$ the total number of clients (with $f/n < 1/2$), convergence holds after an explicit sample burn-in.