Asymptotic Anytime-Valid Inference for Federated Learning

Federated models are often inspected or released repeatedly during training, making uncertainty quantification at a single fixed horizon insufficient. We develop asymptotic anytime-valid inference for smooth, locally identifiable federated $M$-estimation with full-participation Local SGD and deterministic intermittent communication. For the Polyak-Ruppert average of synchronized iterates, we establish an almost-sure Gaussian approximation that accounts for both total local computation and the variance clock of communication-round averages. The approximation retains the exact finite-schedule normalization and controls the accumulated local drift and noise-freezing errors. Combining it with Gaussian time-uniform boundaries and a strongly consistent covariance estimator yields confidence sequences with asymptotic time-uniform coverage after any fixed, pre-specified multiplicative inflation. We also prove strong consistency of a federated sandwich estimator based on fresh samples. Experiments on generalized linear models and a public-health dataset examine coverage, precision, and sensitivity to communication schedules.