Privacy Risk Scales with Effective Dimension in Federated Learning

Privacy mechanisms in federated learning (FL) are often calibrated without explicit regard to model scale, implicitly assuming that privacy risk remains stable as federated models grow. We challenge this assumption by introducing federated leakage, a signal-level mutual-information measure of client-specific information surviving in privatized update trajectories, and by identifying effective model dimension, rather than raw parameter count, as the operative scaling variable. Under a Gaussian signal-channel model with spectral growth of client-informative directions, we establish a regime-conditional scaling law: leakage grows as $\Theta(d/\log d)$ with effective dimension $d$. This result yields a leakage-ratio notion of privacy debt, where fixed-noise deployment can expose increasing client-informative signal even when the accountant-reported DP configuration is unchanged. We therefore propose a scale-aware Gaussian calibration rule that preserves the target leakage regime up to constant factors, and show across vision and language settings that it substantially reduces cross-scale privacy drift relative to fixed-noise baselines.