Regressor-triggered federated gradient updates

1. Integration point: begin with a two-layer network whose feature extractor is frozen and whose trainable final layer is theta, or use online linear probing of a transformer embedding. Each federated client observes feature vectors C_i(t) and labels y_i(t), but does not continuously communicate. The server maintains theta and receives client sufficient statistics H_i=C_i^T C_i and b_i=C_i^T y_i. 2. Pseudocode: initialize H_i_held and b_i_held at the first client event; between events accumulate local minibatch statistics; if spectral_norm(H_i-H_i_held)>epsilon_H or norm(b_i-b_i_held)>epsilon_b, transmit both statistics and reset the held values. At the server, apply theta <- theta + eta * sum_i(b_i_held-H_i_held theta), or integrate the corresponding ODE with Euler steps. If exact convergence is required under changing features, use a decreasing threshold epsilon_H(t)=epsilon_H0 exp(-rho t); a fixed threshold is expected to produce a bounded error floor. 3. Computed quantities: H_i and b_i are exactly computable locally; spectral norms can be estimated with 3-5 power iterations. The persistent-excitation constants alpha and T are generally unknown for neural features, so estimate the minimum eigenvalue of rolling sums of H over logged windows. 4. First cheap experiment: freeze a ResNet-18 or small MLP feature extractor on CIFAR-10, partition data across 10 simulated clients, and train only the linear head. Compare minibatch-periodic communication with epsilon_H in {1e-4,1e-3,1e-2,1e-1}. Measure head error, loss decay, accuracy, rolling excitation eigenvalue, and transmission count. The quantitative prediction is that sufficiently small epsilon_H preserves the full-communication exponential-like loss-decay slope, while the asymptotic parameter or loss error grows approximately linearly with the stale-statistic bound. The largest stable threshold should decrease when the estimated excitation eigenvalue alpha decreases. Accept the mechanism if it achieves at least 5x fewer transmissions with less than 10% degradation in fitted loss-decay rate and if the observed threshold transition tracks the predicted dependence on alpha and estimator gain.