IQC-Certified Training Dynamics

First, integrate the mechanism into a recurrent optimizer or learned training controller whose state contains parameters, momentum, adaptive statistics, or a compact controller state. A cheap baseline is heavy-ball training with state x_k = [theta_k; theta_(k-1)] and a data-dependent gradient operator Phi; the same construction can later be applied to an RNN or state-space model used for long-horizon prediction. Second, use the following MVP procedure: sample a task and two datasets differing by one example; run the controller on both; estimate e_k = Phi(y_k,S) - Phi(y_k,S') after aligning the feedback state; estimate an incremental slope bound L from pairs of outputs and updates; choose an IQC such as Q = diag(L^2 I_p, -I_m) for the condition ||w_k|| <= L ||delta_y_k||; construct the lifted matrices T and H; solve an SDP for P, lambda, rho, and the smallest gamma; and add alpha log(gamma + 1e-8) or alpha max(0, lambda_max(M))^2 to the controller objective. Initially stop gradients through the SDP and alternate controller updates with certificate updates. The matrices A, B, C, and D come from the optimizer realization; L and one-sample disturbance magnitudes are estimated empirically, while the SDP is the formal certificate conditional on those estimates. Test on a d = 20 finite-sum quadratic regression problem with N = 100, comparing vanilla gradient descent, heavy-ball, Nesterov, and a learned two-state optimizer. Sweep learning rate and momentum, measure leave-one-sample-out sensitivity, loss divergence, and certified gamma. The prediction is that the empirical divergence or sensitivity boundary appears close to the first hyperparameter value where the LMI becomes infeasible, and cumulative sensitivity grows approximately linearly with gamma, with empirical sensitivity within a factor of two of the certificate.