Schur Interaction Monitor for Adaptive Hyperparameters

Attach this mechanism controller above an existing optimizer rather than replacing its parameter update. Partition the network into m groups, for example transformer layers or ResNet blocks, and let u_j be a log learning-rate multiplier, momentum interpolation coefficient, or low-rank preconditioner amplitude for group j. Maintain a hidden controller sigma of k coefficients that generates the actual optimizer state or preconditioner. Every T training steps, compute a validation or training hypergradient h=partial L/partial u using one-step unrolling or implicit differentiation. At the current relaxed state, compute G=partial^2E/(partial sigma partial u) with Jacobian-vector products: for each mechanism basis vector e_j, differentiate the controller-gradient with respect to u_j. Compute H=partial^2E/partial sigma^2, add lambda I, and solve k-by-k linear systems (H+lambda I)X=G. Form R=G^T X and update u with delta u=-eta(I+beta R)^{-1}h; use Cholesky if H is positive definite and a small eigendecomposition if not. Do not form full parameter Hessians. The mathematical quantity R is computed from the paper's Schur formula; h, H, G, lambda, and beta are estimated numerically. The first cheap experiment is a 6-layer transformer on WikiText-2 or a CIFAR-10 ResNet with four layer groups. Compare independently tuned group learning rates against the Schur-preconditioned controller at the same hypergradient frequency. Log the largest eigenvalue of R, update norms, loss spikes, and final validation loss. Success is fewer instability spikes and faster validation-loss reduction, especially when mechanisms are deliberately initialized with conflicting learning-rate or momentum values.