Gram-multilevel Gauss–Newton optimizer

Integrate this at the optimizer step for a least-squares model, immediately after computing a minibatch residual r and Jacobian J. Do not form J^T J explicitly. Use parameter blocks omega_i, initially one block per layer or per group of neurons. First compute approximate row supports: for each residual/Jacobian row j, record parameter blocks whose Jacobian norm exceeds a threshold tau; define each block's overlap Omega_i as the union of blocks touched by those rows. Pseudocode: (1) compute r and J or Jacobian-vector products; (2) set A_lambda = J^T J + lambda I conceptually; (3) for every overlapping Omega_i, form the restricted Jacobian J_i and solve (J_i^T J_i + lambda I) z_i = R_i v using a small dense Cholesky or a few CG iterations; combine z = sum_i R_i^T z_i with diagonal overlap normalization to obtain M^{-1}v; (4) build P by taking k low-energy eigenvectors from each local generalized problem (J_i^T J_i + lambda I)u = mu (D_i + lambda I)u, where D_i is the local diagonal or block-diagonal preconditioner; discard duplicate vectors and orthonormalize; (5) apply the coarse correction by solving A_c y = P^T b with A_c represented as (JP)^T(JP) + lambda P^T P; (6) run preconditioned CG on A_lambda Delta_theta = -J^T r, then update theta <- theta + Delta_theta. The paper supplies the Gram preservation, row-closure rule, and multiplicative error-correction structure; the eigenvalue cutoff k, overlap threshold tau, damping lambda, and CG tolerance are empirical. Start with a two-level method and k=2 to 8 modes per layer. The first experiment should use a width-256 MLP on a 20-dimensional nonlinear regression dataset, comparing Adam, ordinary damped Gauss–Newton CG, block-Jacobi-preconditioned CG, and this method at equal Jacobian evaluations. Measure residual decrease per CG iteration, wall-clock time to a fixed training loss, number of CG iterations, and damping sensitivity. Success is a 2x reduction in CG iterations or a lower loss at equal Jacobian/FLOP budget without worsening final residual; a useful negative result is no improvement over block Jacobi when supports are dense.