Local Hermite Action Surrogate

Use this as a derivative-free local planner around a policy action or as an inner optimizer for a learned world-model objective. At each replanning step, obtain the nominal action a0 from the policy. Select coordinate radii h_r and evaluate the expensive objective J at a0, at a0 plus and minus h_r e_r, and, when cross terms are required, at a0 plus signed combinations of two coordinate directions. Build each feature vector phi(u)=[1,u_1,...,u_m,u_1^2,...,u_m^2,u_1u_2,...,u_{m-1}u_m]. Solve the small weighted least-squares system (Phi^T R Phi+lambda I)theta=Phi^T R y, where y contains direct objective values, R contains confidence or recency weights, and lambda is a ridge stabilizer. Optimize the fitted quadratic over a bounded trust region U using projected gradient descent, enumeration for one or two dimensions, or a box-constrained quadratic solver. If the fitted Hessian is indefinite, retain the trust-region minimizer or add diagonal damping eta I. The formulas determine coefficient estimation; model mismatch, trust-region radius, and damping are tuned empirically. First test on Pendulum or a 2D synthetic control task with a neural dynamics model, comparing nine objective evaluations against finite-difference Taylor optimization and random shooting. Measure objective regret after replanning, action quality, and wall-clock time. Success is lower regret at equal evaluations, particularly when rollout noise or imperfect gradients destabilizes the Taylor baseline.