Certified Adaptive Backup Rollouts

1. Integration point: use this as an inference-time safety filter around an RL policy or model-predictive planner, and optionally add the tightened margin as a training penalty. Choose a world model with a parameter-affine final layer, for example F(z,a) = F0(z,a;w) + Psi(z,a;w) theta, while keeping the feature extractor fixed or slowly updated. Define h(z) from physical limits, collision distance, or a separately validated safe-set classifier. 2. Pseudocode: maintain theta_hat and epsilon; at each decision, generate candidate actions, roll out z_hat from the learned model under the backup policy, initialize S_0 = 0, and recursively compute S_(k+1) = F_z S_k + F_theta. Reject or optimize only action sequences satisfying h(z_hat_k) - L_h r_k >= rho. For uncertain action gains, introduce auxiliary variables s, impose s >= c(a) and s >= -c(a), and include epsilon^T s in the robust constraint. 3. Computed from the paper's mechanism: adaptive estimates, component-wise error bounds, nominal backup flows, sensitivity tightening, and dualized box uncertainty. Estimated empirically: L_h, unmodeled rollout error, and calibration of epsilon; inflate epsilon until held-out perturbed trajectories are covered. 4. First cheap experiment: use CartPole or a two-dimensional point-mass obstacle task with a deliberately misspecified linear or MLP world model. Compare nominal MPC, ensemble MPC, and this filter over 1000 episodes while sweeping uncertainty scale and horizon. The quantitative prediction is a safety transition at the computed margin boundary: violations should remain near zero when min_k[h(z_hat_k) - L_h r_k] / rho >= 0 and rise rapidly after it becomes negative. With calibrated uncertainty boxes, at least 95 percent of held-out perturbed rollouts should satisfy the predicted margin; shrinking boxes below calibration should monotonically increase violation probability.