Adaptive Barrier-Margin Regularization

Add the adaptive barrier term to supervised control learning, behavioral cloning, or actor updates in an RL algorithm. Keep the inference-time QP safety filter active, but train the policy to produce actions that satisfy the reconstructed constraint before projection. Initially detach $\hat d$ and $\varepsilon_d$ from policy gradients so the policy cannot manipulate the observer or uncertainty estimate. For every transition, update the observer, estimate the one-step residual from a known or learned dynamics model, update $\varepsilon_d$ using the displayed clipped Robbins-Monro rule, compute $r_t$, and backpropagate $L_{\mathrm{task}}+\beta\operatorname{softplus}(r_t/\tau)^2$. If true disturbances are unavailable, use held-out model residuals in simulation where the perturbation is known. The first experiment should behavior-clone an inverse-dynamics controller for a cart-pole or two-dimensional point mass with bounded unmodeled force. Compare ordinary training, a fixed robust barrier penalty, and the adaptive reconstructed penalty. Record violations, QP intervention norm, tracking error, and empirical uncertainty coverage while sweeping disturbance amplitude. The prediction is a calibration transition: below the observer's capacity, coverage remains above $1-\delta$, violations remain near zero, and intervention norm decreases; once disturbance amplitude causes coverage to fall below $1-\delta$, violations rise sharply unless the margin is enlarged.