Unverified 2026

Representation-Invariant Authority Margin

Implementation & benchmark of arXiv:2609.03319 — On the Degree of Safety: Beyond Safe or Unsafe with Control Barrier Functions

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Dynamics Stability Accuracy Generalization Rl Training dynamics Regularization World model Control theory Dynamical systems Optimization Geometry

Source paper: On the Degree of Safety: Beyond Safe or Unsafe with Control Barrier Functions arXiv:2609.03319 ⓘ · analyzed Sep 4, 2026

AI-generated research hypothesis, automatically tested. Not peer-reviewed.

Idea description

Replace raw control-barrier-function value penalties with an invariance-authority demand computed from boundary geometry and available control authority. For a learned or known control-affine neural dynamical system, penalize states where the uncontrolled vector field points outward more strongly than the actuator can push inward. The resulting quantity is invariant to positive rescaling of the barrier representation and directly predicts the actuator-strength threshold at which controlled invariance becomes feasible.

Formulas

$$\dot{x}=f_\theta(x)+g_\theta(x)u,\qquad u\in\mathcal U,\qquad \mathcal A=\{x:h(x)\ge 0\}.$$

$$r(x)=-\nabla h(x)^\top f_\theta(x),\qquad a(x)=\max_{u\in\mathcal U}\nabla h(x)^\top g_\theta(x)u.$$

$$d(x)=\begin{cases}\dfrac{\max\{0,r(x)\}}{a(x)},&a(x)>0,\\ +\infty,&a(x)=0\ \text{and}\ r(x)>0,\end{cases}\qquad D_{\mathrm{IAD}}=\sup_{x\in\partial\mathcal A}d(x).$$

$$\mathcal U_\rho=\rho\mathcal U:\qquad \text{controlled invariance is feasible only if }\rho\ge D_{\mathrm{IAD}};\qquad \tilde h=q(h)\Rightarrow \tilde d(x)=d(x)\text{ on }\partial\mathcal A.$$

Mathematical statement

Let the controlled dynamics be $\dot{x}=f_\theta(x)+g_\theta(x)u$, with admissible controls $u\in\mathcal U$, and let the safe set be $\mathcal A=\{x:h(x)\ge 0\}$, where $h$ is continuously differentiable and $\nabla h(x)\neq 0$ on the boundary $\partial\mathcal A=\{x:h(x)=0\}$. The outward derivative is $r(x)=-\nabla h(x)^\top f_\theta(x)$, and the maximum inward correction supplied by the actuator is $a(x)=\max_{u\in\mathcal U}\nabla h(x)^\top g_\theta(x)u$. Define the pointwise demand as $d(x)=\max(0,r(x))/a(x)$ when $a(x)>0$, and $d(x)=+\infty$ when $a(x)=0$ and $r(x)>0$. The set-level IAD is $D_{\mathrm{IAD}}=\sup_{x\in\partial\mathcal A}d(x)$. If the control set is scaled as $\mathcal U_\rho=\rho\mathcal U$, the predicted feasibility boundary is $\rho=D_{\mathrm{IAD}}$. Under a defining-function change $\tilde h=q(h)$, with $q(0)=0$ and $q'(0)>0$, both $r$ and $a$ are multiplied by $q'(0)$ on the boundary, so their ratio is unchanged.

Implementation notes

1. Integration point: use this as a safety regularizer and feasibility monitor for an MLP policy controlling a known or learned control-affine model, such as a pendulum, cart-pole, or two-dimensional neural ODE. Keep the policy output inside $\mathcal U$, but compute authority demand independently of the policy; this distinguishes an inadequate policy from an actuator that cannot make the set invariant. Choose a differentiable safe set $h(x)\ge0$, such as a quadratic region or a smooth soft-min encoding of box constraints. 2. Pseudocode: sample boundary states $x_i$ by perturbing replay states toward $h(x)=0$; compute $J_i=\nabla h(x_i)$ with autodiff; compute $r_i=-J_i f_\theta(x_i)$; solve $a_i=\max_{u\in\mathcal U}J_i g_\theta(x_i)u$. For box controls, compute this support function by selecting each control bound according to the sign of the corresponding coefficient. Form $d_i=\operatorname{relu}(r_i)/(a_i+\varepsilon)$, estimate the supremum with $L_{\mathrm{IAD}}=\tau\log\sum_i\exp(d_i/\tau)$, and add $\lambda\operatorname{softplus}(L_{\mathrm{IAD}}-\rho)$ to the policy or model loss. At inference, optionally project the policy through $J_i(f_\theta+g_\theta u)\ge0$. 3. Compute the barrier Jacobian and actuator support function exactly when the model is known; estimate the boundary supremum empirically with adaptive sampling, and estimate $f_\theta,g_\theta$ from a differentiable world model when needed. 4. First experiment: train identical MLP policies on cart-pole with actuator limits scaled by $\rho$, comparing raw barrier, standard CBF, and IAD penalties. Sweep $\rho$ and predict a sharp failure-to-invariance transition near the measured $D_{\mathrm{IAD}}$; the critical actuator scale should agree within 20%. Replace $h$ by $2h$, $h^3$, and $\log(1+h)$ where valid. IAD and its transition should change by less than 5%, whereas raw barrier penalties should change substantially. Track boundary violation rate, estimated IAD, and the minimum one-step safety derivative.

Verification

This idea has not been verified yet.

Verification happens in two stages: Stage 1 — a mechanism check on a toy system confirms the claimed mathematical phenomenon reproduces; Stage 2 — a benchmark implements the idea on a real (small) neural network task and compares it against a tuned baseline over 8 paired seeds with a permutation test.

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