On the Degree of Safety: Beyond Safe or Unsafe with Control Barrier Functions

What the math gives to ML

The paper introduces invariance authority demand (IAD), a representation-independent measure of how much bounded control authority is required to render a set forward invariant. Its key warning is that the numerical value and gradient of a control barrier function are not intrinsic: equivalent defining functions can arbitrarily rescale or reshape them, so CBF values should not be used directly as safety margins. The transferable mechanism is to measure the ratio between unavoidable outward drift and the maximum inward correction available from the actuator, evaluated on the set boundary. In neural controlled systems, this can become a representation-invariant safety regularizer, actuator-sizing diagnostic, or feasibility monitor for neural ODEs, learned dynamics, and reinforcement-learning policies.