  In this paper we seek to quantify the ability of learning to improve safety
guarantees endowed by Control Barrier Functions (CBFs). In particular, we
investigate how model uncertainty in the time derivative of a CBF can be
reduced via learning, and how this leads to stronger statements on the safe
behavior of a system. To this end, we build upon the idea of Input-to-State
Safety (ISSf) to define Projection-to-State Safety (PSSf), which characterizes
degradation in safety in terms of a projected disturbance. This enables the
direct quantification of both how learning can improve safety guarantees, and
how bounds on learning error translate to bounds on degradation in safety. We
demonstrate that a practical episodic learning approach can use PSSf to reduce
uncertainty and improve safety guarantees in simulation and experimentally.
