Equilibrium of Feasible Zone and Uncertain Model in Safe Exploration

Ensuring the safety of environmental exploration is a critical problem in reinforcement learning (RL). While limiting exploration to a feasible zone has become widely accepted as a way to ensure safety, key questions remain unresolved: what is the maximum feasible zone achievable through exploration, and how can it be identified? This paper, for the first time, answers these questions by revealing that the goal of safe exploration is to find the equilibrium between the feasible zone and the environment model. This conclusion is based on the understanding that these two components are interdependent: a larger feasible zone leads to a more accurate environment model, and a more accurate model, in turn, enables exploring a larger zone. We propose the first equilibrium-oriented safe exploration framework called safe equilibrium exploration (SEE), which alternates between finding the maximum feasible zone and the least uncertain model. Using a graph formulation of the uncertain model, we prove that the uncertain model obtained by SEE is monotonically refined, the feasible zones monotonically expand, and both converge to the equilibrium of safe exploration. Experiments on classic control tasks show that our algorithm successfully expands the feasible zones with zero constraint violation, and achieves the equilibrium of safe exploration within a few iterations.