Curvature-Guided Safety Filters: State-Dependent Hessian-Weighted Projection with Provable Performance Bounds

Abstract

Safety filters provide a lightweight mechanism for enforcing state and input safety in learning-enabled control. However, common Euclidean projections onto the safe set disregard long-term performance, while directly optimizing the action-value function within the safe set can be nonconvex and computationally prohibitive. This paper proposes a state-dependent, Hessian-guided projection for safety filtering that preserves convexity while improving performance. The key idea is to select a weighted projection matrix from the curvature of the action-value function, thereby biasing the correction toward action directions with higher value sensitivity. We establish (i) a uniform bound on the performance gap between the weighted projection and the safe value-optimal action, and (ii) a condition under which the weighted projection outperforms the Euclidean projection in long-term value. To support black-box controllers, we further present a data-driven construction of the weighted projection matrix via an iterative Q-function learning algorithm with quadratic feature blocks and regularization that enforces curvature dominance and bounded higher-order terms. Simulations on a quadrotor tracking-and-avoidance task indicate that the proposed filter maintains safety while reducing value degradation relative to Euclidean projection, with computational overhead compatible with real-time operation.