$\lambda$-Reachability: Geometric-Horizon Safety Bellman Equations for Humanoid Safety
About
We introduce $\lambda$-Reachability, a scalable approach to Hamilton--Jacobi safety analysis for high-dimensional robotic systems. Unlike prior discounted formulations that rely on fixed one-step Bellman updates, $\lambda$-Reachability employs a stochastic multi-step estimator of the safety value, using a geometrically distributed rollout horizon together with a randomly absorbed terminal. Conceptually analogous to TD($\lambda$), $\lambda$-Reachability interpolates between local self-consistency updates and long-horizon max-over-trajectory safety targets via an interpretable horizon-control parameter. Unlike TD($\lambda$), where the terminal value is always incorporated in learning targets, the terminal safety value in $\lambda$-Reachability is only used at a probability controlled by parameter $\delta$. We formally show that for $\delta<1$, the update induces a contraction mapping that allows temporal-difference learning; as $\lambda \to 1$, the estimator recovers the undiscounted reachability objective. We apply $\lambda$-Reachability to high-dimensional safety learning problems with both simulated and real humanoid robots under balance and collision avoidance constraints. Experimental results demonstrate that $\lambda$-Reachability significantly improves both safe-set boundary classification and safety margin estimation compared to single-step temporal-difference baselines.