Title: GDC: From Brittle Optimality to Robust Satisfiability  via Riemannian Risk Geometry
Primary area: applications to robotics, autonomy, planning
Keywords: Differential Geometry, Risk-Sensitive Control, Robustness

Abstract: Standard reinforcement learning (RL) often yields brittle policies that fail under hard safety constraints. We propose **Geodesic Duality Control (GDC)**, which adapts an agent’s risk posture endogenously by re-weighting the Bellman target using local geometric cues of the value function—specifically, the gradient magnitude $||\nabla V||$ and a curvature surrogate $\kappa$. To accommodate piecewise-smooth neural critics, we formulate a Sub-Riemannian / generalized-gradient treatment and provide practical, numerically stable curvature surrogates. Our main theoretical result shows that, under explicit regularity and stochastic-model assumptions, GDC induces a curvature-decreasing learning dynamic:

$$
\frac{d\kappa}{dt} < 0,
$$

which in turn increases a quantifiable safety margin $\Delta_s$. We validate the mechanism with proof-of-concept experiments—including a hard-boundary safety environment (*Optimal-Trap*), targeted ablations, and a computational-cost study on Humanoid-Safety—to confirm the intended geometric risk posture. We do not claim broad empirical superiority on all benchmarks; rather, the paper’s primary contribution is theoretical, with key components validated empirically.

TL;DR: We propose Geodesic Duality Control (GDC), a reinforcement learning method that adjusts the Bellman target based on geometric curvature to improve policy safety margins while maintaining performance.