First-order Sobolev Reinforcement Learning
Published in Workshop on Differentiable Systems and Scientific Machine Learning, EurIPS 2025, 2025
Recommended citation: Schramm, F., Perrin-Gilbert, N., & Carpentier, J. (2025). "First-order Sobolev Reinforcement Learning." Workshop on Differentiable Systems and Scientific Machine Learning, EurIPS 2025. https://arxiv.org/abs/2511.19165
We propose a refinement of temporal-difference learning that enforces first-order Bellman consistency: the learned value function is trained to match not only the Bellman targets in value but also their derivatives with respect to states and actions. By differentiating the Bellman backup through differentiable dynamics, we obtain analytically consistent gradient targets. Incorporating these into the critic objective using a Sobolev-type loss encourages the critic to align with both the value and local geometry of the target function. This first-order TD matching principle can be seamlessly integrated into existing algorithms, such as Q-learning or actor-critic methods (e.g., DDPG, SAC), potentially leading to faster critic convergence and more stable policy gradients without altering their overall structure.