First-order Sobolev Reinforcement Learning

Fabian Schramm, Nicolas Perrin-Gilbert, Justin Carpentier

Keywords: Reinforcement Learning, Sobolev Training

TL;DR: We refine TD learning by enforcing first-order Bellman consistency, matching both values and their derivatives through differentiable dynamics to improve critic convergence and policy stability.

Abstract:

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.

Email Sharing: We authorize the sharing of all author emails with Program Chairs.

Data Release: We authorize the release of our submission and author names to the public in the event of acceptance.

Submission Number: 16