First-order Sobolev Reinforcement Learning

TL;DR

This paper introduces a first-order Sobolev approach to reinforcement learning that enforces derivative consistency in value functions, leading to potentially faster convergence and more stable policies.

Contribution

It presents a novel first-order Bellman consistency method that incorporates derivatives into the critic training, enhancing existing RL algorithms.

Findings

Improved critic convergence speed.

Enhanced stability of policy gradients.

Seamless integration with existing algorithms.

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.