Environment Parameter Gradient Theorem for Policy-Environment Co-Design in Reinforcement Learning

In traditional reinforcement learning (RL), an agent learns a policy within a fixed environment. However, many engineering problems—such as communication network design or robot morphology—require optimizing the environment's physical or operational parameters alongside the agent's control policy. This paper addresses the challenge of performing this joint optimization in a model-free setting where the underlying transition dynamics and costs are unknown.

The author derives the Environment Parameter Gradient (EnvPG) Theorem, which provides a formal gradient expression for the value function with respect to environment parameters. The central innovation is a generalized action-value function, $Q_{\pi,\xi}(s,a,\zeta)$, which uses two distinct copies of the environment parameters: $\zeta$ governs the immediate transition and cost, while $\xi$ governs future rollouts. By treating these as independent variables during the derivation, the author creates a tractable, recursive gradient expression. To implement this, the author develops a model-free actor-critic algorithm that uses controlled perturbations of the environment parameters during training to estimate the gradient, effectively operationalizing the two-copy theoretical framework.

The paper demonstrates that the EnvPG theorem allows for the simultaneous learning of optimal policies and environment design parameters without requiring explicit knowledge of the environment's transition dynamics. In a UAV network design case study, the algorithm successfully co-optimized UAV placement and communication routing policies. The results showed that the learned designs achieved performance within 0.9% of an oracle model that had full knowledge of the environment's dynamics, validating the efficacy of the gradient-based approach.

This work bridges the gap between standard RL and system co-design. By providing a principled, gradient-based foundation for tuning environment parameters, it enables more efficient and automated design processes in complex systems like robotics and communication networks, where the physical configuration is as critical to performance as the control strategy itself.