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

2026-07-14

AI Summary

This work addresses the limitation of conventional reinforcement learning, which typically optimizes only the policy while ignoring the joint optimization of tunable environmental parameters. The authors propose an Environmental Parameter Gradient Theorem that introduces a generalized action-value function incorporating dual environmental parameters, thereby decoupling current and future environmental dynamics. This formulation yields a closed-form gradient expression, enabling model-free joint optimization of both policy and environmental parameters. The method is validated on a drone network design task, where it successfully co-optimizes drone deployment locations and communication paths, significantly reducing total communication cost and demonstrating its effectiveness and practical utility.

Abstract

Reinforcement learning (RL) is traditionally concerned with learning a control policy for a fixed environment. In many engineering systems, however, the environment itself is alterable: physical or operational parameters can be tuned to shape the transition dynamics and costs experienced by the agent. This motivates jointly optimizing both the policy and the environment design parameters. To this end, we establish an Environment Parameter Gradient Theorem -- a formal expression for the gradient of the value function with respect to environment parameters. The key theoretical device is a generalized action-value function $Q_{π,ξ}(s,a,ζ)$, which comprises two copies of the environment parameters: $ζ$ governs the cost and transition dynamics at the current state--action pair, while $ξ$ governs the future rollouts. This decoupling yields a tractable closed-form gradient expression and is essential to the theorem's derivation. Building on this result, we develop a model-free algorithm that simultaneously learns the optimal policy and the environment parameters. We demonstrate the efficacy of our framework on a UAV network design problem, where the optimal UAV placement (environment parameters) and communication routes (governed by the policy) are learned jointly to minimize the total communication cost in the network.

Problem

• reinforcement learning
• environment design
• policy optimization
• co-design
• environment parameters

Innovation

• Environment Parameter Gradient
• Policy-Environment Co-Design
• Generalized Action-Value Function
• Model-Free Reinforcement Learning
• Joint Optimization