Operator-Guided Invariance Learning for Continuous Reinforcement Learning

Researchers propose VPSD-RL, a reinforcement learning framework that discovers value-preserving structures in continuous control tasks using Lie-group operators and diffusion models. The method improves data efficiency and robustness by identifying nonlinear transformations that preserve optimal value functions, addressing brittleness in RL systems under environmental variability.

VPSD-RL addresses a fundamental challenge in continuous reinforcement learning: the field's sensitivity to nuisance variability and distributional shifts that degrade performance despite theoretical optimality. Traditional approaches either assume predefined symmetries or require exact equivariance, limiting their applicability to real-world problems where value-preserving structures are unknown and potentially nonlinear. This research extends beyond these constraints by automatically discovering transformation operators through mathematical machinery based on Lie groups and controlled diffusions.

The framework's novelty lies in its principled approach to structure discovery. By modeling continuous RL as a controlled diffusion process, the authors establish conditions under which value-preserving mappings exist—specifically, when pulling back value functions and pushing forward actions commute with the controlled generator and reward functional. The method learns infinitesimal generators through residual minimization, exponentiates them to obtain finite transformations, and incorporates these into learning through augmentation and regularization.

For the broader AI community, this work contributes theoretical grounding to a practical problem: improving sample efficiency and robustness in continuous control tasks. The quantitative stability guarantees tied to generator mismatches and effective horizons provide confidence bounds for approximate structures. On benchmarks, the approach demonstrates improved data efficiency and robustness—critical metrics for real-world deployment where sample collection is expensive.

The research bridges theoretical mathematics (differential geometry, Lie theory) with applied RL, potentially influencing how robotics and autonomous systems are trained. Future validation on complex industrial control problems and comparison with recent equivariance-learning methods will clarify practical advantages. The work also raises questions about computational overhead of generator discovery relative to performance gains.

• →VPSD-RL automatically discovers nonlinear value-preserving structures without requiring predefined symmetries or exact equivariance assumptions.
• →The framework provides rigorous quantitative guarantees for approximate structures when Hamilton-Jacobi-Bellman mismatch is bounded.
• →Experimental results show improved data efficiency and robustness on continuous-control benchmarks through transformation-consistency regularization.
• →The method uses Lie-group operators and controlled diffusions to model continuous RL, connecting differential geometry with practical learning algorithms.
• →Generator learning via determining-equation residual minimization enables scalable discovery of complex transformation operators.