From Ticks to Flows: Dynamics of Neural Reinforcement Learning in Continuous Environments

Researchers present a theoretical framework for deep reinforcement learning in continuous environments using continuous-time stochastic processes and stochastic control theory. The work establishes a two time-scale model for actor-critic algorithms with neural networks, deriving equations that describe how state distributions evolve during training in the infinite width limit.

This research advances theoretical understanding of how neural networks learn in continuous control problems, a foundational challenge in reinforcement learning. The authors bridge discrete algorithmic updates with continuous-time mathematics, providing rigorous theoretical tools previously unavailable for analyzing RL in continuous domains. By modeling the environment and gradient descent as separate time scales, they create a mathematical framework that separates environmental dynamics from learning dynamics, enabling more precise analysis of convergence behavior.

The work builds on recent trends in neural network theory that leverage infinite-width limits and mean-field analysis to derive tractable characterizations of deep learning dynamics. This particular contribution extends those insights to the RL setting with stochastic transitions and exploration noise, making it applicable to realistic control problems. The derivation of infinitesimal change equations for state distributions under small learning rates mirrors techniques from statistical physics and stochastic analysis.

For the broader AI research community, this theoretical framework could accelerate development of more interpretable and stable RL algorithms. Understanding the precise dynamics of actor-critic methods under different conditions may inform algorithm design and hyperparameter selection. The nonparametric formulation for overparametrized networks aligns with observed empirical success in practice, potentially explaining why over-parameterization benefits RL agents. While currently validated only on toy tasks, the theoretical insights may guide development of more efficient continuous control algorithms in robotics and other applications requiring smooth action spaces.

• →Novel continuous-time stochastic framework characterizes deep RL dynamics with unprecedented mathematical rigor.
• →Two time-scale decomposition separates environment dynamics from neural network learning, enabling clearer theoretical analysis.
• →Infinite width limit analysis derives tractable equations describing state distribution evolution during training.
• →Framework applies to actor-critic algorithms with exploration and stochastic transitions in continuous environments.
• →Theoretical predictions validated empirically on continuous control tasks, suggesting practical relevance.