Model-Agnostic Solutions for Deep Reinforcement Learning in Non-Ergodic Contexts

Abstract

Reinforcement Learning (RL) remains a central optimisation framework in machine learning. Although RL agents can converge to optimal solutions, the definition of ``optimality'' depends on the environment's statistical properties. The Bellman equation, central to most RL algorithms, is formulated in terms of expected values of future rewards. However, when ergodicity is broken, long-term outcomes depend on the specific trajectory rather than on the ensemble average. In such settings, the ensemble average diverges from the time-average growth experienced by individual agents, with expected-value formulations yielding systematically suboptimal policies. Prior studies demonstrated that traditional RL architectures fail to recover the true optimum in non-ergodic environments. We extend this analysis to deep RL implementations and show that these, too, produce suboptimal policies under non-ergodic dynamics. Introducing explicit time dependence into the learning process can correct this limitation. By allowing the network's function approximation to incorporate temporal information, the agent can estimate value functions consistent with the process's intrinsic growth rate. This improvement does not require altering the environmental feedback, such as reward transformations or modified objective functions, but arises naturally from the agent's exposure to temporal trajectories. Our results contribute to the growing body of research on reinforcement learning methods for non-ergodic systems.