Xinyi Sheng

Dominik Baumann, Erfaun Noorani, Arsenii Mustafin et al.

In reinforcement learning, we typically aim to optimize the expected value of the sum of rewards an agent collects over a trajectory. However, if the process generating these rewards is non-ergodic, the expected value, i.e., the average over infinitely many trajectories with a given policy, is uninformative for the average over a single, but infinitely long trajectory. Thus, if we care about how the individual agent performs during deployment, the expected value is not a good optimization objective. In this paper, we discuss the impact of non-ergodic reward processes on reinforcement learning agents through an instructive example, relate the notion of ergodic reward processes to more widely used notions of ergodic Markov chains, and present existing solutions that optimize long-term performance of individual trajectories under non-ergodic reward dynamics.

Arsenii Mustafin, Xinyi Sheng, Dominik Baumann

The theoretical analysis of Markov Decision Processes (MDPs) is commonly split into two cases - the average-reward case and the discounted-reward case - which, while sharing similarities, are typically analyzed separately. In this work, we extend a recently introduced geometric interpretation of MDPs for the discounted-reward case to the average-reward case, thereby unifying both. This allows us to extend a major result known for the discounted-reward case to the average-reward case: under a unique and ergodic optimal policy, the Value Iteration algorithm achieves a geometric convergence rate.

Xinyi Sheng, Dominik Baumann

Reinforcement learning (RL) algorithms typically optimize the expected cumulative reward, i.e., the expected value of the sum of scalar rewards an agent receives over the course of a trajectory. The expected value averages the performance over an infinite number of trajectories. However, when deploying the agent in the real world, this ensemble average may be uninformative for the performance of individual trajectories. Thus, in many applications, optimizing the long-term performance of individual trajectories might be more desirable. In this work, we propose a novel RL algorithm that combines the standard ensemble average with the time-average growth rate, a measure for the long-term performance of individual trajectories. We first define the Bellman operator for the time-average growth rate. We then show that, under multiplicative reward dynamics, the geometric mean aligns with the time-average growth rate. To address more general and unknown reward dynamics, we propose a modified geometric mean with $N$-sliding window that captures the path-dependency as an estimator for the time-average growth rate. This estimator is embedded as a regularizer into the objective, forming a practical algorithm and enabling the policy to benefit from ensemble average and time-average simultaneously. We evaluate our algorithm in challenging simulations, where it outperforms conventional RL methods.