From Task Distributions to Expected Paths Lengths Distributions: Value Function Initialization in Sparse Reward Environments for Lifelong Reinforcement Learning

This paper studies value function transfer within reinforcement learning frameworks, focusing on tasks continuously assigned to an agent through a probabilistic distribution. Specifically, we focus on environments characterized by sparse rewards with a terminal goal. Initially, we propose and theoretically demonstrate that the distribution of the computed value function from such environments, whether in cases where the goals or the dynamics are changing across tasks, can be reformulated as the distribution of the number of steps to the goal generated by their optimal policies, which we name expected optimal path length. To test our propositions, we hypothesize that the distribution of the expected optimal path lengths resulting from the task distribution is normal. This claim leads us to propose that if the distribution is normal, then the distribution of the value function follows a log-normal pattern. Leveraging this insight, we introduce "LogQInit" as a novel value function transfer method, based on the properties of log-normality. Finally, we run experiments on a scenario of goals and dynamics distributions, validate our proposition by providing an a dequate analysis of the results, and demonstrate that LogQInit outperforms existing methods of value function initialization, policy transfer, and reward shaping.