Compositionality and Bounds for Optimal Value Functions in Reinforcement Learning

An agent's ability to reuse solutions to previously solved problems is critical for learning new tasks efficiently. Recent research using composition of value functions in reinforcement learning has shown that agents can utilize solutions of primitive tasks to obtain solutions for exponentially many new tasks. However, previous work has relied on restrictive assumptions on the dynamics, the method of composition, and the structure of reward functions. Here we consider the case of general composition functions without any restrictions on the structure of reward functions, applicable to both deterministic and stochastic dynamics. For this general setup, we provide bounds on the corresponding optimal value functions and characterize the value of corresponding policies. The theoretical results derived lead to improvements in training for both entropy-regularized and standard reinforcement learning, which we validate with numerical simulations.