Pushdown Reward Machines for Reinforcement Learning

Reward machines (RMs) are automata structures that encode (non-Markovian) reward functions for reinforcement learning (RL). RMs can reward any behaviour representable in regular languages and, when paired with RL algorithms that exploit RM structure, have been shown to significantly improve sample efficiency in many domains. In this work, we present pushdown reward machines (pdRMs), an extension of reward machines based on deterministic pushdown automata. pdRMs can recognize and reward temporally extended behaviours representable in deterministic context-free languages, making them more expressive than reward machines. We introduce two variants of pdRM-based policies, one which has access to the entire stack of the pdRM, and one which can only access the top $k$ symbols (for a given constant $k$) of the stack. We propose a procedure to check when the two kinds of policies (for a given environment, pdRM, and constant $k$) achieve the same optimal expected reward. We then provide theoretical results establishing the expressive power of pdRMs, and space complexity results about the proposed learning problems. Finally, we provide experimental results showing how agents can be trained to perform tasks representable in deterministic context-free languages using pdRMs.