Pushdown Reward Machines for Reinforcement Learning

TL;DR

This paper introduces pushdown reward machines (pdRMs), an extension of reward machines using pushdown automata, enabling the encoding of more complex, context-free language-based behaviors in reinforcement learning, with theoretical and experimental validation.

Contribution

The work extends reward machines to pushdown automata, increasing expressiveness for representing complex behaviors in reinforcement learning, and provides theoretical analysis and practical algorithms.

Findings

pdRMs are more expressive than reward machines.

Theoretical bounds on policy equivalence with limited stack access.

Experimental results demonstrate successful training on context-free language tasks.

Abstract

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 recognise 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 state values. We then provide theoretical results establishing the expressive power of pdRMs, and space complexity results for the proposed learning problems. Lastly, we propose an approach for off-policy RL algorithms that exploits counterfactual experiences with pdRMs. We conclude by providing experimental results showing how agents can be trained to perform tasks representable in deterministic context-free languages using pdRMs.