The main focus of Hierarchical Reinforcement Learning (HRL) is studying how large Markov Decision Processes (MDPs) can be more efficiently solved when addressed in a modular way, by combining partial solutions computed for smaller subtasks. Despite their very intuitive role for learning, most notions of MDP abstractions proposed in the HRL literature have limited expressive power or do not possess formal efficiency guarantees. This work addresses these fundamental issues by defining Realizable Abstractions, a new relation between generic low-level MDPs and their associated high-level decision processes. The notion we propose avoids non-Markovianity issues and has desirable near-optimality guarantees. Indeed, we show that any abstract policy for Realizable Abstractions can be translated into near-optimal policies for the low-level MDP, through a suitable composition of options. As demonstrated in the paper, these options can be expressed as solutions of specific constrained MDPs. Based on these findings, we propose RARL, a new HRL algorithm that returns compositional and near-optimal low-level policies, taking advantage of the Realizable Abstraction given in the input. We show that RARL is Probably Approximately Correct, it converges in a polynomial number of samples, and it is robust to inaccuracies in the abstraction.

We will add this discussion.

Abstraction has been widely studied as a way to improve the efficiency and generalization of reinforcement learning algorithms.

How can a scientist use a Reinforcement Learning (RL) algorithm to design experiments over a dynamical system's state space? In the case of finite and Markovian systems, an area called Active Exploration (AE) relaxes the optimization problem of experiments design into Convex RL, a generalization of RL admitting a wider notion of reward. Unfortunately, this framework is currently not scalable and the potential of AE is hindered by the vastness of experiment spaces typical of scientific discovery applications. However, these spaces are often endowed with natural geometries, e.g., permutation invariance in molecular design, that an agent could leverage to improve the statistical and computational efficiency of AE. To achieve this, we bridge AE and MDP homomorphisms, which offer a way to exploit known geometric structures via abstraction. Towards this goal, we make two fundamental contributions: we extend MDP homomorphisms formalism to Convex RL, and we present, to the best of our knowledge, the first analysis that formally captures the benefit of abstraction via homomorphisms on sample efficiency. Ultimately, we propose the Geometric Active Exploration (GAE) algorithm, which we analyse theoretically and experimentally in environments motivated by problems in scientific discovery.

Animals are able to rapidly infer from limited experience when sets of state action pairs have equivalent reward and transition dynamics. On the other hand, modern reinforcement learning systems must painstakingly learn through trial and error that sets of state action pairs are value equivalent -- requiring an often prohibitively large amount of samples from their environment. MDP homomorphisms have been proposed that reduce the observed MDP of an environment to an abstract MDP, which can enable more sample efficient policy learning. Consequently, impressive improvements in sample efficiency have been achieved when a suitable MDP homomorphism can be constructed a priori -- usually by exploiting a practioner's knowledge of environment symmetries. We propose a novel approach to constructing a homomorphism in discrete action spaces, which uses a partial model of environment dynamics to infer which state action pairs lead to the same state -- reducing the size of the state-action space by a factor equal to the cardinality of the action space. We call this method equivalent effect abstraction. In a gridworld setting, we demonstrate empirically that equivalent effect abstraction can improve sample efficiency in a model-free setting and planning efficiency for modelbased approaches. Furthermore, we show on cartpole that our approach outperforms an existing method for learning homomorphisms, while using 33x less training data.