Reducing Planning Complexity of General Reinforcement Learning with Non-Markovian Abstractions

TL;DR

This paper introduces a new non-Markovian abstraction for general reinforcement learning that significantly reduces the complexity of planning by providing tighter bounds on the number of states needed for effective surrogate MDPs.

Contribution

The paper proposes a novel non-MDP abstraction that improves upon the existing ESA framework by offering much tighter upper bounds on the state complexity for planning in GRL.

Findings

New non-MDP abstraction with improved upper bounds

Bound reduced from exponential to near-logarithmic in actions

Action-sequentialization further tightens the bound

Abstract

The field of General Reinforcement Learning (GRL) formulates the problem of sequential decision-making from ground up. The history of interaction constitutes a "ground" state of the system, which never repeats. On the one hand, this generality allows GRL to model almost every domain possible, e.g.\ Bandits, MDPs, POMDPs, PSRs, and history-based environments. On the other hand, in general, the near-optimal policies in GRL are functions of complete history, which hinders not only learning but also planning in GRL. The usual way around for the planning part is that the agent is given a Markovian abstraction of the underlying process. So, it can use any MDP planning algorithm to find a near-optimal policy. The Extreme State Aggregation (ESA) framework has extended this idea to non-Markovian abstractions without compromising on the possibility of planning through a (surrogate) MDP. A distinguishing feature of ESA is that it proves an upper bound of $O\left(\varepsilon^{-A} \cdot (1-\gamma)^{-2A}\right)$ on the number of states required for the surrogate MDP (where $A$ is the number of actions, $\gamma$ is the discount-factor, and $\varepsilon$ is the optimality-gap) which holds \emph{uniformly} for \emph{all} domains. While the possibility of a universal bound is quite remarkable, we show that this bound is very loose. We propose a novel non-MDP abstraction which allows for a much better upper bound of $O\left(\varepsilon^{-1} \cdot (1-\gamma)^{-2} \cdot A \cdot 2^{A}\right)$. Furthermore, we show that this bound can be improved further to $O\left(\varepsilon^{-1} \cdot (1-\gamma)^{-2} \cdot \log^3 A \right)$ by using an action-sequentialization method.