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# Options (Sutton, Precup, Singh)

## Background

As background, when this paper was published, the RL community had been exploring different approaches for considering abstractions over actions. The hope was that agents don’t necessarily need to plan at a granular level, but could instead form some notion of temporally extended, multi-step sequences of actions and use those instead. This paper introduced the notion of options to present a minimal extension of RL that allows for a general treatment of abstract actions.

Disclaimer: In my opinion, this paper is something of a letdown. It introduces a new mathematical object and then spends the paper showing that the new object doesn’t introduce any mischief. However, this new object fails to (1) provide new conceptual insight or (2) direct the creation of better RL agents. Later work has certainly addressed these questions, but this paper does not.

## Notation

$$\mathcal{A}_s$$ is the set of actions available in state $$s \in \mathcal{S}$$. $$\mathcal{S}^+$$ is the set of states $$\mathcal{S}$$ unioned with a special terminal state, indicating the end of the trajectory.

## Contents

### Options

An option is defined as a 3 tuple consisting of

1. A policy $$\pi: \mathcal{S} \times \mathcal{A} \rightarrow [0, 1]$$
2. A termination condition $$\beta: \mathcal{S}^+ \rightarrow [0, 1]$$
3. An initiation set $$\mathcal{I} \subseteq S$$

We say that an option $$(\pi, \beta, \mathcal{I})$$ is available in state $$s \in \mathcal{S}$$ iff the state is an element of that option’s initiation set i.e. $$s \in \mathcal{I}$$. We suppose that at every state $$s$$, there is a set of available options denoted $$\mathcal{O}_s$$, and the agent selects one. The selected option’s policy is then followed until the option is terminated randomly, as dictated by the termination condition.