Summary. This teaser post sketches our current ideas for dealing with more complex environments. It will ultimately be replaced by one or more longer posts describing these in more detail. Reach out if you would like to collaborate on these issues.

Multi-dimensional aspirations

For real-world tasks that are specified in terms of more than a single evaluation metric, e.g., how much apples to buy and how much money to spend at most, we can generalize Algorithm 2 as follows from aspiration intervals to convex aspiration sets:

Assume there are $d > 1$ many evaluation metrics $u_{i}$ , combined into a vector-valued evaluation metric $u = (u_{1}, \dots, u_{d})$ .
Preparation: Pick $d + 1$ many linear combinations $f_{j}$ in the space spanned by these metrics so that their convex hull is full-dimensional and contains the origin, and consider the $d + 1$ many policies $π_{j}$ each of which maximizes the expected value of the corresponding function $f_{j}$ . Let $V_{j} (s)$ and $Q_{j} (s, a)$ be the expected values of $u$ when using $π_{j}$ in state $s$ or after using action $a$ in state $s$ , respectively (see Fig. 1). Let the admissibility simplices $V (s)$ and $Q (s, a)$ be the simplices spanned by the vertices $V_{j} (s)$ and $Q_{j} (s, a)$ , respectively (red and violet triangles in Fig. 1). They replace the feasibility intervals used in Algorithm 2.
Policy: Given a convex state-aspiration set $E (s) \subseteq V (s)$ (central green polyhedron in Fig. 1), compute its midpoint (centre of mass) $m$ and consider the $d + 1$ segments $ℓ_{j}$ from $m$ to the corners $V_{j} (s)$ of $V (s)$ (dashed black lines in Fig. 1). For each of these segments $ℓ_{j}$ , let $A_{j}$ be the (nonempty!) set of actions for which $ℓ_{j}$ intersects $Q (s, a)$ . For each $a \in A_{j}$ , compute the action-aspiration $E (s, a) \subseteq Q (s, a)$ by shifting a copy $C_{j}$ of $E (s)$ along $ℓ_{j}$ towards $V_{j} (s)$ until the intersection of $C_{j}$ and $ℓ_{j}$ is contained in the intersection of $Q (s, a)$ and $ℓ_{j}$ (half-transparent green polyhedra in Fig. 1), and then intersecting $C_{j}$ with $Q (s, a)$ to give $E (s, a)$ (yellow polyhedra in Fig. 1). Then pick one candidate action from each $A_{j}$ and randomi

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