Interval Markov Decision Processes with Continuous Action-Spaces

TL;DR

This paper introduces continuous-action IMDPs (caIMDPs), extending traditional IMDPs to handle continuous action spaces, and develops efficient value iteration algorithms for control synthesis in uncertain stochastic systems.

Contribution

The paper proposes caIMDPs with action-dependent transition bounds and provides algorithms for their value iteration, enabling control synthesis with continuous actions.

Findings

Efficient value iteration algorithms for caIMDPs using linear or convex programming.

Synthesis over discrete actions at vertices of the action polytope can be optimal.

Demonstrated methods on a numerical example.

Abstract

Interval Markov Decision Processes (IMDPs) are finite-state uncertain Markov models, where the transition probabilities belong to intervals. Recently, there has been a surge of research on employing IMDPs as abstractions of stochastic systems for control synthesis. However, due to the absence of algorithms for synthesis over IMDPs with continuous action-spaces, the action-space is assumed discrete a-priori, which is a restrictive assumption for many applications. Motivated by this, we introduce continuous-action IMDPs (caIMDPs), where the bounds on transition probabilities are functions of the action variables, and study value iteration for maximizing expected cumulative rewards. Specifically, we decompose the max-min problem associated to value iteration to $|\mathcal{Q}|$ max problems, where $|\mathcal{Q}|$ is the number of states of the caIMDP. Then, exploiting the simple form of these max problems, we identify cases where value iteration over caIMDPs can be solved efficiently (e.g., with linear or convex programming). We also gain other interesting insights: e.g., in certain cases where the action set $\mathcal{A}$ is a polytope, synthesis over a discrete-action IMDP, where the actions are the vertices of $\mathcal{A}$, is sufficient for optimality. We demonstrate our results on a numerical example. Finally, we include a short discussion on employing caIMDPs as abstractions for control synthesis.