Measurement Simplification in ρ-POMDP with Performance Guarantees

TL;DR

This work tackles the computational explosion of belief-space planning in ρ-POMDPs by introducing observation-space partitioning to derive tractable, adaptive bounds on the expected information-theoretic reward. By partitioning future observations into subsets and employing a hierarchical partitioning tree, the authors derive upper and lower bounds on conditional entropy that converge to the true reward as the partition depth increases, while reducing per-step computation. The approach is specialized to Gaussian beliefs with a closed-form entropy bound using determinants of the augmented information matrix, and leverages efficient determinant updates via rAMDL. Empirical results in active SLAM show substantial planning-speedups and guaranteed performance, both in simulation and real-world experiments, compared to state-of-the-art methods like rAMDL and iSAM2. The framework is general, scalable, and extensible to other belief distributions and information-theoretic rewards, offering a practical path toward fast, guaranteed decision-making under uncertainty.

Abstract

Decision making under uncertainty is at the heart of any autonomous system acting with imperfect information. The cost of solving the decision making problem is exponential in the action and observation spaces, thus rendering it unfeasible for many online systems. This paper introduces a novel approach to efficient decision-making, by partitioning the high-dimensional observation space. Using the partitioned observation space, we formulate analytical bounds on the expected information-theoretic reward, for general belief distributions. These bounds are then used to plan efficiently while keeping performance guarantees. We show that the bounds are adaptive, computationally efficient, and that they converge to the original solution. We extend the partitioning paradigm and present a hierarchy of partitioned spaces that allows greater efficiency in planning. We then propose a specific variant of these bounds for Gaussian beliefs and show a theoretical performance improvement of at least a factor of 4. Finally, we compare our novel method to other state of the art algorithms in active SLAM scenarios, in simulation and in real experiments. In both cases we show a significant speed-up in planning with performance guarantees.

Figures (22)

• Figure 1: A prior factor graph is shown in the gray blob; considering an action that leads to the factor $f_3$, we see the posterior factor graph in the yellow blob. The posterior graph includes the future +random measurements $Z_1,Z_2,Z_3$. Different sets of random measurements are assigned different colors which represents one possible partitioning. These sets are used to bound the conditional Entropy of the entire posterior graph.
• Figure 2: The expected reward is shown in red within its bounds. On the left we can select the optimal action based on the bounds alone, on the right the worst-case loss is shaded in gray.
• Figure 3: An Illustration of a possible partition tree. At each level of partitioning, we split a measurement set into two. For $Z\in \mathbb R^m$, the depth of the tree is $d=\log_2m$.
• Figure 4: Visualizing the marginal Entropies of two measurement variables, the lower bound double counts the overlapping region which is the mutual information between those variables.
• Figure 5: Any combination of children nodes, that their union equals to the parent node, can make up a lower bound on that parent node. Any child node at any depth can make up an upper bound on a parent node. For instance, the node highlighted in yellow can form an upper bound, while the nodes highlighted in gray and yellow can form a lower bound.

Theorems & Definitions (17)

• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 1
• proof
• Theorem 2
• proof
• Corollary 1
• Theorem 3