Fast Action Elimination for Efficient Decision Making and Belief Space Planning Using Bounded Approximations

Abstract

In this work, we introduce a new approach for the efficient solution of autonomous decision and planning problems, with a special focus on decision making under uncertainty and belief space planning (BSP) in high-dimensional state spaces. Usually, to solve the decision problem, we identify the optimal action, according to some objective function. Instead, we claim that we can sometimes generate and solve an analogous yet simplified decision problem, which can be solved more efficiently. Furthermore, a wise simplification method can lead to the same action selection, or one for which the maximal loss can be guaranteed. This simplification is separated from the state inference, and does not compromise its accuracy, as the selected action would finally be applied on the original state. At first, we develop the concept for general decision problems, and provide a theoretical framework of definitions to allow a coherent discussion. We then practically apply these ideas to BSP problems, in which the problem is simplified by considering a sparse approximation of the initial belief. The scalable sparsification algorithm we provide is able to yield solutions which are guaranteed to be consistent with the original problem. We demonstrate the benefits of the approach in the solution of a highly realistic active-SLAM problem, and manage to significantly reduce computation time, with practically no loss in the quality of solution. This rigorous and fundamental work is conceptually novel, and holds numerous possible extensions.

Probabilistic Loss and its Online Characterization for Simplified Decision Making Under Uncertainty

“…which is Dirac's delta function in the regular setting of POMDP and ρ-POMDP. If B is a Dirac function, a sample from (10) uniquely defines the corresponding posterior beliefs b k+1:k+L . This, therefore, corresponds to the classical belief tree.…”

“…This, therefore, corresponds to the classical belief tree. In contrast, our Pρ-POMDP (9), corresponds to an extended belief tree, which, due to (6), allows many samples of the beliefs b k+1:k+L for each sample of z k+1:k+L from (10). We illustrate this in Fig.…”

“…In other words, in this paper we assume that operator ν affects exclusively (11). However, the measurements are sampled as in the original problem as in (10).…”

“…Moreover, if, utilizing the simplified problem, we can find online bounds over the returns or objective function of the original problem, it is possible to account for the simplification loss. Replacement of various parts of the decision making problem to ease the computation burden while preserving the precedence of objectives for potential action plans recently appeared in the literature under the name action consistency [10], [9], [19]. Yet, these works considered a limited setting of a specific projection operator, Gaussian distributions, and maximum likelihood observations.…”

Topological Information-Theoretic Belief Space Planning with Optimality Guarantees

“…More importantly, if we can quantify the error of making such approximation, optimality guarantees can be established. If needed, the optimal solution can still be obtained, but this time, by evaluating only a subset of candidate actions, while discarding the rest, as proposed in [10]. Doing so will generally reduce the number of variables for which the marginal covariance needs to be recovered.…”