Simplified decision making in the belief space using belief sparsification

Abstract

In this work, we introduce a new and efficient solution approach for the problem of decision making under uncertainty, which can be formulated as decision making in a belief space, over a possibly high-dimensional state space. Typically, to solve a decision problem, one should identify the optimal action from a set of candidates, according to some objective. We claim that one can often generate and solve an analogous yet simplified decision problem, which can be solved more efficiently. A wise simplification method can lead to the same action selection, or one for which the maximal loss in optimality can be guaranteed. Furthermore, such 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. First, we present the concept for general decision problems and provide a theoretical framework for a coherent formulation of the approach. We then practically apply these ideas to decision problems in the belief space, which can be simplified by considering a sparse approximation of their initial belief. The scalable belief 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 realistic active-SLAM problem and manage to significantly reduce computation time, with no loss in the quality of solution. This work is both fundamental and practical and holds numerous possible extensions.

Nonmyopic Distilled Data Association Belief Space Planning Under Budget Constraints

“…The notion of simplification was introduced in [4], where, the authors formulated the loss in solution quality in BSP problems via bounds over the objective function. However, they only considered the Gaussian case and a maximum likelihood assumption.…”

“…Case 3 In this setting we consider the heuristic in (4) to be given within planning, i.e. posterior belief tree nodes exactly represent how the belief would evolve in inference under (4). In contrast to Section 4.3, as the number of components does not grow exponentially, we sample future observations according to (8) and construct the belief tree explicitly, i.e.…”

“…The bounds over the considered cost are now a function of the belief in the previous time step under (4). Specifically, we rewrite them as…”

“…We now relax the assumption that the planning algorithm is confined to the specific heuristic in (4). Unlike in Case 2, where each subset of components can be used in each node to calculate the bounds, this setting has an additional constraint.…”

Simplified Continuous High Dimensional Belief Space Planning with Adaptive Probabilistic Belief-dependent Constraints

“…This fact discards many early approaches [10], [11] to include belief-dependent rewards to POMDP. Another line of simplification works alleviate curse of dimensionality in the setting of multivariate Gaussian distributions utilizing sparsification [17] and topological [18] aspects. The simplification paradigm was also applied with Gaussian-mixture distributed beliefs [19], [20].…”