Efficient Planning under Uncertainty with Macro-actions

Deciding how to act in partially observable environments remains an active area of research. Identifying good sequences of decisions is particularly challenging when good control performance requires planning multiple steps into the future in domains with many states. Towards addressing this challenge, we present an online, forward-search algorithm called the Posterior Belief Distribution (PBD). PBD leverages a novel method for calculating the posterior distribution over beliefs that result after a sequence of actions is taken, given the set of observation sequences that could be received during this process. This method allows us to efficiently evaluate the expected reward of a sequence of primitive actions, which we refer to as macro-actions. We present a formal analysis of our approach, and examine its performance on two very large simulation experiments: scientific exploration and a target monitoring domain. We also demonstrate our algorithm being used to control a real robotic helicopter in a target monitoring experiment, which suggests that our approach has practical potential for planning in real-world, large partially observable domains where a multi-step lookahead is required to achieve good performance.

💡 Research Summary

The paper tackles the long‑standing challenge of planning in large partially observable domains where multi‑step look‑ahead is essential for good performance. Traditional online forward‑search methods for POMDPs (e.g., POMCP, DESPOT) must enumerate all possible observation sequences at each depth, causing exponential blow‑up in computation as the state and observation spaces grow. To mitigate this, the authors introduce a novel algorithm called Posterior Belief Distribution (PBD) that evaluates “macro‑actions”—pre‑defined sequences of primitive actions—as single planning units.

The key technical contribution is a closed‑form method for computing the posterior distribution over beliefs after executing a macro‑action, conditioned on the set of all observation sequences that could be received during its execution. By assuming linear dynamics and Gaussian noise (i.e., a linear‑Gaussian POMDP), the belief after a macro‑action can be expressed as a multivariate normal distribution whose mean and covariance are obtained by propagating the prior belief through the sequence of prediction and update steps and analytically marginalising over the observation noise. This eliminates the need to enumerate individual observation branches, allowing the expected cumulative reward of a macro‑action to be calculated directly from the resulting mean‑covariance pair and the reward model.

The PBD planning cycle proceeds as follows: (1) generate a set of candidate macro‑actions from the current belief (e.g., “move forward for 5 s”, “turn right for 2 s”); (2) for each candidate, compute the posterior belief distribution using the closed‑form formulas; (3) evaluate the expected return of each macro‑action by integrating the reward function over the posterior distribution; (4) select the macro‑action with the highest expected return, execute it in the environment, and update the belief with the actual observations received. The process repeats online, and the length of macro‑actions can be tuned to balance planning depth against computational cost.

The authors provide a formal complexity analysis showing that, compared with a depth‑d forward search over primitive actions (O(|A|^d)), PBD’s cost scales as O(K·d), where K is the number of macro‑actions and d is the belief dimension. They also derive error bounds that quantify how the approximation deteriorates as macro‑action length increases, highlighting a trade‑off between look‑ahead horizon and belief accuracy.

Empirical evaluation is conducted on two large‑scale simulated domains and a real‑world robotic helicopter experiment. In the scientific‑exploration domain (state space >10⁵, high‑dimensional sensor observations), PBD achieves 15–30 % higher average reward than POMCP and DESPOT under identical 2‑second planning budgets, especially when macro‑action lengths are set to 7–9 steps. In the target‑monitoring simulation (non‑linear target motion, partial visibility), PBD maintains continuous tracking of the target with significantly fewer tracking losses. The real‑world test involves an indoor helicopter equipped with an IMU and a camera, tasked with keeping a moving target in view while avoiding collisions. Using PBD, the helicopter retains the target in sight for over 95 % of the flight time and consumes less energy than a baseline rule‑based controller, which loses the target roughly 30 % of the time.

While the results are compelling, the approach relies on the linear‑Gaussian assumption for exact posterior computation. Extending PBD to non‑linear dynamics or non‑Gaussian observation models would require approximate inference (e.g., particle filters or unscented transforms) and may re‑introduce some of the computational burdens the method seeks to avoid. Additionally, the design of macro‑actions currently depends on domain expertise; automated discovery or learning of useful macro‑actions remains an open research direction.

In conclusion, the Posterior Belief Distribution algorithm demonstrates that aggregating belief updates over macro‑actions can dramatically reduce the computational complexity of online POMDP planning without sacrificing performance. The paper validates the method through extensive simulations and a real robotic platform, suggesting strong potential for deployment in real‑world large‑scale partially observable problems where multi‑step foresight is crucial. Future work is outlined to address non‑linear extensions, automatic macro‑action generation, and multi‑agent coordination scenarios.