The Importance of a Suitable Distance Function in Belief-Space Planning

Many methods for planning under uncertainty operate in the belief space, i.e., the set of probability distributions over states. Although the problem is computationally hard, recent advances have shown that belief-space planning is becoming practical for many medium size problems. Some of the most successful methods utilize sampling and often rely on distances between beliefs to partially guide the search process. This paper deals with the question of what is a suitable distance function for belief space planning, which despite its importance remains unanswered. This work indicates that the rarely used Wasserstein distance (also known as Earth Mover’s Distance (\(\mathtt{EMD} \))) is a more suitable metric than the commonly used \(\mathtt{L1} \) and Kullback–Leibler (\(\mathtt{KL} \)) for belief-space planning. Simulation results on Non-Observable Markov Decision Problems, i.e., the simplest class of belief-space planning, indicate that as the problem becomes more complex, the differences on the effectiveness of different distance functions become quite prominent. In fact, in state spaces with more than 4 dimensions, by just replacing \(\mathtt{L1} \) or \(\mathtt{KL} \) distance with \(\mathtt{EMD} \), the problems become from virtually unsolvable to solvable within a reasonable time frame. Furthermore, preliminary results on Partially Observable Markov Decision Processes indicate that point-based solvers with \(\mathtt{EMD} \) use a smaller number of samples to generate policies with similar qualities, compared to those with \(\mathtt{L1} \) and \(\mathtt{KL} \). This paper also shows that \(\mathtt{EMD} \) carries the Lipschitz continuity of the state’s cost function to Lipschitz continuity of the expected cost of the beliefs. Such a continuity property is often critical for convergence to optimal solutions.