We study fundamental theoretical aspects of probabilistic roadmaps (PRM) in the finite time (non-asymptotic) regime. In particular, we investigate how completeness and optimality guarantees of the approach are influenced by the underlying deterministic sampling distribution X and connection radius r > 0. We develop the notion of (δ, ε)-completeness of the parameters X, r, which indicates that for every motion-planning problem of clearance at least δ > 0, PRM using X, r returns a solution no longer than 1+ε times the shortest δ-clear path. Leveraging the concept of e-nets, we characterize in terms of lower and upper bounds the number of samples needed to guarantee (δ, ε)-completeness. This is in contrast with previous work which mostly considered the asymptotic regime in which the number of samples tends to infinity. In practice, we propose a sampling distribution inspired by e-nets that achieves nearly the same coverage as grids while using fewer samples.

This paper presents a model predictive control (MPC) approach to optimize routes for Ride-sharing Autonomous Mobility-on-Demand (RAMoD) systems, whereby self-driving vehicles provide coordinated on-demand mobility, possibly allowing multiple customers to share a ride. Specifically, we first devise a time-expanded network flow model for RAMoD. Second, leveraging this model, we design a real-time MPC algorithm to optimize the routes of both empty and customer-carrying vehicles, with the goal of optimizing social welfare, namely, a weighted combination of customers' travel time and vehicles' mileage. Finally, we present a real-world case study for the city of San Francisco, CA, by using the micro-scopic traffic simulator MATSim. The simulation results show that a RAMoD system can significantly improve social welfare with respect to a single-occupancy Autonomous Mobility-on-Demand (AMoD) system, and that the predictive structure of the proposed MPC controller allows it to outperform existing reactive ride-sharing coordination algorithms for RAMoD.

Machine learning algorithms can be viewed as stochastic transformations that map training data to hypotheses. Following Bousquet and Elisseeff, we say that such an algorithm is stable if its output does not depend too much on any individual training example. Since stability is closely connected to generalization capabilities of learning algorithms, it is of theoretical and practical interest to obtain sharp quantitative estimates on the generalization bias of machine learning algorithms in terms of their stability properties. We propose several information-theoretic measures of algorithmic stability and use them to upper-bound the generalization bias of learning algorithms. Our framework is complementary to the information-theoretic methodology developed recently by Russo and Zou.