An autonomous robot with a limited vision range finds a path to the goal in an unknown environment in 2D avoiding polygonal obstacles. In the process of discovering the environmental map, the robot has to return to some positions marked previously, the regions where the robot traverses to return are defined as sequences of bundles of line segments. This paper presents a novel algorithm for finding approximately shortest paths along the sequences of bundles of line segments based on the method of multiple shooting. Three factors of the approach including bundle partition, collinear condition, and update of shooting points are presented. We then show that if the collinear condition holds, the exactly shortest paths of the problems are determined, otherwise, the sequence of paths obtained by the update of the method converges to the shortest path. The algorithm is implemented in Python and some numerical examples show that the running time of path-planning for autonomous robots using our method is faster than that using the rubber band technique of Li and Klette in Euclidean Shortest Paths, Springer, 53-89 (2011).

Active learning seeks to build the best possible model with a budget of labelled data by sequentially selecting the next point to label. However the training set is no longer \textit{iid}, violating the conditions required by existing consistency results. Inspired by the success of Stone's Theorem we aim to regain consistency for weighted averaging estimators under active learning. Based on ideas in \citet{dasgupta2012consistency}, our approach is to enforce a small amount of random sampling by running an augmented version of the underlying active learning algorithm. We generalize Stone's Theorem in the noise free setting, proving consistency for well known classifiers such as $k$-NN, histogram and kernel estimators under conditions which mirror classical results. However in the presence of noise we can no longer deal with these estimators in a unified manner; for some satisfying this condition also guarantees sufficiency in the noisy case, while for others we can achieve near perfect inconsistency while this condition holds. Finally we provide conditions for consistency in the presence of noise, which give insight into why these estimators can behave so differently under the combination of noise and active learning.