Informative Path Planning for Active Regression With Gaussian Processes via Sparse Optimization

Abstract

We study informative path planning for active regression in Gaussian Processes (GP). Here, a resource constrained robot team collects measurements of an unknown function, assumed to be a sample from a GP, with the goal of minimizing the trace of the $M$-weighted expected squared estimation error covariance (where $M$ is a positive semidefinite matrix) resulting from the GP posterior mean. While greedy heuristics are a popular solution in the case of length constrained paths, it remains a challenge to compute optimal solutions in the discrete setting subject to routing constraints. We show that this challenge is surprisingly easy to circumvent. Using the optimality of the posterior mean for a class of functions of the squared loss yields an exact formulation as a mixed integer program. We demonstrate that this approach finds optimal solutions in a variety of settings in seconds and when terminated early, it finds sub-optimal solutions of higher quality than existing heuristics.