Learning with rare data: Using active importance sampling to optimize objectives dominated by rare events

Deep neural networks, when optimized with sufficient data, provide accurate representations of high-dimensional functions; in contrast, function approximation techniques that have predominated in scientific computing do not scale well with dimensionality. While the promise of unparalleled accuracy may suggest a renaissance for applications that require parameterizing representations of complex systems, in many applications gathering sufficient data to develop such a representation remains a significant challenge. Here we introduce an approach that combines rare events sampling techniques with neural network optimization to optimize objective functions that are dominated by rare events. We show that importance sampling reduces the asymptotic variance of the solution to a learning problem, suggesting benefits for generalization. We study our algorithm for identifying dynamical transition pathways between two states of a system, a problem with applications in statistical and chemical physics. Our numerical experiments demonstrate convergence even with the compounding difficulties of high-dimension and rare data.