Title: Provable hierarchical lifelong learning with a sketch-based modular architecture. Abstract: We propose a modular architecture for lifelong learning of hierarchically structured tasks. Specifically, we prove that our architecture is theoretically able to learn tasks that can be solved by functions that are learnable given access to functions for other, previously learned tasks as subroutines. We show that some tasks that we can learn in this way are not learned by standard training methods in practice; indeed, prior work suggests that some such tasks cannot be learned by \emph{any} efficient method without the aid of the simpler tasks. We also consider methods for identifying the tasks automatically, without relying on explicitly given indicators.