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Natural target functions and tasks typically exhibit hierarchical modularity-- they can be broken down into simpler sub-functions that are organized in ahierarchy. Such sub-functions have two important features: they have a distinctset of inputs (input-separability) and they are reused as inputs higher in thehierarchy (reusability). Previous studies have established that hierarchicallymodular neural networks, which are inherently sparse, offer benefits such aslearning efficiency, generalization, multi-task learning, and transfer.However, identifying the underlying sub-functions and their hierarchicalstructure for a given task can be challenging. The high-level question in thiswork is: if we learn a task using a sufficiently deep neural network, how canwe uncover the underlying hierarchy of sub-functions in that task? As astarting point, we examine the domain of Boolean functions, where it is easierto determine whether a task is hierarchically modular. We propose an approachbased on iterative unit and edge pruning (during training), combined withnetwork analysis for module detection and hierarchy inference. Finally, wedemonstrate that this method can uncover the hierarchical modularity of a widerange of Boolean functions and two vision tasks based on the MNIST digitsdataset.

Neural Sculpting: Uncovering hierarchically modular task structure in neural networks through pruning and network analysis