In predictive modeling applications, it is often of interest to determine the relative contribution of subsets of features in explaining the variability of an outcome. It is useful to consider this variable importance as a function of the unknown, underlying data-generating mechanism rather than the specific predictive algorithm used to fit the data. In this paper, we connect these ideas in nonparametric variable importance to machine learning, and provide a method for efficient estimation of variable importance when building a predictive model using a neural network. We show how a single augmented neural network with multi-task learning simultaneously estimates the importance of many feature subsets, improving on previous procedures for estimating importance. We demonstrate on simulated data that our method is both accurate and computationally efficient, and apply our method to both a study of heart disease and for predicting mortality in ICU patients.