Active Learning for Regression with Aggregated Outputs

Due to the privacy protection or the difficulty of data collection, we cannot observe individual outputs for each instance, but we can observe aggregated outputs that are summed over multiple instances in a set in some real-world applications. To reduce the labeling cost for training regression models for such aggregated data, we propose an active learning method that sequentially selects sets to be labeled to improve the predictive performance with fewer labeled sets. For the selection measurement, the proposed method uses the mutual information, which quantifies the reduction of the uncertainty of the model parameters by observing the aggregated output. With Bayesian linear basis functions for modeling outputs given an input, which include approximated Gaussian processes and neural networks, we can efficiently calculate the mutual information in a closed form. With the experiments using various datasets, we demonstrate that the proposed method achieves better predictive performance with fewer labeled sets than existing methods.