Learning from Aggregate responses: Instance Level versus Bag Level Loss Functions

Due to the rise of privacy concerns, in many practical applications, the training data is aggregated before being shared with the learner to protect the privacy of users' sensitive responses. In an aggregate learning framework, the dataset is grouped into bags of samples, where each bag is available only with an aggregate response, providing a summary of individuals' responses in that bag. In this paper, we study two natural loss functions for learning from aggregate responses: the bag-level loss and the instance-level loss. In the former, the model is learned by minimizing a loss between the aggregate responses and aggregate model predictions, while in the latter, the model aims to fit individual predictions to the aggregate responses. In this work, we show that the instance-level loss can be perceived as a regularized form of the bag-level loss. This observation allows us to compare the two approaches with respect to the bias and variance of the resulting estimators and to introduce a novel interpolating estimator that combines the two approaches. For linear regression tasks, we provide a precise characterization of the risk of the interpolating estimator in an asymptotic regime where the size of the training set grows in proportion to the feature dimension. Our analysis enables us to theoretically understand the effect of different factors, such as bag size, on the model's prediction risk. Additionally, we propose a mechanism for differentially private learning from aggregate responses and derive the optimal bag size in terms of the prediction risk-privacy trade-off. We also carry out thorough experiments to corroborate our theory and show the efficacy of the interpolating estimator.