In modern machine learning, users often have to collaborate to learn distributions that generate the data. Communication can be a significant bottleneck. Prior work has studied homogeneous users---i.e., whose data follow the same discrete distribution---and has provided optimal communication-efficient methods. However, these methods rely heavily on homogeneity, and are less applicable in the common case when users' discrete distributions are heterogeneous. Here we consider a natural and tractable model of heterogeneity, where users' discrete distributions only vary sparsely, on a small number of entries. We propose a novel two-stage method named SHIFT: First, the users collaborate by communicating with the server to learn a central distribution; relying on methods from robust statistics. Then, the learned central distribution is fine-tuned to estimate the individual distributions of users. We show that our method is minimax optimal in our model of heterogeneity and under communication constraints. Further, we provide experimental results using both synthetic data and $n$-gram frequency estimation in the text domain, which corroborate its efficiency.

Fairness concerns are increasingly critical as machine learning models are deployed in high-stakes applications. While existing fairness-aware methods typically intervene at the model level, they often suffer from high computational costs, limited scalability, and poor generalization. To address these challenges, we propose a Bayesian data selection framework that ensures fairness by aligning group-specific posterior distributions of model parameters and sample weights with a shared central distribution. Our framework supports flexible alignment via various distributional discrepancy measures, including Wasserstein distance, maximum mean discrepancy, and $f$-divergence, allowing geometry-aware control without imposing explicit fairness constraints. This data-centric approach mitigates group-specific biases in training data and improves fairness in downstream tasks, with theoretical guarantees. Experiments on benchmark datasets show that our method consistently outperforms existing data selection and model-based fairness methods in both fairness and accuracy.

Communication can be a significant bottleneck.

In modern machine learning, users often have to collaborate to learn distributions that generate the data. Communication can be a significant bottleneck. Prior work has studied homogeneous users---i.e., whose data follow the same discrete distribution---and has provided optimal communication-efficient methods. However, these methods rely heavily on homogeneity, and are less applicable in the common case when users' discrete distributions are heterogeneous. Here we consider a natural and tractable model of heterogeneity, where users' discrete distributions only vary sparsely, on a small number of entries.

In modern machine learning, users often have to collaborate to learn the distribution of the data. Communication can be a significant bottleneck. Prior work has studied homogeneous users -- i.e., whose data follow the same discrete distribution -- and has provided optimal communication-efficient methods for estimating that distribution. However, these methods rely heavily on homogeneity, and are less applicable in the common case when users' discrete distributions are heterogeneous. Here we consider a natural and tractable model of heterogeneity, where users' discrete distributions only vary sparsely, on a small number of entries. We propose a novel two-stage method named SHIFT: First, the users collaborate by communicating with the server to learn a central distribution; relying on methods from robust statistics. Then, the learned central distribution is fine-tuned to estimate their respective individual distribution. We show that SHIFT is minimax optimal in our model of heterogeneity and under communication constraints. Further, we provide experimental results using both synthetic data and $n$-gram frequency estimation in the text domain, which corroborate its efficiency.