Title: Learning Mixture Models with Simultaneous Data Partitioning and Parameter Estimation. Abstract: We study a new framework of learning mixture models via data partitioning called PRESTO, wherein we optimize a joint objective function on the model parameters and the partitioning, with each model tailored to perform well on its specific partition. We connect PRESTO to a number of past works in data partitioning, mixture models, and clustering, and show that PRESTO generalizes several loss functions including the k-means and Bregman clustering objective, the Gaussian mixture model objective, mixtures of support vector machines, and mixtures of linear regression. We then propose a new joint discrete-continuous optimization algorithm which achieves a bounded approximation guarantee for any general loss function, thereby achieving guarantees for the afore-mentioned problems as well. We study PRESTO in the context of resource efficient deep learning, where we train smaller resource constrained models on each partition and show that it outperforms existing data partitioning and model pruning/knowledge distillation approaches, which in contrast to PRESTO, require large initial (teacher) models.