Title: Private and Efficient Meta-Learning with Low Rank and Sparse decomposition. Abstract: Meta-learning is critical for a variety of practical ML systems -- like personalized recommendations systems -- that are  required to generalize to new tasks despite a small number of task-specific training points. Existing meta-learning techniques use two complementary approaches of either learning a low-dimensional representation of points for all tasks, or task-specific fine-tuning of a global model trained using all the tasks. In this work, we propose a novel meta-learning framework that combines both the techniques to enable handling of a large number of data-starved tasks. Our framework models network weights as a sum of low-rank  and sparse matrices. This allows us to capture information from multiple domains together in the low-rank part while still allowing task specific personalization using the sparse part.  We instantiate and study the framework in the linear setting, where the problem reduces to that of estimating the sum of a rank-$r$ and a $k$-column sparse matrix using a small number of linear measurements. We propose an alternating minimization method with hard thresholding -- AMHT-LRS -- to learn the low-rank and sparse part effectively and efficiently. For the realizable, Gaussian data setting, we show that AMHT-LRS indeed solves the problem efficiently with nearly optimal samples.  We extend  AMHT-LRS  to ensure that it preserves privacy of each individual user in the dataset, while still ensuring strong  generalization with nearly optimal number of samples. Finally, on multiple datasets, we demonstrate that the framework allows personalized models to obtain superior performance in the data-scarce regime.