Abstract: In $D$istributed optimization and $L$earning, and even more in the modern framework of federated learning, communication, which is slow and costly, is critical. We introduce LoCoDL, a communication-efficient algorithm that leverages the two popular and effective techniques of $Lo$cal training, which reduces the communication frequency, and $Co$mpression, in which short bitstreams are sent instead of full-dimensional vectors of floats. LoCoDL works with a large class of unbiased compressors that includes widely-used sparsification and quantization methods. LoCoDL provably benefits from local training and compression and enjoys a doubly-accelerated communication complexity, with respect to the condition number of the functions and the model dimension, in the general heterogeneous regime with strongly convex functions. This is confirmed in practice, with LoCoDL outperforming existing algorithms.