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Bringing the success of modern machine learning (ML) techniques to mobile devices can enable many new services and businesses, but also poses significant technical and research challenges. Two factors that are critical for the success of ML algorithms are massive amounts of data and processing power, both of which are plentiful, yet highly distributed at the network edge. Moreover, edge devices are connected through bandwidth- and power-limited wireless links that suffer from noise, time-variations, and interference. Information and coding theory have laid the foundations of reliable and efficient communications in the presence of channel imperfections, whose application in modern wireless networks have been a tremendous success. However, there is a clear disconnect between the current coding and communication schemes, and the ML algorithms deployed at the network edge. In this paper, we challenge the current approach that treats these problems separately, and argue for a joint communication and learning paradigm for both the training and inference stages of edge learning.