While machine-type communication (MTC) devices generate considerable amounts of data, they often cannot process the data due to limited energy and computational power. To empower MTC with intelligence, edge machine learning has been proposed. However, power allocation in this paradigm requires maximizing the learning performance instead of the communication throughput, for which the celebrated water-filling and max-min fairness algorithms become inefficient. To this end, this paper proposes learning centric power allocation (LCPA), which provides a new perspective on radio resource allocation in learning driven scenarios. By employing 1) an empirical classification error model that is supported by learning theory and 2) an uncertainty sampling method that accounts for different distributions at users, LCPA is formulated as a nonconvex nonsmooth optimization problem, and is solved using a majorization minimization (MM) framework. To get deeper insights into LCPA, asymptotic analysis shows that the transmit powers are inversely proportional to the channel gains, and scale exponentially with the learning parameters. This is in contrast to traditional power allocations where quality of wireless channels is the only consideration. Last but not least, a large-scale optimization algorithm termed mirror-prox LCPA is further proposed to enable LCPA in large-scale settings. Extensive numerical results demonstrate that the proposed LCPA algorithms outperform traditional power allocation algorithms, and the large-scale optimization algorithm reduces the computation time by orders of magnitude compared with MM-based LCPA but still achieves competing learning performance.