We consider the problem of communicating data from energy-constrained distributed sensors. To reduce energy requirements, we go beyond the source reconstruction problem classically addressed, and focus on the problem where the recipient wants to perform supervised learning and classification on the data received from the sensors. Restricting our attention to a noiseless communication setting under simplistic Gaussian source assumptions, we study supervised learning and classification under total energy limitations. The energy constraints are modeled in two ways: 1) a linear scaling and 2) an exponential scaling of energy with number of bits used for compression at sensors. We first assume that the underlying parameters for Gaussian distributions have already been learned, and obtain (with linear scaling, reverse-waterfilling-type) strategies for allocating energy, and thus, bits, across different sensors under these two models. Intuitively, these strategies allocate larger rates and energies to sensors that are more “relevant” for the classification goal. These strategies are used to obtain an achievable bound on the tradeoff between energy and error-probability (classification risk). We then provide an algorithm for learning the distribution-parameters of the sensor-data under energy constraints to arrive at high-reliability energy-allocation strategies, while enabling the energy-allocation algorithm to backtrack when the underlying distributions change, or when there is noise in sensed data that can push the algorithm toward a local minimum. Finally, we provide numerical results on energy-savings for classification of simulated data as well as neural data acquired from electrocorticography (ECoG) experiments.