Edge signal processing facilitates distributed learning and inference in the client-server model proposed in federated learning. In traditional machine learning, clients (IoT devices) that acquire raw signal samples can aid a data center (server) learn a global signal model by pooling these distributed samples at a third-party location. Despite the promising capabilities of IoTs, these distributed deployments often face the challenge of sensitive private data and communication rate constraints. This necessitates a learning approach that communicates a processed approximation of the distributed samples instead of the raw signals. Such a decentralized learning approach using signal approximations will be termed distributed signal analytics in this work. Overpredictive signal approximations may be desired for distributed signal analytics, especially in network demand (capacity) planning applications motivated by federated learning. In this work, we propose algorithms that compute an overpredictive signal approximation at the client devices using an efficient convex optimization framework. Tradeoffs between communication cost, sampling rate, and the signal approximation error are quantified using mathematical analysis. We also show the performance of the proposed distributed algorithms on a publicly available residential energy consumption dataset.