Minimax Learning for Remote Prediction

The classical problem of supervised learning is to infer an accurate predictor of a target variable Y from a measured variable X by using a finite number of labeled training samples. Motivated by the increasingly distributed nature of data and decision making, in this paper we consider a variation of this classical problem in which the prediction is performed remotely based on a rate-constrained description M of X. Upon receiving M, the remote node computes an estimate Y of Y. We follow the recent minimax approach to study this learning problem and show that it corresponds to a one-shot minimax noisy source coding problem. We then establish information theoretic bounds on the risk-rate Lagrangian cost and a general method to design a near-optimal descriptor-estimator pair, which can be viewed as a rate-constrained analog to the maximum conditional entropy principle used in the classical minimax learning problem. Our results show that a naive estimate-compress scheme for rate-constrained prediction is not in general optimal.