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In this paper, we analyze the impact of data freshness on real-time supervised learning, where a neural network is trained to infer a time-varying target (e.g., the position of the vehicle in front) based on features (e.g., video frames) observed at a sensing node (e.g., camera or lidar). One might expect that the performance of real-time supervised learning degrades monotonically as the feature becomes stale. Using an information-theoretic analysis, we show that this is true if the feature and target data sequence can be closely approximated as a Markov chain; it is not true if the data sequence is far from Markovian. Hence, the prediction error of real-time supervised learning is a function of the Age of Information (AoI), where the function could be non-monotonic. Several experiments are conducted to illustrate the monotonic and non-monotonic behaviors of the prediction error. To minimize the inference error in real-time, we propose a new "selection-from-buffer" model for sending the features, which is more general than the "generate-at-will" model used in earlier studies. By using Gittins and Whittle indices, low-complexity scheduling strategies are developed to minimize the inference error, where a new connection between the Gittins index theory and Age of Information (AoI) minimization is discovered. These scheduling results hold (i) for minimizing general AoI functions (monotonic or non-monotonic) and (ii) for general feature transmission time distributions. Data-driven evaluations are presented to illustrate the benefits of the proposed scheduling algorithms.