Risk-Aware Estimation from Compressed Data Beyond the Bayes Risk

Inference often relies on compressed data due to communication, storage, or privacy constraints. In order to minimize degradation in the quality of inference, it is desirable to tailor compression schemes to the inference task. The compression scheme should therefore account for the statistic of the loss relevant for the task. While the expected loss is widely considered, in applications sensitive to large losses—such as in safe control and learning—alternative statistics are relevant. A key family of these alternative statistics are obtained via risk measures. In this paper, we characterize the increase in risk measure criteria for inference tasks as a function of the code size. Our characterization applies for general data statistics, loss functions, and number of samples. In the special case of i.i.d. data, we also establish asymptotics and a connection between our characterization for risk measure criteria and the rate-distortion function, which was previously only known for expected loss and excess distortion criteria.