Asymptotic and Non-Asymptotic Rate-Loss Bounds for Linear Regression with Side Information

In the framework of goal-oriented communications, this paper investigates the fundamental achievable rate-loss function of a learning task performed on compressed data. It considers the setup where the data, collected remotely, are compressed and sent over a noiseless channel to a server that aims at applying linear regression on compressed data and side information. The paper contributions are threefold: i) the rateloss region is first derived in the asymptotic regime, i.e., when the length of the source tends to infinity, (ii) the tradeoff between data reconstruction and linear regression is investigated from the asymptotic rate-loss region, and iii) the approach is extended to the finite blocklength regime.