  In the context of goal-oriented communications, this paper addresses the
achievable rate versus generalization error region of a learning task applied
on compressed data. The study focuses on the distributed setup where a source
is compressed and transmitted through a noiseless channel to a receiver
performing polynomial regression, aided by side information available at the
decoder. The paper provides the asymptotic rate generalization error region,
and extends the analysis to the non-asymptotic regime.Additionally, it
investigates the asymptotic trade-off between polynomial regression and data
reconstruction under communication constraints. The proposed achievable scheme
is shown to achieve the minimum generalization error as well as the optimal
rate-distortion region.
