  We consider a crowdsourcing data acquisition scenario, such as federated
learning, where a Center collects data points from a set of rational Agents,
with the aim of training a model. For linear regression models, we show how a
payment structure can be designed to incentivize the agents to provide
high-quality data as early as possible, based on a characterization of the
influence that data points have on the loss function of the model. Our
contributions can be summarized as follows: (a) we prove theoretically that
this scheme ensures truthful data reporting as a game-theoretic equilibrium and
further demonstrate its robustness against mixtures of truthful and heuristic
data reports, (b) we design a procedure according to which the influence
computation can be efficiently approximated and processed sequentially in
batches over time, (c) we develop a theory that allows correcting the
difference between the influence and the overall change in loss and (d) we
evaluate our approach on real datasets, confirming our theoretical findings.
