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There have been a number of recent proposals to enhance the performance ofmachine learning strategies for collider physics by combining many distinctevents into a single ensemble feature. To evaluate the efficacy of theseproposals, we study the connection between single-event classifiers andmulti-event classifiers under the assumption that collider events areindependent and identically distributed (IID). We show how one can buildoptimal multi-event classifiers from single-event classifiers, and we also showhow to construct multi-event classifiers such that they produce optimalsingle-event classifiers. This is illustrated for a Gaussian example as well asfor classification tasks relevant for searches and measurements at the LargeHadron Collider. We extend our discussion to regression tasks by showing howthey can be phrased in terms of parametrized classifiers. Empirically, we findthat training a single-event (per-instance) classifier is more effective thantraining a multi-event (per-ensemble) classifier, as least for the cases westudied, and we relate this fact to properties of the loss function gradient inthe two cases. While we did not identify a clear benefit from using multi-eventclassifiers in the collider context, we speculate on the potential value ofthese methods in cases involving only approximate independence, as relevant forjet substructure studies.

E Pluribus Unum Ex Machina: Learning from Many Collider Events at Once