  Consider two data providers, each maintaining records of different feature
sets about common entities. They aim to learn a linear model over the whole set
of features. This problem of federated learning over vertically partitioned
data includes a crucial upstream issue: entity resolution, i.e. finding the
correspondence between the rows of the datasets. It is well known that entity
resolution, just like learning, is mistake-prone in the real world. Despite the
importance of the problem, there has been no formal assessment of how errors in
entity resolution impact learning.
  In this paper, we provide a thorough answer to this question, answering how
optimal classifiers, empirical losses, margins and generalisation abilities are
affected. While our answer spans a wide set of losses --- going beyond proper,
convex, or classification calibrated ---, it brings simple practical arguments
to upgrade entity resolution as a preprocessing step to learning. One of these
suggests that entity resolution should be aimed at controlling or minimizing
the number of matching errors between examples of distinct classes. In our
experiments, we modify a simple token-based entity resolution algorithm so that
it indeed aims at avoiding matching rows belonging to different classes, and
perform experiments in the setting where entity resolution relies on noisy
data, which is very relevant to real world domains. Notably, our approach
covers the case where one peer \textit{does not} have classes, or a noisy
record of classes. Experiments display that using the class information during
entity resolution can buy significant uplift for learning at little expense
from the complexity standpoint.
