  We consider weighted structures, which extend ordinary relational structures
by assigning weights, i.e. elements from a particular group or ring, to tuples
present in the structure. We introduce an extension of first-order logic that
allows to aggregate weights of tuples, compare such aggregates, and use them to
build more complex formulas. We provide locality properties of fragments of
this logic including Feferman-Vaught decompositions and a Gaifman normal form
for a fragment called FOW1, as well as a localisation theorem for a larger
fragment called FOWA1. This fragment can express concepts from various machine
learning scenarios. Using the locality properties, we show that concepts
definable in FOWA1 over a weighted background structure of at most
polylogarithmic degree are agnostically PAC-learnable in polylogarithmic time
after pseudo-linear time preprocessing.
