  The Shapley-Folkman theorem shows that Minkowski averages of uniformly
bounded sets tend to be convex when the number of terms in the sum becomes much
larger than the ambient dimension. In optimization, Aubin and Ekeland [1976]
show that this produces an a priori bound on the duality gap of separable
nonconvex optimization problems involving finite sums. This bound is highly
conservative and depends on unstable quantities, and we relax it in several
directions to show that non convexity can have a much milder impact on finite
sum minimization problems such as empirical risk minimization and multi-task
classification. As a byproduct, we show a new version of Maurey's classical
approximate Carath\'eodory lemma where we sample a significant fraction of the
coefficients, without replacement, as well as a result on sampling constraints
using an approximate Helly theorem, both of independent interest.

Optimization and Control