  For many optimization problems in machine learning, finding an optimal
solution is computationally intractable and we seek algorithms that perform
well in practice. Since computational intractability often results from
pathological instances, we look for methods to benchmark the performance of
algorithms against optimal solutions on real-world instances. The main
challenge is that an optimal solution cannot be efficiently computed for
intractable problems, and we therefore often do not know how far a solution is
from being optimal. A major question is therefore how to measure the
performance of an algorithm in comparison to an optimal solution on instances
we encounter in practice.
  In this paper, we address this question in the context of submodular
optimization problems. For the canonical problem of submodular maximization
under a cardinality constraint, it is intractable to compute a solution that is
better than a $1-1/e \approx 0.63$ fraction of the optimum. Algorithms like the
celebrated greedy algorithm are guaranteed to achieve this $1-1/e$ bound on any
instance and are used in practice.
  Our main contribution is not a new algorithm for submodular maximization but
an analytical method that measures how close an algorithm for submodular
maximization is to optimal on a given problem instance. We use this method to
show that on a wide variety of real-world datasets and objectives, the
approximation of the solution found by greedy goes well beyond $1-1/e$ and is
often at least 0.95. We develop this method using a novel technique that lower
bounds the objective of a dual minimization problem to obtain an upper bound on
the value of an optimal solution to the primal maximization problem.
