In this paper we consider the following question: can we optimize objective
functions from the training data we use to learn them? We formalize this
question through a novel framework we call optimization from samples (OPS). In
OPS, we are given sampled values of a function drawn from some distribution and
the objective is to optimize the function under some constraint.
While there are interesting classes of functions that can be optimized from
samples, our main result is an impossibility. We show that there are classes of
functions which are statistically learnable and optimizable, but for which no
reasonable approximation for optimization from samples is achievable. In
particular, our main result shows that there is no constant factor
approximation for maximizing coverage functions under a cardinality constraint
using polynomially-many samples drawn from any distribution.
We also show tight approximation guarantees for maximization under a
cardinality constraint of several interesting classes of functions including
unit-demand, additive, and general monotone submodular functions, as well as a
constant factor approximation for monotone submodular functions with bounded
curvature
