6 research outputs found
The Limitations of Optimization from Samples
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
Testing Submodularity
We show that for any constants and , given oracle access to an unknown function it is possible to determine if the function is submodular or is -far from every submodular function, in distance, with a \emph{constant} number of queries to the oracle. We refer to the process of determining if an unknown function has a property, or is far from every function having the property, as \emph{property testing}, and we refer to the algorithm that does that as a tester or a testing algorithm.
A function is a \emph{-junta} if there is a set of cardinality such that the value of on any input is completely determined by the values for . For any constant and a set of -juntas , we give an algorithm which determines if an unknown function is -close to some function in or is -far from every function in , in distance, with a constant number of queries to the unknown function. This result, combined with a recent junta theorem of Feldman and \Vondrak (2016) in which they show every submodular function is -close, in distance, to another submodular function which is a -junta, yields the constant-query testing algorithm for submodular functions.
We also give constant-query testing algorithms for a variety of other natural properties of valuation functions, including fractionally additive (XOS) functions, OXS functions, unit demand functions, coverage functions, and self-bounding functions