The Limitations of Optimization from Samples

TL;DR

This paper introduces a formal framework called optimization from samples (OPS) to analyze the feasibility of optimizing functions based on training data, revealing fundamental limitations and providing approximation guarantees for certain classes.

Contribution

It formalizes the OPS framework, proves an impossibility result for coverage functions, and offers tight approximation bounds for various subclasses of functions.

Findings

No constant factor approximation exists for coverage functions with polynomial samples.

Tight approximation guarantees are achieved for unit-demand, additive, and monotone submodular functions.

A constant factor approximation is possible for monotone submodular functions with bounded curvature.

Abstract

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.