Optimization from Structured Samples for Coverage Functions

TL;DR

This paper introduces a new structured sampling model for optimizing coverage functions, enabling constant approximation algorithms where previous models failed, under certain assumptions.

Contribution

The paper proposes the OPSS model that encodes structural information in samples, allowing constant approximation algorithms for coverage functions under specific assumptions.

Findings

Efficient algorithms achieve constant approximation under three assumptions.

A matching constant lower bound is established, showing tightness.

Removing any assumption leads to no constant approximation.

Abstract

We revisit the optimization from samples (OPS) model, which studies the problem of optimizing objective functions directly from the sample data. Previous results showed that we cannot obtain a constant approximation ratio for the maximum coverage problem using polynomially many independent samples of the form $\{S_i, f(S_i)\}_{i=1}^t$ (Balkanski et al., 2017), even if coverage functions are $(1 - \epsilon)$-PMAC learnable using these samples (Badanidiyuru et al., 2012), which means most of the function values can be approximately learned very well with high probability. In this work, to circumvent the impossibility result of OPS, we propose a stronger model called optimization from structured samples (OPSS) for coverage functions, where the data samples encode the structural information of the functions. We show that under three general assumptions on the sample distributions, we can design efficient OPSS algorithms that achieve a constant approximation for the maximum coverage problem. We further prove a constant lower bound under these assumptions, which is tight when not considering computational efficiency. Moreover, we also show that if we remove any one of the three assumptions, OPSS for the maximum coverage problem has no constant approximation.