Adaptive Submodularity: Theory and Applications in Active Learning and Stochastic Optimization

TL;DR

This paper introduces adaptive submodularity, a property that enables efficient decision-making in stochastic optimization problems with partial information, providing theoretical guarantees and practical applications in various fields.

Contribution

The paper generalizes submodularity to adaptive policies, offering a new framework with performance guarantees and faster algorithms for stochastic optimization tasks.

Findings

Adaptive greedy algorithm is competitive with the optimal policy.

Adaptive submodularity applies to diverse applications like sensor placement and viral marketing.

Proves improved approximation guarantees for several problems.

Abstract

Solving stochastic optimization problems under partial observability, where one needs to adaptively make decisions with uncertain outcomes, is a fundamental but notoriously difficult challenge. In this paper, we introduce the concept of adaptive submodularity, generalizing submodular set functions to adaptive policies. We prove that if a problem satisfies this property, a simple adaptive greedy algorithm is guaranteed to be competitive with the optimal policy. In addition to providing performance guarantees for both stochastic maximization and coverage, adaptive submodularity can be exploited to drastically speed up the greedy algorithm by using lazy evaluations. We illustrate the usefulness of the concept by giving several examples of adaptive submodular objectives arising in diverse applications including sensor placement, viral marketing and active learning. Proving adaptive submodularity for these problems allows us to recover existing results in these applications as special cases, improve approximation guarantees and handle natural generalizations.