Online SuBmodular + SuPermodular (BP) Maximization with Bandit Feedback

TL;DR

This paper extends online submodular maximization to non-submodular objectives, including BP decompositions and weakly submodular functions, with scalable algorithms and theoretical guarantees, applicable to recommendation and data selection tasks.

Contribution

It introduces algorithms for online maximization of non-submodular objectives with BP decomposition, including separate feedback for each term, and employs Nystrom sketching for scalability.

Findings

Achieves sub-linear regret bounds in Gaussian process bandits.

Demonstrates effectiveness in recommendation systems.

Reduces computational costs with Nystrom sketching.

Abstract

In the context of online interactive machine learning with combinatorial objectives, we extend purely submodular prior work to more general non-submodular objectives. This includes: (1) those that are additively decomposable into a sum of two terms (a monotone submodular and monotone supermodular term, known as a BP decomposition); and (2) those that are only weakly submodular. In both cases, this allows representing not only competitive (submodular) but also complementary (supermodular) relationships between objects, enhancing this setting to a broader range of applications (e.g., movie recommendations, medical treatments, etc.) where this is beneficial. In the two-term case, moreover, we study not only the more typical monolithic feedback approach but also a novel framework where feedback is available separately for each term. With real-world practicality and scalability in mind, we integrate Nystrom sketching techniques to significantly reduce the computational cost, including for the purely submodular case. In the Gaussian process contextual bandits setting, we show sub-linear theoretical regret bounds in all cases. We also empirically show good applicability to recommendation systems and data subset selection.