Beyond $1/2$-Approximation for Submodular Maximization on Massive Data Streams

TL;DR

This paper introduces SALSA, a streaming algorithm that surpasses the 0.5-approximation barrier for submodular maximization under random element arrival, with practical benefits demonstrated in real-world data tasks.

Contribution

The paper presents SALSA, the first low-memory, single-pass streaming algorithm achieving better than 0.5-approximation for submodular maximization under random order assumptions.

Findings

SALSA outperforms previous methods in experiments.

Improves approximation factor beyond 0.5 under random order.

No better approximation is possible in arbitrary order.

Abstract

Many tasks in machine learning and data mining, such as data diversification, non-parametric learning, kernel machines, clustering etc., require extracting a small but representative summary from a massive dataset. Often, such problems can be posed as maximizing a submodular set function subject to a cardinality constraint. We consider this question in the streaming setting, where elements arrive over time at a fast pace and thus we need to design an efficient, low-memory algorithm. One such method, proposed by Badanidiyuru et al. (2014), always finds a $0.5$-approximate solution. Can this approximation factor be improved? We answer this question affirmatively by designing a new algorithm SALSA for streaming submodular maximization. It is the first low-memory, single-pass algorithm that improves the factor $0.5$, under the natural assumption that elements arrive in a random order. We also show that this assumption is necessary, i.e., that there is no such algorithm with better than $0.5$-approximation when elements arrive in arbitrary order. Our experiments demonstrate that SALSA significantly outperforms the state of the art in applications related to exemplar-based clustering, social graph analysis, and recommender systems.