Coresets remembered and items forgotten: submodular maximization with deletions

TL;DR

This paper addresses robust submodular maximization under item deletions, proposing algorithms that maintain approximation guarantees with minimal memory, applicable in privacy-sensitive and dynamic data scenarios.

Contribution

It introduces the first streaming and offline algorithms for robust submodular maximization with optimal or near-optimal coresets against adversarial deletions.

Findings

Streaming algorithm achieves a (1-2ε)/(4p)-approximation with coreset size k + d/ε.

Offline algorithm guarantees stronger approximation with coreset size O(d log(k)/ε).

Algorithms demonstrate superior empirical performance in real-world applications.

Abstract

In recent years we have witnessed an increase on the development of methods for submodular optimization, which have been motivated by the wide applicability of submodular functions in real-world data-science problems. In this paper, we contribute to this line of work by considering the problem of robust submodular maximization against unexpected deletions, which may occur due to privacy issues or user preferences. Specifically, we consider the minimum number of items an algorithm has to remember, in order to achieve a non-trivial approximation guarantee against adversarial deletion of up to $d$ items. We refer to the set of items that an algorithm has to keep before adversarial deletions as a deletion-robust coreset. Our theoretical contributions are two-fold. First, we propose a single-pass streaming algorithm that yields a $(1-2\epsilon)/(4p)$-approximation for maximizing a non-decreasing submodular function under a general p-matroid constraint and requires a coreset of size $k + d/\epsilon$, where $k$ is the maximum size of a feasible solution. To the best of our knowledge, this is the first work to achieve an (asymptotically) optimal coreset, as no constant-factor approximation is possible with a coreset of size sublinear in $d$. Second, we devise an effective offline algorithm that guarantees stronger approximation ratios with a coreset of size $O(d \log(k)/\epsilon)$. We also demonstrate the superior empirical performance of the proposed algorithms in real-life applications.