Deletion Robust Submodular Maximization over Matroids

TL;DR

This work tackles deletion-robust submodular maximization under general matroid constraints by formulating a two-phase robustness model where a compact summary $W$ is prepared before deletions, and a solution is produced from $W\setminus D$ after up to $d$ elements are deleted. It introduces threshold-based, random-bundling techniques to build $W$ with near-optimal memory $\tilde{O}(k+d)$ and proves constant-factor guarantees: $2+\beta+O(\varepsilon)$-type bounds in the centralized setting and $4+\beta+O(\varepsilon)$ in streaming, which translate to $(3.582+O(\varepsilon))$ and $(5.582+O(\varepsilon))$-approximation respectively when using standard submodular maximization subroutines. The methods are validated on real datasets (MovieLens, Facebook, RunInRome, Uber), showing they achieve high value with significantly smaller summaries and often competitive with omniscient baselines that know deletions in advance. Together, the results close a long-standing gap for memory-efficient robust submodular optimization under general matroids and offer practical knobs for privacy-aware data summarization and robust selection tasks. The work thus provides theoretically principled, scalable tools for making submodular maximization robust to adversarial deletions in both centralized and streaming contexts.

Abstract

Maximizing a monotone submodular function is a fundamental task in machine learning. In this paper, we study the deletion robust version of the problem under the classic matroids constraint. Here the goal is to extract a small size summary of the dataset that contains a high value independent set even after an adversary deleted some elements. We present constant-factor approximation algorithms, whose space complexity depends on the rank $k$ of the matroid and the number $d$ of deleted elements. In the centralized setting we present a $(3.582+O(\varepsilon))$-approximation algorithm with summary size $O(k + \frac{d \log k}{\varepsilon^2})$. In the streaming setting we provide a $(5.582+O(\varepsilon))$-approximation algorithm with summary size and memory $O(k + \frac{d \log k}{\varepsilon^2})$. We complement our theoretical results with an in-depth experimental analysis showing the effectiveness of our algorithms on real-world datasets.

Figures (7)

• Figure 1: The value of the objective (submodular function $f$) for Centralized and Streaming algorithms compared to the benchmarks Omniscent-Greedy and Omniscent-Swapping with respect to $|D|$. Average and standard deviation over three runs are reported.
• Figure 2: Results of the Influence Maximization experiments on the Facebook dataset for different values of $\varepsilon$. Note how as $\varepsilon$ decreases the performances of our algorithms improve, while being always comparable with the benchmarks. The lines corresponding to $\textsc{Omniscent-Swapping}\xspace$ changes in different plots since the random permutation considered changes; this however does not change its qualitative performance.
• Figure 3: Results of the Interactive Personalized Movie Recommendation on the MovieLens dataset for different values of $\varepsilon$. Note how as $\varepsilon$ decreases the performances of our algorithms improve, while being always comparable with the benchmarks. The performances of the benchmarks change for different values of $\varepsilon$ because in each experiment a new user feature vector is drawn uniformly at random as well as a new permutation is considered in the streaming setting. The results are however qualitatively comparable.
• Figure 4: Results of the kernel logdet experiment on the RunInRome dataset for different values of $\varepsilon$. Note how as $\varepsilon$ decreases the performances of our algorithms improve, while being always comparable with the benchmarks.
• Figure 5: Results of the $k$-medoid experiment on the RunInRome dataset for different values of $\varepsilon$. Note how as $\varepsilon$ decreases the performances of our algorithms improve, starting from a worst case of $\approx 10 \%$ for $\varepsilon = 0.99$ and a small number of deletions to a nearly identical performance for $\varepsilon = 0.3.$

Theorems & Definitions (27)

• Theorem 3.1
• proof
• Corollary 3.2
• proof
• Theorem 4.1
• proof
• Corollary 4.2
• proof
• Lemma A.1
• proof