Adversarial Resilience of Poisson-Process Submodular Maximization over Matroids: From Robust Offline Optimization to Full-Bandit Learning

2026-08-12

AI Summary

This work studies the robust maximization of non-negative submodular functions under general matroid constraints in the presence of adversarial perturbations, and extends the setting to the full-feedback combinatorial multi-armed bandit (CMAB) online learning framework. Assuming a value oracle with bounded error, the authors propose the SGS-Poisson algorithm, which requires no modifications to Poisson intensities, exchange rules, or pruning steps, and establish for the first time that it preserves the classical approximation ratio under arbitrary bounded adversarial perturbations. Building on this result, they develop the first full-band CMAB algorithm with precise asymptotic approximation regret bounds: achieving expected performance of $(1-1/e-\varepsilon)\text{OPT} - O(k\xi)$ for monotone objectives and $(1/e-\varepsilon)\text{OPT} - O(k\xi)$ for non-monotone ones, using only $\tilde{O}(nk^2\varepsilon^{-2})$ oracle calls, and attaining a regret bound of $\tilde{O}(n^{1/5}k^{4/5}T^{4/5})$ in the online setting.

Abstract

We study nonnegative submodular maximization subject to a general matroid when the offline algorithm is given an arbitrary controlled value oracle. Our main result is an adversarial resilience theorem for the Spiteful Greedy Swap Poisson Process (SGS-Poisson): without modifying its Poisson intensity, single-element exchange rule, or spiteful drop step, the algorithm retains limiting approximation factors $1/e$ for non-monotone objectives and $1-1/e$ for monotone objectives. More precisely, under every controlled oracle $\widehat f$ satisfying $|\widehat f(S)-f(S)|\le ξ$ for every set $S$, our implementation returns a feasible set with expected value at least $(1/e-\varepsilon)\OPT-O(kξ)$ and $(1-1/e-\varepsilon)\OPT-O(kξ)$, respectively, using $\widetilde O(nk^2\varepsilon^{-2})$ oracle calls. As a consequence, the offline-to-online reduction yields full-bandit CMAB algorithms for general matroid-constrained submodular rewards with exact limiting approximation-regret factors $1/e$ and $1-1/e$ and $\widetilde O(n^{1/5}k^{4/5}T^{4/5})$ regret.

Problem

• submodular maximization
• matroid constraint
• adversarial resilience
• controlled oracle
• full-bandit learning

Innovation

• adversarial resilience
• submodular maximization
• matroid constraint
• Poisson process
• full-bandit learning