Adversarial Resilience of Poisson-Process Submodular Maximization over Matroids: From Robust Offline Optimization to Full-Bandit Learning
A paper on arXiv (cs.AI) presents an adversarial resilience theorem for the Spiteful Greedy Swap Poisson Process (SGS-Poisson) algorithm. Under a controlled value oracle with error bounded by ξ, the algorithm retains approximation factors 1/e for non-monotone and 1-1/e for monotone submodular objectives, using O(nk^2 ε^{-2}) oracle calls. The offline-to-online reduction yields full-bandit CMAB algorithms with regret O(n^{1/5} k^{4/5} T^{4/5}).
The paper studies nonnegative submodular maximization subject to a general matroid when the offline algorithm is given an arbitrary controlled value oracle. The 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. Under every controlled oracle with error bounded by ξ, the implementation returns a feasible set with expected value at least (1/e - ε)OPT - O(kξ) and (1-1/e - ε)OPT - O(kξ), respectively, using O(nk^2 ε^{-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 O(n^{1/5} k^{4/5} T^{4/5}) regret.
The SGS-Poisson algorithm maintains its approximation guarantees under adversarial perturbations to the value oracle, with error scaling linearly in kξ. The regret bound O(n^{1/5} k^{4/5} T^{4/5}) suggests a trade-off between exploration and exploitation that is sublinear in T, indicating efficient learning in full-bandit settings.
This work advances robust optimization for submodular functions, which are common in summarization, sensor placement, and recommendation. The adversarial resilience could make such algorithms more practical in noisy or adversarial environments.
The algorithm's robustness to oracle errors and its regret guarantees could reduce risk in deploying submodular optimization in production systems where evaluation is noisy or costly, potentially lowering operational costs and improving decision quality.
Potential next steps include empirical validation on real-world matroid-constrained submodular problems, extension to other combinatorial constraints, and tighter regret bounds. The offline-to-online reduction may inspire new bandit algorithms for structured decision-making.