Online Algorithm Matches Offline Factor for Submodular Maximization.

Vaneet Aggarwal, Yiyang Lu· September 3, 2026 View original

Key takeaways

  • An online algorithm now matches the 0.401 offline approximation factor for DR-submodular maximization.
  • This improves online guarantees for non-monotone optimization problems.
  • The algorithm uses a weighted online learner and an asymmetric balance theorem.
  • It offers sublinear regret and efficient oracle calls in the post-decision model.

Who benefits

E-commerceAdvertising TechnologyResource ManagementMachine Learning PlatformsLogistics

Summary

Researchers developed an online algorithm for maximizing non-monotone DR-submodular functions that achieves the same 0.401 approximation factor as the best known constructive offline methods. This breakthrough significantly improves adversarial online guarantees for this class of problems.

A new online algorithm has been developed for maximizing nonnegative, non-monotone DR-submodular functions, a class of optimization problems relevant to various machine learning and resource allocation tasks. Previously, the best known constructive offline approximation factor for these problems was 0.401, while comparable adversarial online guarantees remained at 1/e (approximately 0.368). This research closes that gap, demonstrating that the 0.401 factor is now achievable in an online setting. The algorithm operates in a post-decision full-information value-oracle model, achieving a 0.401 factor with sublinear approximate regret under conditions where oracle feedback is conditionally unbiased and bounded. Instead of directly adapting the offline construction, the online algorithm replaces the objective-dependent box step with a weighted online learner. This learner cumulatively controls residual terms, and an exact asymmetric balance theorem helps preserve the offline coefficients despite adversarial variations in the online environment. The direct implementation offers O(T^(3/4)) regret and uses O(dT^(1/4)) oracle calls per round, with further optimizations for different batching and bandit settings.

Why it matters

This advancement provides more robust and efficient online optimization capabilities for complex decision-making processes where information arrives sequentially and objectives are non-monotone. Professionals in areas like dynamic resource allocation, recommendation systems, or online advertising can leverage this for improved performance.

How to implement this in your domain

  1. 1Identify online decision-making problems within your domain that can be modeled as non-monotone DR-submodular maximization.
  2. 2Review the algorithmic details to understand how the weighted online learner and balance theorem can be adapted.
  3. 3Develop a prototype implementation to test the algorithm's performance on a specific online optimization task.
  4. 4Compare the new algorithm's performance against existing online methods, particularly in adversarial or dynamic environments.

Original post by Vaneet Aggarwal, Yiyang Lu

"arXiv:2609.02145v1 Announce Type: new Abstract: We study online maximization of nonnegative, non-monotone DR-submodular functions over compact convex down-closed subsets of the $d$-dimensional unit cube. The best known constructive offline approximation factor is $0.401$ under th…"

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