Upper-Linearizability of Online Non-Monotone DR-Submodular Maximization over Down-Closed Convex Sets

2026-07-13 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, extended

Summary

The research introduces a novel structural result demonstrating that online non-monotone Diminishing-Return (DR)-submodular functions over down-closed convex sets are 1/e-linearizable. This is achieved through a carefully designed exponential reparametrization and surrogate potential, effectively reducing the complex non-convex problem to online linear optimization. This breakthrough yields O(T^{1/2}) static regret with only a single gradient query per round, significantly improving upon the previous state-of-the-art O(T^{2/3}) rate. Furthermore, the framework provides the first adaptive and dynamic regret guarantees for this challenging setting, alongside enhanced rates for semi-bandit, bandit, and zeroth-order feedback models, surpassing existing bounds across all feedback types.

Key takeaway

For Research Scientists developing online optimization algorithms for complex, non-monotone systems, this work provides a robust framework to achieve superior regret bounds and adaptability. You should investigate integrating the 1/e-linearizable reduction with your existing online linear optimization solvers. This approach offers O(T^{1/2}) static regret with single gradient queries and unlocks adaptive and dynamic regret guarantees, crucial for non-stationary environments in applications like revenue maximization or supply chain management.

Key insights

Non-monotone DR-submodular maximization over down-closed sets is 1/e-linearizable, enabling online linear optimization reduction.

Principles

• Linearizability simplifies non-convex online optimization.
• Jacobian-corrected gradients enable single-query efficiency.
• Exponential reparametrization preserves optimal approximation.

Method

The method involves an exponential reparametrization and a surrogate potential to transform non-monotone DR-submodular maximization into an online linear optimization problem, using a Jacobian-corrected gradient estimator (BQND) for single-query efficiency.

In practice

• Apply to price optimization and social network recommendations.
• Use for resource allocation and experimental design.
• Implement with projection-free online linear optimizers.

Topics

• Online Optimization
• DR-Submodular Maximization
• Linearizability
• Regret Bounds
• Gradient Oracles
• Non-Monotone Functions
• Down-Closed Convex Sets

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.