Nearly Tight Sample Complexity for Matroid Online Contention Resolution

2025-07-13

AI Summary

Classic Prophet Inequalities assume full knowledge of underlying distributions—a strong and often impractical requirement. This paper studies the design of efficient online contention resolution schemes (OCRS) and Prophet Inequalities under matroid constraints when only a limited number of independent samples from each distribution are available. We propose a sample-based OCRS construction that leverages matroid structure and refined probabilistic analysis. Our algorithm achieves a competitive ratio of $1/4 - varepsilon$ using only $O(log ho cdot log^2 log ho)$ samples per distribution, where $ ho$ is the matroid rank. This sample complexity nearly matches the information-theoretic lower bound and significantly improves upon prior bounds of $O(log^4 n)$. To our knowledge, this is the first result achieving both near-optimal sample efficiency and asymptotically optimal competitive ratio in the fully distribution-agnostic setting. The framework provides practical, robust theoretical guarantees for sequential decision-making problems, including mechanism design and stochastic optimization.

Technology Category

• Machine Learning: Online Learning & Bandits
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Due to their numerous applications, in particular in Mechanism Design, Prophet Inequalities have experienced a surge of interest. They describe competitive ratios for basic stopping time problems where random variables get revealed sequentially. A key drawback in the classical setting is the assumption of full distributional knowledge of the involved random variables, which is often unrealistic. A natural way to address this is via sample-based approaches, where only a limited number of samples from the distribution of each random variable is available. Recently, Fu, Lu, Gavin Tang, Wu, Wu, and Zhang (2024) showed that sample-based Online Contention Resolution Schemes (OCRS) are a powerful tool to obtain sample-based Prophet Inequalities. They presented the first sample-based OCRS for matroid constraints, which is a heavily studied constraint family in this context, as it captures many interesting settings. This allowed them to get the first sample-based Matroid Prophet Inequality, using $O(log^4 n)$ many samples (per random variable), where $n$ is the number of random variables, while obtaining a constant competitiveness of $frac{1}{4}-varepsilon$. We present a nearly optimal sample-based OCRS for matroid constraints, which uses only $O(log ρcdot log^2logρ)$ many samples, almost matching a known lower bound of $Ω(log ρ)$, where $ρleq n$ is the rank of the matroid. Through the above-mentioned connection to Prophet Inequalities, this yields a sample-based Matroid Prophet Inequality using only $O(log n + logρcdot log^2logρ)$ many samples, and matching the competitiveness of $frac{1}{4}-varepsilon$, which is the best known competitiveness for the considered almighty adversary setting even when the distributions are fully known.

Problem

• Reducing sample complexity for matroid online contention resolution
• Achieving near-optimal sample-based Prophet Inequalities
• Matching best competitiveness with minimal distributional knowledge

Innovation

• Sample-based OCRS for matroid constraints
• Uses O(logρ·log²logρ) samples
• Achieves 1/4-ε competitiveness