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This paper resolves an open problem in learning theory by proving that the state-of-the-art bounds for preferential compression are tight, offering rigorous safety certificates for modern machine learning methods. By providing a simpler proof based on elementary counting arguments, it removes unnecessary complexity from deriving performance guarantees for algorithms like Support Vector machines and the Scenario Approach. This theoretical advancement directly addresses the bottleneck of deploying reliable, certified learning-based systems in critical applications.
Preferent Compression Bounds Are Tight
arXiv:2609.36030v1 Announce Type: new Abstract: The lack of rigorous safety and performance certificates remains a key bottleneck to the deployment of modern learning-based methods. Sample compression has recently emerged as a powerful tool for deriving such certificates, with particularly sharp bounds available for algorithms satisfying a so-called preference property -- also known as stability in the learning theory literature. These bounds find direct application across domains as different as the Scenario Approach, Pick-to-Learn, and Support Vector methods. However, whether they are tight has remained an open problem. In this paper we resolve this question affirmatively and show that the state-of-the-art bound for preferent compressions is provably tight. We establish this by exhibiting an explicit construction based on the uniform distribution and order statistics that attains the bound in the limit. Along the way, we also provide a considerably shorter and more accessible proof of this bound, requiring only elementary counting arguments and no infinite-dimensional duality.
Abstract:The lack of rigorous safety and performance certificates remains a key bottleneck to the deployment of modern learning-based methods. Sample compression has recently emerged as a powerful tool for deriving such certificates, with particularly sharp bounds available for algorithms satisfying a so-called preference property -- also known as stability in the learning theory literature. These bounds find direct application across domains as different as the Scenario Approach, Pick-to-Learn, and Support Vector methods. However, whether they are tight has remained an open problem. In this paper we resolve this question affirmatively and show that the state-of-the-art bound for preferent compressions is provably tight. We establish this by exhibiting an explicit construction based on the uniform distribution and order statistics that attains the bound in the limit. Along the way, we also provide a considerably shorter and more accessible proof of this bound, requiring only elementary counting arguments and no infinite-dimensional duality.
Machine Learning (cs.LG); Systems and Control (eess.SY); Optimization and Control (math.OC); Probability (math.PR)
arXiv:2609.36030 [cs.LG]
(or arXiv:2609.36030v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2609.36030 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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