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Agnostic PAC Learning Rates Differ for Direct Sums

Mihir More, Aritra Das, Debayan Gupta· August 10, 2026 View original

Key takeaways

  • The agnostic PAC learning rate for direct sums is not solely determined by single-instance rates.
  • This finding refines theoretical understanding of learning complexity.
  • It highlights the need for more nuanced analysis of combined learning systems.

Who benefits

AcademiaAI ResearchTheoretical Computer Science

Summary

This research demonstrates that the single-instance learning rate does not solely determine the direct-sum learning rate in agnostic PAC learning, challenging prior assumptions. It uses specific binary function classes to illustrate this separation.

Recent theoretical work in machine learning explores the relationship between the agnostic PAC learning curve of a single concept class and its direct sum. Previous research had posed a question about how these rates depend on each other. This new study provides a definitive answer, showing that the learning rate for a single instance does not necessarily dictate the rate for its direct sum. The authors illustrate this by constructing specific examples of binary function classes that exhibit the same single-instance agnostic learning curve but diverge in their direct-sum behavior, thereby establishing a rate separation.

Why it matters

This theoretical finding is crucial for researchers and practitioners developing and analyzing machine learning algorithms, as it refines the understanding of learning complexity and generalization bounds. It suggests that simply knowing the performance on individual components might not be sufficient to predict the performance of combined systems.

Original post by Mihir More, Aritra Das, Debayan Gupta

"arXiv:2608.06951v1 Announce Type: new Abstract: Hanneke, Moran, and Waknine \cite{HannekeMoranWaknine2024} asked how the agnostic PAC learning curve of the direct sum $C^r$ depends on the single-instance learning curve $\epsagn(n\mid C)$ and on $r$. We show that the single-instan…"

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Originally posted by Mihir More, Aritra Das, Debayan Gupta on X · view source

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