Learnability can be undecidable


The mathematical foundations of machine learning play a key role in the development of the field. They improve our understanding and provide tools for designing new learning paradigms. The advantages of mathematics, however, sometimes come with a cost. Gödel and Cohen showed, in a nutshell, that not everything is provable. Here we show that machine learning shares this fate. We describe simple scenarios where learnability cannot be proved nor refuted using the standard axioms of mathematics. Our proof is based on the fact the continuum hypothesis cannot be proved nor refuted. We show that, in some cases, a solution to the ‘estimating the maximum’ problem is equivalent to the continuum hypothesis. The main idea is to prove an equivalence between learnability and compression.

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The data that support the findings of this study are available from the corresponding author upon request.

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  • 23 January 2019

    In the version of this Article originally published, the following text was missing from the Acknowledgements: ‘Part of the research was done while S.M. was at the Institute for Advanced Study in Princeton and was supported by NSF grant CCF-1412958.’ This has now been corrected.


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The authors thank D. Chodounský, S. Hanneke, R. Honzk and R. Livni for useful discussions. The authors also acknowledge the Simons Institute for the Theory of Computing for support. A.S.’s research has received funding from the Israel Science Foundation (ISF grant no. 552/16) and from the Len Blavatnik and the Blavatnik Family foundation. A.Y.’s research is supported by ISF grant 1162/15. Part of the research was done while S.M. was at the Institute for Advanced Study in Princeton and was supported by NSF grant CCF-1412958.

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Correspondence to Amir Yehudayoff.

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Ben-David, S., Hrubeš, P., Moran, S. et al. Learnability can be undecidable. Nat Mach Intell 1, 44–48 (2019). https://doi.org/10.1038/s42256-018-0002-3

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