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Mathematical Intuition, Deep Learning, and Robbins’ Problem

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Abstract

The present article is an essay about mathematical intuition and Artificial intelligence (A.I.), followed by a guided excursion to Robbins’ Problem of Optimal Stop**. The latter is still open. The first part of the essay (Sect. 1 to Sect. 4), is written in a narrative style with well-known examples and an easily accessible terminology. It is easy to read. Its goal is to motivate mathematicians to think about certain aspects of A.I., and to prepare people in A.I. for the special features of Robbins’ problem. The major objective is presented in the second part (Sect. 5 to Sect. 13). It is to guide readers through the probabilistic intuition behind Robbins’ problem and to show why A.I., and in particular Deep Learning, may contribute an essential part in its solution. This article is an essay, because it contains no new mathematical results, and no implementation of deep learning either. However it presents a clear mission statement which seems important to us. We substantiate it by proposing two concrete challenges to view and solve the main part of the problem.

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Notes

  1. One often ranks in opposite order, namely the largest value is rank 1, the second largest value rank 2 etc. Our way of ranking has the advantage that ranks increase as the \(X_{j}\) increase.

  2. However, some models explain very well why certain results in Mathematics must hold in great generality. See e.g. the elementary approach to Taylor’s polynomial in B. [6].

  3. Robbins announced this problem by saying “Finally, here is the problem which I’d like to see solved before I die”, and he looked into the audience in a way we all felt he meant it. So much for the name. Sadly, Robbins’ wish did not realize. He died February 12th, 2001.

  4. According to S.M. Samuels, L. Shepp, and D. Siegmund, (private communications), Robbins was usually generous in sharing scientific results, but on “his” problem he seemed particular.

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Acknowledgement

The author thanks G. Bontempi and J.-F. Raskin for discussions on deep learning, and M. Duerinckx and P. Ernst for earlier discussions on Robbins’ problem.

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  1. Department of Mathematics, Université Libre de Bruxelles, Campus Plaine, CP 210, B-1050, Brussels, Belgium

    F. Thomas Bruss

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Correspondence to F. Thomas Bruss.

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Bruss, F.T. Mathematical Intuition, Deep Learning, and Robbins’ Problem. Jahresber. Dtsch. Math. Ver. 126, 69–93 (2024). https://doi.org/10.1365/s13291-024-00277-3

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