Fundamental theorems

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Fundamental theorems of Machine Learning with proofs

The goal of the course is to boost the quality of bachelor's and master's thesis works; to make the results of student scientific research well-founded. The course studies techniques and practice of theorem formulations and proofs in machine learning.

Why one needs to convey an important message, a scientific result as a theorem?

  1. Theorems are the most important messages in the field of research.
  2. Theorems present results in the language of mathematics by generality and rigor.
  3. Theorems are at the heart of mathematics and play a central role in its aesthetics.

Theorems present the message immediately and leave reasoning after. The direct narration puts reason first and the results after that.

  • How does direct narration transform into fast narration?
  • How to find, state, and prove theorems in our work?

This course shows both narration styles. It refers to our educational study and our work experience:

  1. Educational mimic progression
    • Definition \(\to\) (Axiom set) \(\to\) Theorem \(\to\) Proof \(\to\) Corollaries \(\to\) Examples \(\to\) Impact to applications
  2. Scientific discovery progression
    • Application problems \(\to\) Problem generalisations \(\to\) Useful algebraic platform \(\to\) Definitions \(\to\) Axiom set

So in our practice, we mimic the first part of the progression, then learn to discover patterns and formulate theorems. The theoretical talks give us a series of good examples.

Theorems

  1. Fundamental theorem of linear algebra S
  2. Singular values decomposition and spectral theorem W
  3. Gauss–Markov-(Aitken) theorem W
  4. Principal component analysis W
  5. Karhunen–Loève theorem W
  6. Kolmogorov–Arnold representation theorem W
  7. Universal approximation theorem by Cybenko W
  8. Deep neural network theorem Mark
  9. Inverse function theorem and Jacobian W
  10. No free lunch theorem by Wolpert W
  11. RKHS by Aronszajn and Mercer's theorem W
  12. Representer theorem by Schölkopf, Herbrich, and Smola W
  13. Convolution theorem (FT, convolution, correlation with CNN examples) W
  14. Fourier inversion theorem W
  15. Wiener–Khinchin theorem about autocorrelation and spectral decomposition W
  16. Parseval's theorem (and uniform, non-uniform convergence) W
  17. Probably approximately correct learning with the theorem about compression means learnability
  18. Bernstein–von Mises theorem W
  19. Holland's schema theorem W
  20. Variational approximation
  21. Convergence of random variables and Kloek's theorem W
  22. Exponential family of distributions and Nelder's theorem
  23. Multi-armed bandit theorem
  24. Copulas and Sklar's theorem W
  25. Boosting theorem Freud, Shapire, 1996, 1995
  26. Bootstrap theorem (statistical estimations): Ergodic theorem

Each class contains

  1. A lecturer's talk on one of fundamental theorems (\(40' = 30' + 10'\) discussion)
  2. Two students' talks (each \(20' = 15' + 5'\) discussion)

Each student delivers two talks

  1. On a theorem, which is formulated in a paper from the list of student thesis work's references
  2. On a theorem, which is formulated and proved by the student

It is welcome to

  • Make variants of our formulations and proofs
  • Re-formulate significant messages of researchers and formulate these messages as theorems

Plan of the talk

  1. Introduction: the main message briefly
  2. If necessary (it could be introduced during the talk)
    1. Axiom sets
    2. Definitions
    3. Algebraic structures
    4. Notations
  3. Theorem formulation and exact proof
    1. The author's variant of the proof could be ameliorated
  4. Corollaries
  5. Theorem significance and applications

Typography

  • As one (or two) text page example, template to download
  • Please
    • set the font size \(\geqslant 14\)pt
    • include plots, diagrams, freehand drawings

The organization

Scoring

  • Talks and text 0-4 points, according to comparison
  • Out-of-schedule drops a half
  • The exam 2 points: schemes of proof of various theorems
    • time-limit test (as Physics state exam) and discussion
    • theorem formulation and poof scheme are hand-written
    • two random theorems from the list below, 10 min to write the text

Theorem types

  • Uniqueness, existence
  • Universality
  • Convergence
  • Complexity
  • Properties of estimations
  • Bounds

Schedule

Spring semester 2021

Student talks

Speaker References
Bishuk Anton 17.2 link
Weiser Kirill 17.2 link, link
Grebenkova Olga 24.2 link
Gunaev Ruslan 24.2 link
Zholobov Vladimir 3.3 link
Islamov Rustem 3.3 link
Pankratov Victor 10.3 link
Savelyev Nikolay 10.3 link
Filatov Andrey 10.3 link
Filippova Anastasia 17.3 link
Khar Alexandra 17.3 link
Khristolyubov Maxim 24.3 link
Shokorov Vyacheslav 24.3 link

Invited talks

Speaker Link
Strijov, Potanin 10.2 link
Mark Potanin 17.2 link
Mark Potanin 24.2 link
Andriy Grabovyi 10.3 link
Andriy Grabovyi 17.3 link
Andriy Grabovyi 24.3 link
Mark Potanin 31.3 link

Out of schedule

  1. Three works by Greg Yang arXiv:1910.12478, arXiv:2006.14548, arXiv:2009.10685 Youtube Rus
  2. Theorems on flows by Johann Brehmera and Kyle Cranmera arXiv:2003.13913v2

References

  1. Mathematical statistics by A.A. Borovkov, 1998
  2. Learning Theory from First Principles by Francis Bach, 2021
  3. Theoretical foundations of potential function method in pattern recognition by M. A. Aizerman, E. M. Braverman, and L. I. Rozonoer // Avtomatica i Telemekhanica, 1964. Vol. 25, pp. 917-936.

Proof techniques

  1. Proofs and Mathematical Reasoning by Agata Stefanowicz, 2014
  2. The nuts and bolts of proofs by Antonella Cupillari, 2013
  3. Theorems, Corollaries, Lemmas, and Methods of Proof by Richard J. Rossi, 1956
  4. Problem Books in Mathematics by P.R. Halmos (editor), 1990
  5. Les contre-exemples en mathématique par Bertrand Hauchecorne, 2007
  6. Kolmogorov and Mathematical Logic by Vladimir A. Uspensky // The Journal of Symbolic Logic, Vol. 57, No. 2 (Jun., 1992), 385-412.
  7. Что такое аксиоматический метод? В.А. Успенский, 2001
  8. Аксиоматический метод. Е.Е. Золин, 2015

Methodology

  1. Introduction to Metamathematics by Stephen Cole Kleene, 1950
  2. Science and Method by Henry Poincare, 1908
  3. A Summary of Scientific Method by Peter Kosso, 2011
  4. Being a Researcher: An Informatics Perspective by Carlo Ghezzi, 2020
  5. The definitive glossary of higher mathematical jargon by Math Vault, 2015
  6. The definitive guide to learning higher mathematics: 10 principles to mathematical transcendence by Math Vault, 2020
  7. List of mathematical jargon on Wikipedia
  8. Пикабу. Типичные методы доказательства, 2018 (если вы чувствуете, что несет не туда)
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