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DOI: 10.4230/LIPIcs.ITP.2024.4

Taming Differentiable Logics with Coq Formalisation

Authors: Reynald Affeldt, Alessandro Bruni, Ekaterina Komendantskaya, Natalia Ślusarz, and Kathrin Stark

Published in: LIPIcs, Volume 309, 15th International Conference on Interactive Theorem Proving (ITP 2024)

Abstract

For performance and verification in machine learning, new methods have recently been proposed that optimise learning systems to satisfy formally expressed logical properties. Among these methods, differentiable logics (DLs) are used to translate propositional or first-order formulae into loss functions deployed for optimisation in machine learning. At the same time, recent attempts to give programming language support for verification of neural networks showed that DLs can be used to compile verification properties to machine-learning backends. This situation is calling for stronger guarantees about the soundness of such compilers, the soundness and compositionality of DLs, and the differentiability and performance of the resulting loss functions. In this paper, we propose an approach to formalise existing DLs using the Mathematical Components library in the Coq proof assistant. Thanks to this formalisation, we are able to give uniform semantics to otherwise disparate DLs, give formal proofs to existing informal arguments, find errors in previous work, and provide formal proofs to missing conjectured properties. This work is meant as a stepping stone for the development of programming language support for verification of machine learning.

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Reynald Affeldt, Alessandro Bruni, Ekaterina Komendantskaya, Natalia Ślusarz, and Kathrin Stark. Taming Differentiable Logics with Coq Formalisation. In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 4:1-4:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{affeldt_et_al:LIPIcs.ITP.2024.4,
  author =	{Affeldt, Reynald and Bruni, Alessandro and Komendantskaya, Ekaterina and \'{S}lusarz, Natalia and Stark, Kathrin},
  title =	{{Taming Differentiable Logics with Coq Formalisation}},
  booktitle =	{15th International Conference on Interactive Theorem Proving (ITP 2024)},
  pages =	{4:1--4:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-337-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{309},
  editor =	{Bertot, Yves and Kutsia, Temur and Norrish, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2024.4},
  URN =		{urn:nbn:de:0030-drops-207325},
  doi =		{10.4230/LIPIcs.ITP.2024.4},
  annote =	{Keywords: Machine Learning, Loss Functions, Differentiable Logics, Logic and Semantics, Interactive Theorem Proving}
}

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Short Paper

DOI: 10.4230/LIPIcs.ITP.2024.39

Robust Mean Estimation by All Means (Short Paper)

Authors: Reynald Affeldt, Clark Barrett, Alessandro Bruni, Ieva Daukantas, Harun Khan, Takafumi Saikawa, and Carsten Schürmann

Published in: LIPIcs, Volume 309, 15th International Conference on Interactive Theorem Proving (ITP 2024)

Abstract

We report the results of a verification experiment on an algorithm for robust mean estimation, i.e., an algorithm that computes a mean in the presence of outliers. We formalize the algorithm in the Coq proof assistant and devise a pragmatic approach for identifying and solving issues related to the choice of bounds. To keep our formalization succinct and generic, we recast the original argument using an existing library for finite probabilities that we extend with reusable lemmas. To formalize the original algorithm, which relies on a subtle convergence argument, we observe that by adding suitable termination checks, we can turn it into a well-founded recursion without losing its original properties. We also exploit a tactic for solving real-valued inequalities by approximation to heuristically fix inaccurate constant values in the original proof.

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Reynald Affeldt, Clark Barrett, Alessandro Bruni, Ieva Daukantas, Harun Khan, Takafumi Saikawa, and Carsten Schürmann. Robust Mean Estimation by All Means (Short Paper). In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 39:1-39:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{affeldt_et_al:LIPIcs.ITP.2024.39,
  author =	{Affeldt, Reynald and Barrett, Clark and Bruni, Alessandro and Daukantas, Ieva and Khan, Harun and Saikawa, Takafumi and Sch\"{u}rmann, Carsten},
  title =	{{Robust Mean Estimation by All Means}},
  booktitle =	{15th International Conference on Interactive Theorem Proving (ITP 2024)},
  pages =	{39:1--39:8},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-337-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{309},
  editor =	{Bertot, Yves and Kutsia, Temur and Norrish, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2024.39},
  URN =		{urn:nbn:de:0030-drops-207679},
  doi =		{10.4230/LIPIcs.ITP.2024.39},
  annote =	{Keywords: robust statistics, probability theory, formal verification}
}

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Documents authored by Bruni, Alessandro

Taming Differentiable Logics with Coq Formalisation

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Robust Mean Estimation by All Means (Short Paper)

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