## Abstract

The operation of trimming data sets is heavily used in AI systems.

Trimming is useful to make AI systems more robust against adversarial or common perturbations. At the core of robust AI systems lies the concept that outliers in a data set occur with low probability, and therefore can be discarded with little loss of precision in

the result. The statistical argument that formalizes this concept of robustness is based on an extension of the Chebyshev’s inequality first proposed by Tukey in 1960.

In this paper we present a mechanized proof of robustness of the trimmed mean algorithm, which is a statistical method underlying many complex applications of deep learning. For this purpose we use the Coq proof assistant to formalize Tukey’s extension to Chebyshev’s inequality, which allows us to verify the robustness of the trimmed mean algorithm. Our contribution shows the viability of mechanized robustness arguments for algorithms that are at the foundation of complex AI systems.

Trimming is useful to make AI systems more robust against adversarial or common perturbations. At the core of robust AI systems lies the concept that outliers in a data set occur with low probability, and therefore can be discarded with little loss of precision in

the result. The statistical argument that formalizes this concept of robustness is based on an extension of the Chebyshev’s inequality first proposed by Tukey in 1960.

In this paper we present a mechanized proof of robustness of the trimmed mean algorithm, which is a statistical method underlying many complex applications of deep learning. For this purpose we use the Coq proof assistant to formalize Tukey’s extension to Chebyshev’s inequality, which allows us to verify the robustness of the trimmed mean algorithm. Our contribution shows the viability of mechanized robustness arguments for algorithms that are at the foundation of complex AI systems.

Original language | English |
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Title of host publication | Proceedings of PPDP (Principles of Declarative Programing Languages) |

Publisher | Association for Computing Machinery |

Publication date | 2021 |

Publication status | Published - 2021 |

## Keywords

- Machine learning
- robustness
- robust mean estimation
- Formal verification
- Proof