Cauchy Mean Theorem

The purpose of this paper was to prove formally, using the Mizar language, Arithmetic Mean/Geometric Mean theorem known maybe better under the name of AM-GM inequality or Cauchy mean theorem. It states that the arithmetic mean of a list of a non-negative real numbers is greater than or equal to the geometric mean of the same list. The formalization was tempting for at least two reasons: one of them, perhaps the strongest, was that the proof of this theorem seemed to be relatively easy to formalize (e.g. the weaker variant of this was proven in [13]). Also Jensen’s inequality is already present in the Mizar Mathematical Library. We were impressed by the beauty and elegance of the simple proof by induction and so we decided to follow this specific way. The proof follows similar lines as that written in Isabelle [18]; the comparison of both could be really interesting as it seems that in both systems the number of lines needed to prove this are really close. This theorem is item #38 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/.Institute of Informatics University of Białystok Akademicka 2, 15-267 Białystok PolandGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.Grzegorz Bancerek. Tarski’s classes and ranks. Formalized Mathematics, 1(3):563–567, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485–492, 1996.Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507–513, 1990.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990.Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661–668, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.Fuguo Ge and Xiquan Liang. On the partial product of series and related basic inequalities. Formalized Mathematics, 13(3):413–416, 2005.Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5):841–845, 1990.Artur Korniłowicz. On the real valued functions. Formalized Mathematics, 13(1):181–187, 2005.Jarosław Kotowicz. Functions and finite sequences of real numbers. Formalized Mathematics, 3(2):275–278, 1992.Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887–890, 1990.Benjamin Porter. Cauchy’s mean theorem and the Cauchy-Schwarz inequality. Archive of Formal Proofs, March 2006. ISSN 2150-914x. http://afp.sf.net/entries/Cauchy.shtml Formal proof development.Konrad Raczkowski and Andrzej Nędzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213–216, 1991.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329–334, 1990.Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4):341–347, 2003.Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445–449, 1990.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990.Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569–573, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990

Preface

Grabowski Adam - Institute of Informatics, University of Białystok, Akademicka 2, 15-267 Białystok, PolandShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.Grzegorz Bancerek. Tarski’s classes and ranks. Formalized Mathematics, 1(3):563–567, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.Agata Darmochwał. Compact spaces. Formalized Mathematics, 1(2):383–386, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599–603, 1991.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.Agata Darmochwał. Families of subsets, subspaces and mappings in topological spaces. Formalized Mathematics, 1(2):257–261, 1990.Ryszard Engelking. Teoria wymiaru. PWN, 1981.Adam Grabowski. Properties of the product of compact topological spaces. Formalized Mathematics, 8(1):55–59, 1999.Zbigniew Karno. Separated and weakly separated subspaces of topological spaces. Formalized Mathematics, 2(5):665–674, 1991.Artur Korniłowicz. Jordan curve theorem. Formalized Mathematics, 13(4):481–491, 2005.Artur Korniłowicz. The definition and basic properties of topological groups. Formalized Mathematics, 7(2):217–225, 1998.Artur Korniłowicz and Yasunari Shidama. Brouwer fixed point theorem for disks on the plane. Formalized Mathematics, 13(2):333–336, 2005.Artur Korniłowicz and Yasunari Shidama. Intersections of intervals and balls in EnT. Formalized Mathematics, 12(3):301–306, 2004.Robert Milewski. Bases of continuous lattices. Formalized Mathematics, 7(2):285–294, 1998.Yatsuka Nakamura and Andrzej Trybulec. Components and unions of components. Formalized Mathematics, 5(4):513–517, 1996.Beata Padlewska. Connected spaces. Formalized Mathematics, 1(1):239–244, 1990.Beata Padlewska. Locally connected spaces. Formalized Mathematics, 2(1):93–96, 1991.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223–230, 1990.Karol Pąk. Tietze extension theorem for n-dimensional spaces. Formalized Mathematics, 22(1):11–19. doi:10.2478/forma-2014-0002.Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441–444, 1990.Marco Riccardi. The definition of topological manifolds. Formalized Mathematics, 19(1): 41–44, 2011. doi:10.2478/v10037-011-0007-4.Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4): 535–545, 1991.Andrzej Trybulec. On the geometry of a Go-Board. Formalized Mathematics, 5(3):347– 352, 1996.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.Mirosław Wysocki and Agata Darmochwał. Subsets of topological spaces. Formalized Mathematics, 1(1):231–237, 1990

Federated Unlearning: A Survey on Methods, Design Guidelines, and Evaluation Metrics

Federated Learning (FL) enables collaborative training of a Machine Learning (ML) model across multiple parties, facilitating the preservation of users' and institutions' privacy by keeping data stored locally. Instead of centralizing raw data, FL exchanges locally refined model parameters to build a global model incrementally. While FL is more compliant with emerging regulations such as the European General Data Protection Regulation (GDPR), ensuring the right to be forgotten in this context - allowing FL participants to remove their data contributions from the learned model - remains unclear. In addition, it is recognized that malicious clients may inject backdoors into the global model through updates, e.g. to generate mispredictions on specially crafted data examples. Consequently, there is the need for mechanisms that can guarantee individuals the possibility to remove their data and erase malicious contributions even after aggregation, without compromising the already acquired "good" knowledge. This highlights the necessity for novel Federated Unlearning (FU) algorithms, which can efficiently remove specific clients' contributions without full model retraining. This survey provides background concepts, empirical evidence, and practical guidelines to design/implement efficient FU schemes. Our study includes a detailed analysis of the metrics for evaluating unlearning in FL and presents an in-depth literature review categorizing state-of-the-art FU contributions under a novel taxonomy. Finally, we outline the most relevant and still open technical challenges, by identifying the most promising research directions in the field.Comment: 23 pages, 8 figures, and 6 table

University of Białystok CC-BY-SA License ver. 3.0 or later ISSN

Summary. The purpose of this paper was to prove formally, using the Mizar language, Arithmetic Mean/Geometric Mean theorem known maybe better under the name of AM-GM inequality or Cauchy mean theorem. It states that the arithmetic mean of a list of a non-negative real numbers is greater than or equal to the geometric mean of the same list. The formalization was tempting for at least two reasons: one of them, perhaps the strongest, was that the proof of this theorem seemed to be relatively easy to formalize (e.g. the weaker variant of this was proven in The proof follows similar lines as that written in Isabelle [18]; the comparison of both could be really interesting as it seems that in both systems the number of lines needed to prove this are really close. This theorem is item #38 from the "Formalizing 100 Theorems" list maintained by Freek Wiedijk a