Russian Mathematical Surveys, 2024, Volume 79, Issue 6, Pages 937–938 DOI: https://doi.org/10.4213/rm10224e (Mi rm10224)

DOI: https://doi.org/10.4213/rm10224e

Bibliographic databases:

By implementing new technologies in science, education, health care, and other areas the mankind opens a new chapter of its history. Deep learning and artificial intelligence (AI) are extremely important contemporary fields where there is currently a large gap between great practical successes and the theoretical understanding of the underlying structures.

This issue contains papers of Russian researchers devoted to optimizing and improving the reliability and the interpretability of AI models. These papers are linked by the topics of investigating models of machine learning, increasing the confidence in AI models, developing theoretical guarantees of their work, and extending the mathematical theory. The studies included in the issue aim at improving the reliability and efficacity of AI methods and at an analysis and development of methods having formal guarantees of convergence and predictive power.

The first article proposes a new approach combining two data similarity assumptions that have previously been investigated separately: using the Hessian matrix and homogeneous gradients. Incorporating both ideas in analyzing a new method allows one to take account of local features of data and reduce incidentally the costs of communication between devices. To address privacy concerns, the authors added noise to local gradients and examined its effect on the convergence of the proposed algorithm. The second article considers approximations in the theory of greedy algorithms and, in particular, the Weak Chebyshev Greedy Algorithm. The authors extend the theory to the case of complex Banach spaces, since the existing theoretical base has only been developed for real Banach spaces. The third paper focuses on improving the stability of the Bayesian classifier by adding artificially generated observations from a background class. This approach enables the classifier to reject the decision in ambiguous situations and improves the detection of anomalies and outliers in input data. In the fourth paper the authors discuss the improvement of theoretical guarantees of the Local SGD method, allowing one to reduce the frequency of communications in distributed systems. The authors propose to abandon the assumption of Lipschitz Hessian by introducing the concept of approximate quadraticity, which enables them to extend the range of applications of the method and analyze its efficiency for a wide class of problems. The last paper presents a new version of Local SGD. This method is adapted to scaling techniques, such as Adam and RMSProp. A characteristic feature of the new approach is its flexibility, allowing one to use various preconditioning matrices and achieve improved convergence in distributed learning problems.

The issue will be interesting to researchers and practitioners engaged in machine learning, combinatorial optimization, or the theoretical aspects of artificial intelligence, to those interested in state-of-the-art approaches to enhancing the reliability of classification, development of effective methods of work with big data, and in new ideas about hardware implementation of AI algorithms. The papers collected here will particularly be valuable for experts in distributed systems, who are interested in formal guarantees of the work of models.

This is not one of the usual issues of the journal Uspekhi Matematicheskikh Nauk, (translated as Russian Mathematical Surveys). Normally, we publish surveys and articles on purely theoretical areas of contemporary mathematics. On the other hand AI is now being actively implemented in our daily life, affecting also the traditional fields of applications of mathematics, related to mathematical modelling, solution of equations of mathematical physics, use of statistical methods, and so on. However, many questions relating to the rigorous mathematical justification of AI algorithms are still open. The editorial board hopes that publishing this issue can attract young researchers from various areas of mathematics to AI problems.

Citation in format AMSBIB

\Bibitem{Koz24}
\by V.~V.~Kozlov
\paper Preface by the Editor-in-Chief
\jour Russian Math. Surveys
\yr 2024
\vol 79
\issue 6
\pages 937--938
\mathnet{http://mi.mathnet.ru/eng/rm10224}
\crossref{https://doi.org/10.4213/rm10224e}
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