Abstract
We formulate an optimization stochastic algorithm convergence theorem, of Solis and Wets type, and we show several instances of its application to concrete algorithms. In this convergence theorem the algorithm is a sequence of random variables and, in order to describe the increasing flow of information associated to this sequence we define a filtration – or flow of \(\sigma \)-algebras – on the probability space, depending on the sequence of random variables and on the function being optimized. We compare the flow of information of two convergent algorithms by comparing the associated filtrations by means of the Cotter distance of \(\sigma \)-algebras. The main result is that two convergent optimization algorithms have the same information content if both their limit minimization functions generate the full \(\sigma \)-algebra of the probability space.
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Acknowledgements
This work was partially supported, for the first and fourth authors, through the project UID/MAT/00297/2020 of the Centro de Matemática e Aplicações, financed by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) and, for the third author, through RFBR (Grant n. 19-01-00451).
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A Appendix
A Appendix
Deduction of Formula (2). Let \( \lambda _x\) denote the Lebesgue measure over \(\mathcal {D}\) applied to the set defined by the variable x.
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Esquível, M.L., Machado, N., Krasii, N.P., Mota, P.P. (2021). On the Information Content of Some Stochastic Algorithms. In: Shiryaev, A.N., Samouylov, K.E., Kozyrev, D.V. (eds) Recent Developments in Stochastic Methods and Applications. ICSM-5 2020. Springer Proceedings in Mathematics & Statistics, vol 371. Springer, Cham. https://doi.org/10.1007/978-3-030-83266-7_5
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