Abstract
We study the problem of empirical minimization for variance-type functionals over functional classes. Sharp non-asymptotic bounds for the excess variance are derived under mild conditions. In particular, it is shown that under some restrictions imposed on the functional class fast convergence rates can be achieved including the optimal non-parametric rates for expressive classes in the non-Donsker regime under some additional assumptions. Our main applications include variance reduction and optimal control.
Funding Statement
This article was prepared within the framework of the HSE University Basic Research Program.
Results of Section 3 were obtained by Denis Belomestny and Leonid Iosipoi under support of the RSF grant 19-71-30020 (HSE University).
Nikita Zhivotovskiy is funded in part by ETH Foundations of Data Science (ETH-FDS).
Citation
Denis Belomestny. Leonid Iosipoi. Quentin Paris. Nikita Zhivotovskiy. "Empirical variance minimization with applications in variance reduction and optimal control." Bernoulli 28 (2) 1382 - 1407, May 2022. https://doi.org/10.3150/21-BEJ1392
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