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Journal of the European Mathematical Society

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Volume 22, Issue 3, 2020, pp. 925–965
DOI: 10.4171/JEMS/937

Published online: 2019-12-16

Risk minimization by median-of-means tournaments

Gábor Lugosi[1] and Shahar Mendelson[2]

(1) Pompeu Fabra University, Barcelona, Spain
(2) The Australian National University, Canberra, Australia

We consider the classical statistical learning/regression problem, when the value of a real random variable $Y$ is to be predicted based on the observation of another random variable $X$. Given a class of functions $\mathcal F$ and a sample of independent copies of $(X,Y)$, one needs to choose a function $\widehat{f}$ from $\mathcal F$ such that $\widehat{f}(X)$ approximates $Y$ as well as possible, in the mean-squared sense. We introduce a new procedure, the so-called median-of-means tournament, that achieves the optimal tradeoff between accuracy and confidence under minimal assumptions, and in particular outperforms classical methods based on empirical risk minimization.

Keywords: Mean estimation, heavy-tailed distributions, empirical processes

Lugosi Gábor, Mendelson Shahar: Risk minimization by median-of-means tournaments. J. Eur. Math. Soc. 22 (2020), 925-965. doi: 10.4171/JEMS/937