• Bernoulli
  • Volume 22, Number 3 (2016), 1520-1534.

Performance of empirical risk minimization in linear aggregation

Guillaume Lecué and Shahar Mendelson

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We study conditions under which, given a dictionary $F=\{f_{1},\ldots,f_{M}\}$ and an i.i.d. sample $(X_{i},Y_{i})_{i=1}^{N}$, the empirical minimizer in $\operatorname{span}(F)$ relative to the squared loss, satisfies that with high probability

\[R(\tilde{f}^{\mathrm{ERM}})\leq\inf_{f\in\operatorname{span}(F)}R(f)+r_{N}(M),\] where $R(\cdot)$ is the squared risk and $r_{N}(M)$ is of the order of $M/N$.

Among other results, we prove that a uniform small-ball estimate for functions in $\operatorname{span}(F)$ is enough to achieve that goal when the noise is independent of the design.

Article information

Bernoulli, Volume 22, Number 3 (2016), 1520-1534.

Received: March 2014
Revised: February 2015
First available in Project Euclid: 16 March 2016

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aggregation theory empirical processes empirical risk minimization learning theory


Lecué, Guillaume; Mendelson, Shahar. Performance of empirical risk minimization in linear aggregation. Bernoulli 22 (2016), no. 3, 1520--1534. doi:10.3150/15-BEJ701.

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