• Bernoulli
  • Volume 19, Number 5B (2013), 2153-2166.

Empirical risk minimization is optimal for the convex aggregation problem

Guillaume Lecué

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Let $F$ be a finite model of cardinality $M$ and denote by $\operatorname{conv}(F)$ its convex hull. The problem of convex aggregation is to construct a procedure having a risk as close as possible to the minimal risk over $\operatorname{conv} (F)$. Consider the bounded regression model with respect to the squared risk denoted by $R(\cdot)$. If ${ \widehat{f}}_{n}^{\mathit{ERM}\mbox{-}C}$ denotes the empirical risk minimization procedure over $\operatorname{conv}(F)$, then we prove that for any $x>0$, with probability greater than $1-4\exp(-x)$,


where $c_{0}>0$ is an absolute constant and $\psi_{n}^{(C)}(M)$ is the optimal rate of convex aggregation defined in (In Computational Learning Theory and Kernel Machines (COLT-2003) (2003) 303–313 Springer) by $\psi _{n}^{(C)}(M)=M/n$ when $M\leq\sqrt{n}$ and $\psi _{n}^{(C)}(M)=\sqrt{\log (\mathrm{e}M/\sqrt{n})/n}$ when $M>\sqrt{n}$.

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Bernoulli, Volume 19, Number 5B (2013), 2153-2166.

First available in Project Euclid: 3 December 2013

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aggregation empirical processes theory empirical risk minimization learning theory


Lecué, Guillaume. Empirical risk minimization is optimal for the convex aggregation problem. Bernoulli 19 (2013), no. 5B, 2153--2166. doi:10.3150/12-BEJ447.

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