## Bernoulli

• Bernoulli
• Volume 24, Number 3 (2018), 2176-2203.

### On optimality of empirical risk minimization in linear aggregation

#### Abstract

In the first part of this paper, we show that the small-ball condition, recently introduced by (J. ACM 62 (2015) Art. 21, 25), may behave poorly for important classes of localized functions such as wavelets, piecewise polynomials or for trigonometric polynomials, in particular leading to suboptimal estimates of the rate of convergence of ERM for the linear aggregation problem. In a second part, we recover optimal rates of convergence for the excess risk of ERM when the dictionary is made of trigonometric functions. Considering the bounded case, we derive the concentration of the excess risk around a single point, which is an information far more precise than the rate of convergence. In the general setting of a $L_{2}$ noise, we finally refine the small ball argument by rightly selecting the directions we are looking at, in such a way that we obtain optimal rates of aggregation for the Fourier dictionary.

#### Article information

Source
Bernoulli, Volume 24, Number 3 (2018), 2176-2203.

Dates
First available in Project Euclid: 2 February 2018

https://projecteuclid.org/euclid.bj/1517540472

Digital Object Identifier
doi:10.3150/17-BEJ925

Mathematical Reviews number (MathSciNet)
MR3757527

Zentralblatt MATH identifier
06839264

#### Citation

Saumard, Adrien. On optimality of empirical risk minimization in linear aggregation. Bernoulli 24 (2018), no. 3, 2176--2203. doi:10.3150/17-BEJ925. https://projecteuclid.org/euclid.bj/1517540472

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