  This paper deals with recovering an unknown vector $\theta$ from the noisy
data $Y=A\theta+\sigma\xi$, where $A$ is a known $(m\times n)$-matrix and $\xi$
is a white Gaussian noise. It is assumed that $n$ is large and $A$ may be
severely ill-posed. Therefore, in order to estimate $\theta$, a spectral
regularization method is used, and our goal is to choose its regularization
parameter with the help of the data $Y$. For spectral regularization methods
related to the so-called ordered smoothers [see Kneip Ann. Statist. 22 (1994)
835--866], we propose new penalties in the principle of empirical risk
minimization. The heuristical idea behind these penalties is related to
balancing excess risks. Based on this approach, we derive a sharp oracle
inequality controlling the mean square risks of data-driven spectral
regularization methods.
