  $l^q$-regularization has been demonstrated to be an attractive technique in
machine learning and statistical modeling. It attempts to improve the
generalization (prediction) capability of a machine (model) through
appropriately shrinking its coefficients. The shape of a $l^q$ estimator
differs in varying choices of the regularization order $q$. In particular,
$l^1$ leads to the LASSO estimate, while $l^{2}$ corresponds to the smooth
ridge regression. This makes the order $q$ a potential tuning parameter in
applications. To facilitate the use of $l^{q}$-regularization, we intend to
seek for a modeling strategy where an elaborative selection on $q$ is
avoidable. In this spirit, we place our investigation within a general
framework of $l^{q}$-regularized kernel learning under a sample dependent
hypothesis space (SDHS). For a designated class of kernel functions, we show
that all $l^{q}$ estimators for $0< q < \infty$ attain similar generalization
error bounds. These estimated bounds are almost optimal in the sense that up to
a logarithmic factor, the upper and lower bounds are asymptotically identical.
This finding tentatively reveals that, in some modeling contexts, the choice of
$q$ might not have a strong impact in terms of the generalization capability.
From this perspective, $q$ can be arbitrarily specified, or specified merely by
other no generalization criteria like smoothness, computational complexity,
sparsity, etc..
