  Sparse additive models are families of $d$-variate functions that have the
additive decomposition $f^* = \sum_{j \in S} f^*_j$, where $S$ is an unknown
subset of cardinality $s \ll d$. In this paper, we consider the case where each
univariate component function $f^*_j$ lies in a reproducing kernel Hilbert
space (RKHS), and analyze a method for estimating the unknown function $f^*$
based on kernels combined with $\ell_1$-type convex regularization. Working
within a high-dimensional framework that allows both the dimension $d$ and
sparsity $s$ to increase with $n$, we derive convergence rates (upper bounds)
in the $L^2(\mathbb{P})$ and $L^2(\mathbb{P}_n)$ norms over the class
$\MyBigClass$ of sparse additive models with each univariate function $f^*_j$
in the unit ball of a univariate RKHS with bounded kernel function. We
complement our upper bounds by deriving minimax lower bounds on the
$L^2(\mathbb{P})$ error, thereby showing the optimality of our method. Thus, we
obtain optimal minimax rates for many interesting classes of sparse additive
models, including polynomials, splines, and Sobolev classes. We also show that
if, in contrast to our univariate conditions, the multivariate function class
is assumed to be globally bounded, then much faster estimation rates are
possible for any sparsity $s = \Omega(\sqrt{n})$, showing that global
boundedness is a significant restriction in the high-dimensional setting.
