  We present tight lower bounds on the number of kernel evaluations required to
approximately solve kernel ridge regression (KRR) and kernel $k$-means
clustering (KKMC) on $n$ input points. For KRR, our bound for relative error
approximation to the minimizer of the objective function is
$\Omega(nd_{\mathrm{eff}}^\lambda/\varepsilon)$ where
$d_{\mathrm{eff}}^\lambda$ is the effective statistical dimension, which is
tight up to a $\log(d_{\mathrm{eff}}^\lambda/\varepsilon)$ factor. For KKMC,
our bound for finding a $k$-clustering achieving a relative error approximation
of the objective function is $\Omega(nk/\varepsilon)$, which is tight up to a
$\log(k/\varepsilon)$ factor. Our KRR result resolves a variant of an open
question of El Alaoui and Mahoney, asking whether the effective statistical
dimension is a lower bound on the sampling complexity or not. Furthermore, for
the important practical case when the input is a mixture of Gaussians, we
provide a KKMC algorithm which bypasses the above lower bound.
