  We investigate regularized algorithms combining with projection for
least-squares regression problem over a Hilbert space, covering nonparametric
regression over a reproducing kernel Hilbert space. We prove convergence
results with respect to variants of norms, under a capacity assumption on the
hypothesis space and a regularity condition on the target function. As a
result, we obtain optimal rates for regularized algorithms with randomized
sketches, provided that the sketch dimension is proportional to the effective
dimension up to a logarithmic factor. As a byproduct, we obtain similar results
for Nystr\"{o}m regularized algorithms. Our results are the first ones with
optimal, distribution-dependent rates that do not have any saturation effect
for sketched/Nystr\"{o}m regularized algorithms, considering both the
attainable and non-attainable cases.
