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Oral
Optimal Rates of Sketched-regularized Algorithms for Least-Squares Regression over Hilbert Spaces
Junhong Lin · Volkan Cevher

Wed Jul 11 05:10 AM -- 05:20 AM (PDT) @ A6

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 provide optimal, distribution-dependent rates for sketched/Nystr\"{o}m regularized algorithms, considering both the attainable and non-attainable cases.