# Optimal Rates of Sketched-regularized Algorithms for Least-Squares Regression over Hilbert Spaces

## TL;DR

This paper establishes optimal, distribution-dependent convergence rates for sketched and Nyström regularized algorithms in Hilbert space least-squares regression, without saturation effects, under capacity and regularity assumptions.

## Contribution

It provides the first optimal, distribution-dependent convergence rates for sketched and Nyström regularized algorithms in Hilbert spaces, addressing both attainable and non-attainable cases.

## Key findings

- Optimal rates achieved with sketch dimension proportional to effective dimension
- Results apply to both sketched and Nyström regularized algorithms
- No saturation effect in the derived convergence rates

## Abstract

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.