Optimal Rates for Spectral Algorithms with Least-Squares Regression over Hilbert Spaces
This provides theoretical guarantees for regression methods in machine learning, but it is incremental as it builds on existing frameworks.
The paper tackles regression problems in Hilbert spaces, proving optimal convergence rates for spectral algorithms like ridge regression and principal component regression, with results showing almost sure convergence and improvements over prior work.
In this paper, we study regression problems over a separable Hilbert space with the square loss, covering non-parametric regression over a reproducing kernel Hilbert space. We investigate a class of spectral/regularized algorithms, including ridge regression, principal component regression, and gradient methods. We prove optimal, high-probability convergence results in terms of variants of norms for the studied algorithms, considering a capacity assumption on the hypothesis space and a general source condition on the target function. Consequently, we obtain almost sure convergence results with optimal rates. Our results improve and generalize previous results, filling a theoretical gap for the non-attainable cases.